SOURCES IN RECREATIONAL MATHEMATICS

AN ANNOTATED BIBLIOGRAPHY

 

EIGHTH PRELIMINARY EDITION

 

DAVID SINGMASTER

 

Contact via the Puzzle Museum

http://puzzlemuseum.com

Last updated on 19 March 2004.

 

          This is a copy of the current version from my source files.  I had intended to reorganise the material before producing a Word version, but have decided to produce this version for G4G6 and to renumber it as the Eighth Preliminary Edition.

 

         

 

 

          If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things.  [Fibonacci, translated by Grimm.])

 

CONTENTS        

 

INTRODUCTION 16

NATURE OF THIS WORK 16

SIMILAR WORKS 17

COVERAGE 18

STATUS OF THE PROJECT 19

TECHNICAL NOTES 21

ACKNOWLEDGMENTS 22

ABBREVIATIONS 25

DIACRITICAL MARKS AND NOTATION 25

SOME OTHER RECURRING REFERENCES 89

1.       BIOGRAPHICAL MATERIAL  --  in chronological order 91

2.       GENERAL PUZZLE COLLECTIONS AND SURVEYS 99

3.       GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL 100

3.A.       GENERAL HISTORICAL MATERIAL 100

3.B.       BIBLIOGRAPHICAL MATERIAL 100

4.       MATHEMATICAL GAMES 105

4.A.       GENERAL THEORY AND NIM‑LIKE GAMES 105

4.A.1.    ONE PILE GAME 106

4.A.1.a.        THE 31 GAME 109

4.A.2.        SYMMETRY ARGUMENTS 110

4.A.3.    KAYLES 111

4.A.4.    NIM 112

4.A.5.    GENERAL THEORY 113

4.B.       PARTICULAR GAMES 115

4.B.1.    TIC‑TAC‑TOE  =  NOUGHTS AND CROSSES 115

4.B.1.a     IN HIGHER DIMENSIONS 123

4.B.2.    HEX 124

4.B.3.    DOTS AND BOXES 125

4.B.4.    SPROUTS 126

4.B.5.    OVID'S GAME AND NINE MEN'S MORRIS 126

4.B.6.    PHUTBALL 132

4.B.7.    BRIDG‑IT 132

4.B.8.    CHOMP 133

4.B.9.    SNAKES AND LADDERS 133

4.B.10.      MU TORERE 135

4.B.11.      MASTERMIND, ETC. 136

4.B.12. RITHMOMACHIA  =  THE PHILOSOPHERS' GAME 136

4.B.13.      MANCALA GAMES 137

4.B.14.DOMINOES, ETC. 138

4.B.15.      SVOYI KOSIRI 138

5.COMBINATORIAL RECREATIONS 139

5.A.       THE 15 PUZZLE, ETC. 139

GENERAL 139

EARLY ALPHABETIC VERSIONS 139

LOYD 140

THE 15 PUZZLE 140

5.A.1.    NON‑SQUARE PIECES 147

5.A.2.    THREE DIMENSIONAL VERSIONS 148

5.A.3.    ROLLING PIECE PUZZLES 149

5.A.4.    PANEX PUZZLE 150

5.B.       CROSSING PROBLEMS 150

5.B.1.    LOWERING FROM TOWER PROBLEM 159

5.B.2.    CROSSING A BRIDGE WITH A TORCH 160

5.C.       FALSE COINS WITH A BALANCE 160

5.C.1RANKING COINS WITH A BALANCE 163

5.D. MEASURING PROBLEMS 163

5.D.1.    JUGS & BOTTLES 163

5.D.2.    RULER WITH MINIMAL NUMBER OF MARKS 169

5.D.3     FALSE COINS WITH A WEIGHING SCALE 170

5.D.4.    TIMING WITH HOURGLASSES 170

5.D.5.    MEASURE HALF A BARREL 170

5.E.       EULER CIRCUITS AND MAZES 171

5.E.1.    MAZES 175

5.E.2.    MEMORY WHEELS  =  CHAIN CODES 180

5.E.2.a.    PANTACTIC SQUARES 181

5.F.    HAMILTONIAN CIRCUITS 182

5.F.1.     KNIGHT'S TOURS AND PATHS 182

5.F.2.     OTHER HAMILTONIAN CIRCUITS 189

5.F.3.     KNIGHT'S TOURS IN HIGHER DIMENSIONS 190

5.F.4.     OTHER CIRCUITS IN AND ON A CUBE 190

5.G.       CONNECTION PROBLEMS 191

5.G.1.    GAS, WATER AND ELECTRICITY 191

5.H.       COLOURED SQUARES AND CUBES, ETC. 192

5.H.1.    INSTANT INSANITY  =  THE TANTALIZER 192

5.H.2.    MACMAHON PIECES 192

5.H.3.    PATH FORMING PUZZLES 194

5.H.4.    OTHER AND GENERAL 195

5.I.     LATIN SQUARES AND EULER SQUARES 197

5.I.1.     EIGHT QUEENS PROBLEM 199

5.I.2.     COLOURING CHESSBOARD WITH NO REPEATS IN A LINE 202

5.J.    SQUARED SQUARES, ETC. 202

5.J.1.     MRS PERKINS'S QUILT 205

5.J.2.     CUBING THE CUBE 205

5.J.3.     TILING A SQUARE OF SIDE 70 WITH SQUARES OF SIDES  1, 2, ..., 24 205

5.K.       DERANGEMENTS 205

5.K.1.    DERANGED BOXES OF  A,  B  AND  A & B 206

5.K.2.    OTHER LOGIC PUZZLES BASED ON DERANGEMENTS 207

5.K.3.    CAYLEY'S MOUSETRAP 207

5.L.       MÉNAGE PROBLEM 207

5.M.      SIX PEOPLE AT A PARTY  --  RAMSEY THEORY 208

5.N.       JEEP OR EXPLORER'S PROBLEM 209

5.O.       TAIT'S COUNTER PUZZLE:  BBBBWWWW  TO  WBWBWBWB 212

5.P.    GENERAL MOVING PIECE PUZZLES 214

5.P.1.    SHUNTING PUZZLES 214

5.P.2.    TAQUIN 216

5.Q.       NUMBER OF REGIONS DETERMINED BY  N  LINES OR PLANES 216

5.Q.1.      NUMBER OF INTERSECTIONS DETERMINED BY  N  LINES 217

5.R.       JUMPING PIECE GAMES 217

5.R.1.    PEG SOLITAIRE 217

5.R.1.a.    TRIANGULAR VERSION 220

5.R.1.b.    OTHER SHAPES 221

5.R.2.    FROGS AND TOADS:  BBB_WWW  TO  WWW_BBB 222

5.R.3.    FORE AND AFT  --  3 BY 3  SQUARES MEETING AT A CORNER 224

5.R.4.    REVERSING FROGS AND TOADS:  _12...n  TO  _n...21 , ETC. 225

5.R.5.    FOX AND GEESE, ETC. 225

5.R.6.    OCTAGRAM PUZZLE 228

5.R.7.    PASSING OVER COUNTERS 230

5.S.    CHAIN CUTTING AND REJOINING 233

5.S.1.     USING CHAIN LINKS TO PAY FOR A ROOM 234

5.T.       DIVIDING A CAKE FAIRLY 234

5.U.       PIGEONHOLE RECREATIONS 235

5.V.       THINK‑A‑DOT, ETC. 236

5.W.      MAKING THREE PIECES OF TOAST 237

5.W.1.       BOILING EGGS 237

5.X.       COUNTING FIGURES IN A PATTERN 237

5.X.1.    COUNTING TRIANGLES 237

5.X.2.    COUNTING RECTANGLES OR SQUARES 240

5.X.3.    COUNTING HEXAGONS 241

5.X.4.    COUNTING CIRCLES 242

5.Y.       NUMBER OF ROUTES IN A LATTICE 242

5.Z.       CHESSBOARD PLACING PROBLEMS 245

5.Z.1.    KINGS 246

5.Z.2.    QUEENS 246

5.Z.3.    BISHOPS 247

5.Z.4.    KNIGHTS 248

5.Z.5.    ROOKS 248

5.Z.6.    MIXTURES 248

5.AA.   CARD SHUFFLING 249

5.AB.     FOLDING A STRIP OF STAMPS 251

5.AC.      PROPERTIES OF THE SEVEN BAR DIGITAL DISPLAY 251

5.AD.    STACKING A DECK TO PRODUCE A SPECIAL EFFECT 252

5.AE.     REVERSING CUPS 252

5.AF.     SPOTTING DICE 252

5.AG.    RUBIK'S CUBE AND SIMILAR PUZZLES 253

5.AG.1.     RUBIK'S CUBE 253

5.AG.2.     HUNGARIAN RINGS, ETC. 253

6.       GEOMETRIC RECREATIONS 254

6.A.       PI 254

6.B.       STRAIGHT LINE LINKAGES 255

6.C.       CURVES OF CONSTANT WIDTH 257

6.D.       FLEXAGONS 257

6.E.       FLEXATUBE 259

6.F.    POLYOMINOES, ETC. 259

6.F.1.        OTHER CHESSBOARD DISSECTIONS 268

6.F.2.     COVERING DELETED CHESSBOARD WITH DOMINOES 270

6.F.3.     DISSECTING A CROSS INTO  Zs  AND  Ls 270

6.F.4.     QUADRISECT AN  L‑TROMINO, ETC. 272

6.F.5.     OTHER DISSECTIONS INTO POLYOMINOES 274

6.G.       SOMA CUBE 275

6.G.1.    OTHER CUBE DISSECTIONS 275

6.G.2.    DISSECTION OF  63  INTO  33,  43  AND  53,  ETC. 276

6.G.3.    DISSECTION OF A DIE INTO NINE  1 x 1 x 3 277

6.G.4.    USE OF OTHER POLYHEDRAL PIECES 277

6.H.       PICK'S THEOREM 278

6.I.     SYLVESTER'S PROBLEM OF COLLINEAR POINTS 278

6.J.    FOUR BUGS AND OTHER PURSUIT PROBLEMS 279

6.K.       DUDENEY'S SQUARE TO TRIANGLE DISSECTION 281

6.L.       CROSSED LADDERS 281

6.L.1.        LADDER OVER BOX 283

6.M.      SPIDER & FLY PROBLEMS 283

6.N.       DISSECTION OF A  1 x 1 x 2  BLOCK TO A CUBE 284

6.O.       PASSING A CUBE THROUGH AN EQUAL OR SMALLER CUBE. PRINCE RUPERT'S PROBLEM 285

6.P.    GEOMETRICAL VANISHING 286

6.P.1.    PARADOXICAL DISSECTIONS OF THE CHESSBOARD BASED ON FIBONACCI NUMBERS 286

6.P.2.    OTHER TYPES 288

6.Q.       KNOTTING A STRIP TO MAKE A REGULAR PENTAGON 290

6.R.       GEOMETRIC FALLACIES 291

6.R.1.    EVERY TRIANGLE IS ISOSCELES 292

6.R.2.    A RIGHT ANGLE IS OBTUSE 292

6.R.3.    LINES APPROACHING BUT NOT MEETING 292

6.R.4.    OTHERS 293

6.S.    TANGRAMS, ET AL. 293

GENERAL HISTORIES. 293

SPECIFIC ITEMS 295

6.S.1.     LOCULUS OF ARCHIMEDES 304

6.S.2.     OTHER SETS OF PIECES 305

6.T.       NO THREE IN A LINE PROBLEM 306

6.U.       TILING 307

6.U.1.    PENROSE PIECES 307

6.U.2.    PACKING BRICKS IN BOXES 308

6.V.       SILHOUETTE AND VIEWING PUZZLES 308

6.W.      BURR PUZZLES 311

6.W.1.       THREE PIECE BURR 311

6.W.2.       SIX PIECE BURR  =  CHINESE CROSS 312

6.W.3.       THREE PIECE BURR WITH IDENTICAL PIECES 314

6.W.4.       DIAGONAL SIX PIECE BURR  =  TRICK STAR 314

6.W.5.       SIX PIECE BURR WITH IDENTICAL PIECES 315

6.W.6.       ALTEKRUSE PUZZLE 315

6.W.7.  OTHER BURRS 316

6.X.       ROTATING RINGS OF POLYHEDRA 316

6.Y.       ROPE ROUND THE EARTH 317

6.Z.       LANGLEY'S ADVENTITIOUS ANGLES 319

6.AA.    NETS OF POLYHEDRA 320

6.AB.     SELF‑RISING POLYHEDRA 321

6.AC.    CONWAY'S LIFE 321

6.AD.    ISOPERIMETRIC PROBLEMS 322

6.AD.1.     LARGEST PARCEL ONE CAN POST 323

6.AE.     6"  HOLE THROUGH SPHERE LEAVES CONSTANT VOLUME 324

6.AF.     WHAT COLOUR WAS THE BEAR? 325

6.AG.    MOVING AROUND A CORNER 327

6.AH.    TETHERED GOAT 328

6.AI.      TRICK JOINTS 329

6.AJ.     GEOMETRIC ILLUSIONS 330

6.AJ.1       TWO PRONGED TRIDENT 333

6.AJ.2.      TRIBAR AND IMPOSSIBLE STAIRCASE 334

6.AJ.3.      CAFÉ WALL ILLUSION 335

6.AJ.4.      STEREOGRAMS 336

6.AJ.5.      IMPOSSIBLE CRATE. 336

6.AK.    POLYGONAL PATH COVERING  N x N LATTICE OF POINTS, QUEEN'S TOURS, ETC. 336

6.AL.     STEINER‑LEHMUS THEOREM 339

6.AM.       MORLEY'S THEOREM 340

6.AN.    VOLUME OF THE INTERSECTION OF TWO CYLINDERS 341

6.AO.    CONFIGURATION PROBLEMS 341

6.AO.1.     PLACE FOUR POINTS EQUIDISTANTLY  =  MAKE FOUR TRIANGLES WITH SIX MATCHSTICKS 349

6.AO.2.     PLACE AN EVEN NUMBER ON EACH LINE 351

6.AP      . DISSECTIONS OF A TETRAHEDRON 352

6.AP.1.     TWO PIECES 352

6.AP.2.     FOUR PIECES 352

6.AQ.    DISSECTIONS OF A CROSS,  T  OR  H 352

6.AR.    QUADRISECTED SQUARE PUZZLE 354

6.AS.     DISSECTION OF SQUARES INTO A SQUARE 354

6.AS.1.      TWENTY 1, 2, Ö5 TRIANGLES MAKE A SQUARE OR FIVE EQUAL SQUARES TO A SQUARE 355

6.AS.1.a.      GREEK CROSS TO A SQUARE 358

6.AS.1.b.      OTHER GREEK CROSS DISSECTIONS 358

6.AS.2.      TWO (ADJACENT) SQUARES TO A SQUARE 359

6.AS.2.a.      TWO EQUAL SQUARES TO A SQUARE 361

6.AS.3.      THREE EQUAL SQUARES TO A SQUARE 361

6.AS.3.a.      THREE EQUAL 'SQUARES' TO A HEXAGON 362

6.AS.4.      EIGHT EQUAL SQUARES TO A SQUARE 362

6.AS.5.      RECTANGLE TO A SQUARE OR OTHER RECTANGLE 362

6.AT.     POLYHEDRA AND TESSELLATIONS 363

6.AT.1.     REGULAR POLYHEDRA 363

6.AT.2      STAR AND STELLATED POLYHEDRA 367

6.AT.3.     ARCHIMEDEAN POLYHEDRA 369

6.AT.4.     UNIFORM POLYHEDRA 374

6.AT.5.     REGULAR‑FACED POLYHEDRA 374

6.AT.6.     TESSELLATIONS 374

6.AT.6.a.     TESSELLATING WITH CONGRUENT FIGURES 375

6.AT.7.     PLAITING OF POLYHEDRA 376

6.AT.8.     DÜRER'S OCTAHEDRON 376

6.AT.9.     OTHER POLYHEDRA 377

6.AU.    THREE RABBITS, DEAD DOGS AND TRICK MULES 377

CHINA 378

OTHER ASIA 381

PADERBORN 383

MEDIEVAL EUROPE 385

MODERN VERSIONS OF THE THREE RABBITS PUZZLE 393

DEAD DOGS 394

TRICK MULES 398

6.AV.    CUTTING UP IN FEWEST CUTS 399

6.AW.       DIVISION INTO CONGRUENT PIECES 399

6.AW.1.    MITRE PUZZLE 399

6.AW.2.    REP‑TILES 400

6.AW.3.    DIVIDING A SQUARE INTO CONGRUENT PARTS 401

6.AW.4.    DIVIDING AN  L-TROMINO INTO CONGRUENT PARTS 402

6.AX.     THE PACKER'S SECRET 402

6.AY.     DISSECT  3A x 2B  TO MAKE  2A x 3B,  ETC. 402

6.AY.1.     O'BEIRNE'S STEPS 405

6.AY.2.     SWISS FLAG PUZZLE 405

6.AZ.     BALL PYRAMID PUZZLES 406

6.BA.     CUTTING A CARD SO ONE CAN PASS THROUGH IT 407

6.BB.     DOUBLING AN AREA WITHOUT CHANGING ITS HEIGHT OR WIDTH 407

6.BC.     HOFFMAN'S CUBE 408

6.BD.  BRIDGE A MOAT WITH PLANKS 408

6.BE.     REVERSE A TRIANGULAR ARRAY OF TEN CIRCLES 410

6.BF.     PYTHAGOREAN RECREATIONS 410

6.BF.1       THE BROKEN BAMBOO 411

6.BF.2.      SLIDING SPEAR  =  LEANING REED 412

6.BF.3.      WELL BETWEEN TWO TOWERS 413

6.BF.4.      RAIL BUCKLING. 416

6.BF.5.      TRAVELLING ON SIDES OF A RIGHT TRIANGLE. 417

6.BG.    QUADRISECT A PAPER SQUARE WITH ONE CUT 417

6.BH.    MOIRÉ PATTERNS 417

6.BI.  VENN DIAGRAMS FOR  N  SETS 418

6.BJ.     3D DISSECTION PUZZLES 420

6.BK     SUPERELLIPSE 420

6.BL.     TAN-1 ⅓ + TAN-1 ½  =  TAN-1 1, ETC. 421

6.BM.       DISSECT CIRCLE INTO TWO HOLLOW OVALS 421

6.BN.     ROUND PEG IN SQUARE HOLE OR VICE VERSA 422

6.BO.    BUTTERFLY PROBLEM 423

6.BP.     EARLY MATCHSTICK PUZZLES 423

6.BQ.    COVERING A DISC WITH DISCS 424

6.BR.     WHAT IS A GENERAL TRIANGLE? 424

6.BS.     FORM SIX COINS INTO A HEXAGON 425

6.BT.     PLACING OBJECTS IN CONTACT 425

6.BU.     CONSTRUCTION OF  N-GONS 426

6.BV.     GEOMETRIC CONSTRUCTIONS 428

6.BW.       DISTANCES TO CORNERS OF A SQUARE 429

7.       ARITHMETIC  &  NUMBER‑THEORETIC RECREATIONS 430

7.A.       FIBONACCI NUMBERS 430

7.B.       JOSEPHUS OR SURVIVOR PROBLEM 433

7.C.       EGYPTIAN FRACTIONS 448

7.D.       THE FIRST DIGIT PROBLEM 449

7.E.       MONKEY AND COCONUTS PROBLEMS 449

7.E.1.    VERSIONS WITH ALL GETTING THE SAME 463

7.F.        ILLEGAL OPERATIONS GIVING CORRECT RESULT 465

7.G.       INHERITANCE PROBLEMS 465

7.G.1.    HALF + THIRD + NINTH, ETC. 465

7.G.2.    POSTHUMOUS TWINS, ETC. 471

7.H.       DIVISION AND SHARING PROBLEMS  --  CISTERN PROBLEMS 474

7.H.1.    WITH GROWTH  --  NEWTON'S CATTLE PROBLEM 493

7.H.2.    DIVISION OF CASKS 495

7.H.3.    SHARING UNEQUAL RESOURCES  --  PROBLEM OF THE PANDECTS 497

7.H.4.    EACH DOUBLES OTHERS' MONEY TO MAKE ALL EQUAL, ETC. 500

7.H.5.    SHARING COST OF STAIRS, ETC. 503

7.H.6.    SHARING A GRINDSTONE 504

7.H.7.    DIGGING PART OF A WELL. 505

7.I.        FOUR FOURS, ETC. 507

7.I.1.     LARGEST NUMBER USING FOUR ONES, ETC. 514

7.J.        SALARY PUZZLE 516

7.K.       CONGRUENCES 517

7.K.1.    CASTING OUT NINES 517

7.L.       GEOMETRIC PROGRESSIONS 520

7.L.1.    1 + 7 + 49 + ...  &  ST. IVES 523

7.L.2.    1 + 2 + 4 + ... 525

7.L.2.a.    CHESSBOARD PROBLEM 526

7.L.2.b.    HORSESHOE NAILS PROBLEM 530

7.L.2.c.    USE OF 1, 2, 4, ... AS WEIGHTS, ETC. 531

7.L.3.    1 + 3 + 9 + ... AND OTHER SYSTEMS OF WEIGHTS 533

7.M.      BINARY SYSTEM AND BINARY RECREATIONS 535

7.M.1.       CHINESE RINGS 536

7.M.2.       TOWER OF HANOI 540

7.M.2.a.       TOWER OF HANOI WITH MORE PEGS 544

7.M.3.       GRAY CODE 544

7.M.4.       BINARY DIVINATION 545

7.M.4.a.       TERNARY DIVINATION 547

7.M.4.b.   OTHER DIVINATIONS USING BINARY OR TERNARY 547

7.M.5.  LOONY LOOP  =  GORDIAN KNOT 551

7.M.6.       BINARY BUTTON GAMES 551

7.N.       MAGIC SQUARES 554

SURVEYS 554

POSSIBLE EARLY REFERENCES 555

7.N.1     MAGIC CUBES 569

7.N.2.    MAGIC TRIANGLES 571

7.N.3.    ANTI‑MAGIC SQUARES AND TRIANGLES 573

7.N.4.    MAGIC KNIGHT'S TOUR 574

7.N.5.    OTHER MAGIC SHAPES 575

7.O.       MAGIC HEXAGON 578

7.O.1.    OTHER MAGIC HEXAGONS 579

7.P.    DIOPHANTINE RECREATIONS 582

7.P.1.    HUNDRED FOWLS AND OTHER LINEAR PROBLEMS 582

7.P.2.    CHINESE REMAINDER THEOREM 602

7.P.3.    ARCHIMEDES' CATTLE PROBLEM 608

7.P.4.    PRESENT OF GEMS 609

7.P.5     . SELLING DIFFERENT AMOUNTS 'AT SAME PRICES' YIELDING THE SAME 610

7.P.6.    CONJUNCTION OF PLANETS, ETC. 616

7.P.7.    ROBBING AND RESTORING 617

7.Q.       BLIND ABBESS AND HER NUNS -- REARRANGEMENT ALONG SIDES OF A  3 x 3  SQUARE TO CONSERVE SIDE TOTALS 619

7.Q.1.    REARRANGEMENT ON A CROSS 622

7.Q.2.    REARRANGE A CROSS OF SIX TO MAKE TWO LINES OF FOUR, ETC. 623

7.R.       "IF I HAD ONE FROM YOU, I'D HAVE TWICE YOU" 624

7.R.1.    MEN FIND A PURSE AND 'BLOOM OF THYMARIDAS' 631

7.R.2.    "IF I HAD  1/3  OF YOUR MONEY, I COULD BUY THE HORSE" 638

7.R.3.    SISTERS AND BROTHERS 648

7.R.4.    "IF I SOLD YOUR EGGS AT MY PRICE, I'D GET ...." 649

7.S.    DILUTION AND MIXING PROBLEMS 650

7.S.1.     DISHONEST BUTLER DRINKING SOME AND REPLACING WITH WATER 650

7.S.2.     WATER IN WINE VERSUS WINE IN WATER 651

7.T.       FOUR NUMBER GAME 652

7.U.       POSTAGE STAMP PROBLEM 653

7.V.       XY  =  YX  AND ITERATED EXPONENTIALS 653

7.W.      CARD PILING OVER A CLIFF 654

7.X.       HOW OLD IS ANN? AND OTHER AGE PROBLEMS 655

7.Y.       COMBINING AMOUNTS AND PRICES INCOHERENTLY 666

7.Y.1.    REVERSAL OF AVERAGES PARADOX 668

7.Y.2.    UNFAIR DIVISION 669

7.Z.       MISSING DOLLAR AND OTHER ERRONEOUS ACCOUNTING 669

7.AA.    NEGATIVE DIGITS 671

7.AA.1.     NEGATIVE BASES, ETC. 672

7.AB.     PERFECT NUMBERS, ETC. 673

7.AC.    CRYPTARITHMS, ALPHAMETICS AND SKELETON ARITHMETIC 675

7.AC.1.     CRYPTARITHMS:  SEND + MORE  =  MONEY,  ETC. 676

7.AC.2.     SKELETON ARITHMETIC:  SOLITARY SEVEN, ETC. 678

7.AC.3.     PAN‑DIGITAL SUMS 681

7.AC.3.a      INSERTION OF SIGNS TO MAKE 100, ETC. 682

7.AC.4.     PAN‑DIGITAL PRODUCTS 683

7.AC.5.     PAN‑DIGITAL FRACTIONS 685

7.AC.6.     OTHER PAN‑DIGITAL AND SIMILAR PROBLEMS 686

7.AC.7.     SELF-DESCRIPTIVE NUMBERS, PANGRAMS, ETC. 689

7.AD. SELLING, BUYING AND SELLING SAME ITEM 690

7.AD.1.     PAWNING MONEY 691

7.AE.     USE OF COUNTERFEIT BILL OR FORGED CHEQUE 691

7.AF.     ARITHMETIC PROGRESSIONS 693

7.AF.1.      COLLECTING STONES 693

7.AF.2.      CLOCK STRIKING 696

7.AG.    2592 697

7.AH.    MULTIPLYING BY REVERSING 697

7.AH.1.     OTHER REVERSAL PROBLEMS 698

7.AI.      IMPOSSIBLE EXCHANGE RATES 699

7.AJ.     MULTIPLYING BY SHIFTING 699

7.AJ.1.      MULTIPLYING BY APPENDING DIGITS 701

7.AK.    LAZY WORKER 701

7.AL.     IF  A  IS  B,  WHAT IS  C? 704

7.AM.       CROSSNUMBER PUZZLES 707

7.AN.    THREE ODDS MAKE AN EVEN, ETC. 709

7.AO.    DIVINATION OF A PERMUTATION 711

7.AP.     KNOWING SUM VS KNOWING PRODUCT 716

7.AQ.    NUMBERS IN ALPHABETIC ORDER 719

7.AR.    1089 719

7.AS.     CIGARETTE BUTTS 722

7.AT.     BOOKWORM'S DISTANCE 722

7.AU.    NUMBER OF CUTS TO MAKE  N  PIECES 723

7.AV.    HOW LONG TO STRIKE TWELVE? 724

7.AW.       28/7  =  13 724

7.AX.     SUM  =  PRODUCT,  ETC. 725

7.AY.     SUM OF POWERS OF DIGITS 725

7.AZ.     DIVINATION OF A PAIR OF CARDS FROM ITS ROWS 727

7.BA.     CYCLE OF NUMBERS WITH EACH CLOSER TO TEN THAN THE PREVIOUS 729

7.BB.     ITERATED FUNCTIONS OF INTEGERS 729

7.BC.     UNUSUAL DIFFICULTY IN GIVING CHANGE 730

8.       PROBABILITY RECREATIONS 731

8.A.       BUFFON'S NEEDLE PROBLEM 731

8.B.       BIRTHDAY PROBLEM 732

8.D.       ATTEMPTS TO MODIFY BOY‑GIRL RATIO 734

8.E.       ST. PETERSBURG PARADOX 734

8.F.    PROBLEM OF POINTS 735

8.G.       PROBABILITY THAT THREE LENGTHS FORM A TRIANGLE 737

8.H.       PROBABILITY PARADOXES 738

8.H.1.    BERTRAND'S BOX PARADOX 738

8.H.2.    BERTRAND'S CHORD PARADOX 738

8.I.     TAKING THE NEXT TRAIN 738

8.J.    CLOCK PATIENCE OR SOLITAIRE 738

8.K.       SUCKER BETS 739

8.L.       NONTRANSITIVE GAMES 740

9.       LOGICAL RECREATIONS 741

9.A.       ALL CRETANS ARE LIARS, ETC. 741

9.B.       SMITH -- JONES -- ROBINSON PROBLEM 744

9.C.       FORTY UNFAITHFUL WIVES 744

9.D.       SPOTS ON FOREHEADS 744

9.E.       STRANGE FAMILIES 746

GENERAL STUDIES OF KINSHIP RELATIONS 746

DECEASED WIFE'S SISTER, ETC. 747

GENERAL FAMILY RIDDLES 748

9.E.1.    THAT MAN'S FATHER IS MY FATHER'S SON, ETC. 755

9.E.2.    IDENTICAL SIBLINGS WHO ARE NOT TWINS 759

9.F.    THE UNEXPECTED HANGING 759

9.G.       TRUTHTELLERS AND LIARS 760

9.H.       PRISONER'S DILEMMA 763

9.I.     HEMPEL'S RAVEN PARADOX 763

9.J.    FALLEN SIGNPOST 763

9.K.       CARROLL'S BARBER PARADOX 764

10.     PHYSICAL RECREATIONS 764

10.A.     OVERTAKING AND MEETING PROBLEMS 765

10.A.1.      CIRCLING AN ARMY 780

10.A.2.      NUMBER OF BUSES MET 781

10.A.3.      TIMES FROM MEETING TO FINISH GIVEN 782

10.A.4.      THE EARLY COMMUTER 783

10.A.5.      HEAD START PROBLEMS 784

10.A.6.      DOUBLE CROSSING PROBLEMS 785

10.A.7.      TRAINS PASSING 785

10.A.8.      TOO SLOW, TOO FAST 785

10.B.     FLY BETWEEN TRAINS 786

10.C.     LEWIS CARROLL'S MONKEY PROBLEM 787

10.D.     MIRROR PROBLEMS 789

10.D.1       MIRROR REVERSAL PARADOX 789

10.D.2.      OTHER MIRROR PROBLEMS 790

10.D.3       MAGIC MIRRORS 790

10.E.      WHEEL PARADOXES 791

10.E.1.      ARISTOTLE'S WHEEL PARADOX 791

10.E.2.      ONE WHEEL ROLLING AROUND ANOTHER 792

10.E.3.      HUNTER AND SQUIRREL 792

10.E.4.      RAILWAY WHEELS PARADOX 793

10.F.      FLOATING BODY PROBLEMS 794

10.H.     SNAIL CLIMBING OUT OF WELL 797

10.I.      LIMITED MEANS OF TRANSPORT -- TWO MEN AND A BIKE, ETC. 802

10.J.      RESISTOR NETWORKS 803

10.K.     PROBLEM OF THE DATE LINE 803

10.L.     FALLING DOWN A HOLE THROUGH THE EARTH 806

10.M.    CELTS  =  RATTLEBACKS 808

10.M.1.     TIPPEE TOPS 809

10.N.     SHIP'S LADDER IN RISING TIDE 810

10.O.     ERRONEOUS AVERAGING OF VELOCITIES 810

10.P.      FALSE BALANCE 811

10.Q.     PUSH A BICYCLE PEDAL 812

10.R.     CLOCK HAND PROBLEMS 812

10.S.      WALKING IN THE RAIN 815

10.T.      CENTRIFUGAL PUZZLES 816

10.U.     SHORTEST ROUTE VIA A WALL, ETC. 816

10.V.     PICK UP PUZZLES  =  PLUCK IT 817

10.W.    PUZZLE VESSELS 817

10.X.     HOW FAR DOES A PHONOGRAPH NEEDLE TRAVEL? 823

10.Y.     DOUBLE CONE ROLLING UPHILL 824

10.Z.      THE WOBBLER 824

10.AA.      NON-REGULAR DICE 825

10.AB.      BICYCLE TRACK PROBLEMS. 828

10.AC.      ROBERVAL'S BALANCE. 830

10.AD.      POUND OF FEATHERS. 830

10.AE.      JUGGLING OVER A BRIDGE 830

11.     TOPOLOGICAL RECREATIONS 831

11.A.     SCISSORS ON STRING 831

11.B.     TWO PEOPLE JOINED BY ROPES AT WRISTS 832

11.C.     TWO BALLS ON STRING THROUGH LEATHER HOLE AND STRAP  =  CHERRIES PUZZLE 833

11.D.     SOLOMON'S SEAL 835

11.E.      LOYD'S PENCIL PUZZLE 837

11.F.      THE IMPERIAL SCALE 838

11.G.     TRICK PURSES 838

11.H.     REMOVING WAISTCOAT WITHOUT REMOVING COAT 839

11.H.1.      REMOVING LOOP FROM ARM 840

11.I.      HEART AND BALL PUZZLE AND OTHER LOOP PUZZLES 840

11.J.      MÖBIUS STRIP 843

11.K.     WIRE PUZZLES 846

11.K.1.      RING AND SPRING PUZZLE 848

11.K.2.      STRING AND SPRING PUZZLE 849

11.K.3.      MAGIC CHAIN  =  TUMBLE RINGS 849

11.K.4.      PUZZLE RINGS 849

11.K.5.      RING MAZES 850

11.K.6.      INTERLOCKED NAILS, HOOKS, HORNS, ETC. 850

11.K.7.      HORSESHOES PUZZLE 851

11.K.8.      THE CAUGHT HEART 851

11.L.     JACOB'S LADDER AND OTHER HINGING DEVICES 852

11.M.    PUZZLE BOXES 853

11.N.     THREE KNIVES MAKE A SUPPORT 855

11.O.     BORROMEAN RINGS 857

11.P.      THE LONELY MONK 859

11.Q.     TURNING AN INNER TUBE INSIDE OUT 859

11.R.     STRING FIGURES 860

11.S.      PUZZLE KNIVES 861

 

 

                              INTRODUCTION

 

                    NATURE OF THIS WORK

 

          Recreational mathematics is as old as mathematics itself.  Recreational problems already occur in the oldest extant sources -- the Rhind Papyrus and Old Babylonian tablets.  The Rhind Papyrus has an example of a purely recreational problem -- Problem 79 is like the "As I was going to St. Ives" nursery rhyme.  The Babylonians give fairly standard practical problems with a recreational context -- a man knows the area plus the difference of the length and width of his field, a measurement which no surveyor would ever make!  There is even some prehistoric mathematics which could not have been practical -- numerous 'carved stone balls' have been found in eastern Scotland, dating from the Neolithic period and they include rounded forms of all the regular polyhedra and some less regular ones.  Since these early times, recreations have been a feature of mathematics, both as pure recreations and as pedagogic tools.  In this work, I use recreational in a fairly broad sense, but I tend to omit the more straightforward problems and concentrate on those which 'stimulate the curiosity' (as Montucla says). 

          In addition, recreational mathematics is certainly as diffuse as mathematics.  Every main culture and many minor ones have contributed to the history.  A glance at the Common References below, or at almost any topic in the text, will reveal the diversity of sources which are relevant to this study.  Much information arises from material outside the purview of the ordinary historian of mathematics -- e.g.  patents;  articles in newspapers, popular magazines and minor journals;  instruction leaflets;  actual artifacts  and even oral tradition.

          Consequently, it is very difficult to determine the history of any recreational topic and the history given in popular books is often extremely dubious or even simply fanciful.  For example, Nim, Tangrams, and Magic Squares are often traced back to China of about  2000 BC.  The oldest known reference to Nim is in America in 1903.  Tangrams appear in China and Europe at essentially the same time, about 1800, though there are related puzzles in 18C Japan and in the Hellenistic world.  Magic Squares seem to be genuinely a Chinese invention, but go back to perhaps a few centuries BC and are not clearly described until about 80AD.  Because of the lack of a history of the field, results are frequently rediscovered.

          When I began this bibliography in 1982, I had the the idea of producing a book (or books) of the original sources, translated into English, so people could read the original material.  This bibliography began as the table of contents of such a book.  I thought that this would be an easy project, but it has become increasingly apparent that the history of most recreations is hardly known.  I have recently realised that mathematical recreations are really the folklore of mathematics and that the historical problems are similar to those of folklore.  One might even say that mathematical recreations are the urban myths or the jokes or the campfire stories of mathematics.  Consequently I decided that an annotated bibliography was the first necessity to make the history clearer.  This bibliography alone has grown into a book, something like Dickson's History of the Theory of Numbers.  Like that work, the present work divides the subject into a number of topics and treats them chronologically. 

          I have printed six preliminary editions of this work, with slightly varying titles.  The first version of 4 Jul 1986 had 224 topics and was spaced out so entries would not be spread over two pages and to give room for page numbers.  This stretched the text from 110pp to 129pp and was printed for the Strens Memorial Conference at the Univ. of Calgary in Jul/Aug 1986.  I no longer worry about page breaks.  The following editions had:  250 topics on 152 pages;  290 topics on 192 pages;  307 topics on 223 pages;  357 topics on 311 pages  and 392 topics on 456 pages.  The seventh edition was never printed, but was a continually changing computer file.  It had about 419 topics (as of 20 Oct 95) and 587 pages, as of 20 Oct 1995.  I then carried out the conversion to proportional spacing and this reduced the total length from 587 to 488 pages, a reduction of 16.87% which is conveniently estimated as 1/6.  This reduction was fairly consistent throughout the conversion process. 

          This eighth edition is being prepared for the Gathering for Gardner 6 in March 2004.  The text is 818 pages as of 18 Mar 2004.  There are about 457 topics as of 18 Mar 2004.

          A fuller description of this project in 1984-1985 is given in my article Some early sources in recreational mathematics, in: C. Hay et al., eds.; Mathematics from Manuscript to Print; Oxford Univ. Press, 1988, pp. 195‑208.  A more recent description is in my article: Recreational mathematics; in: Encyclopedia of the History and Philosophy of the Mathematical Sciences; ed. by I. Grattan-Guinness; Routledge & Kegan Paul, 1993; pp. 1568-1575.

          Below I compare this work with Dickson and similar works and discuss the coverage of this work.

 

                    SIMILAR WORKS

 

          As already mentioned, the work which the present most resembles is Dickson's History of the Theory of Numbers.

 

                    The history of science can be made entirely impartial, and perhaps that is what it should be, by merely recording who did what, and leaving all "evaluations" to those who like them.  To my knowledge there is only one history of a scientific subject (Dickson's, of the Theory of Numbers) which has been written in this coldblooded, scientific way.  The complete success of that unique example -- admitted by all who ever have occasion to use such a history in their work -- seems to indicate that historians who draw morals should have their own morals drawn.

 

                    E. T. Bell.  The Search for Truth.  George Allen & Unwin, London, 1935, p. 131.

 

          Dickson attempted to be exhaustive and certainly is pretty much so.  Since his time, many older sources have been published, but their number-theoretic content is limited and most of Dickson's topics do not go back that far, so it remains the authoritative work in its field.

 

          The best previous book covering the history of recreational mathematics is the second edition of Wilhelm Ahrens's Mathematische Unterhaltungen und Spiele in two volumes.  Although it is a book on recreations, it includes extensive histories of most of the topics covered, far more than in any other recreational book.  He also gives a good index and a bibliography of 762 items, often with some bibliographical notes.  I will indicate the appropriate pages at the beginning of any topic that Ahrens covers.  This has been out of print for many years but Teubner has some plans to reissue it.

          Another similar book is the 4th edition of J. Tropfke's Geschichte der Elementarmathematik, revised by Vogel, Reich and Gericke.  This is quite exhaustive, but is concerned with older problems and sources.  It presents the material on a topic as a history with references to the sources, but it doesn't detail what is in each of the sources.  Sadly, only one volume, on arithmetic and algebra, appeared before Vogel's death.  A second volume, on geometry, is being prepared.  For any topic covered in Tropfke, it should be consulted for further references to early material which I have not seen, particularly material not available in any western language.  I cite the appropriate pages of Tropfke at the beginning of any topic covered by Tropfke.

          Another book in the field is W. L. Schaaf's Bibliography of Recreational Mathematics, in four volumes.  This is a quite exhaustive bibliography of recent articles, but it is not chronological, is without annotation and is somewhat less classified than the present work.  Nonetheless it is a valuable guide to recent material.

 

          Collecting books on magic has been popular for many years and quite notable collections and bibliographies have been made.  Magic overlaps recreational mathematics, particularly in older books, and I have now added references to items listed in the bibliographies of Christopher, Clarke & Blind, Hall, Heyl, Toole Stott and Volkmann & Tummers -- details of these works are given in the list of Common References below.  There is a notable collection of Harry Price at Senate House, University of London, and a catalogue was printed in 1929 & 1935 -- see HPL in Common References.

          Another related bibliography is Santi's Bibliografia della Enigmistica, which is primarily about word puzzles, riddles, etc., but has some overlap with recreational mathematics -- again see the entry in the list of Common References.  I have not finished working through this.

          Other relevant bibliographies are listed in Section 3.B.

 

                    COVERAGE

 

          In selecting topics, I tend to avoid classical number theory and classical geometry.  These are both pretty well known.  Dickson's History of the Theory of Numbers and Leveque's and Guy's Reviews in Number Theory cover number theory quite well.  I also tend to avoid simple exercises, e.g. in the rule of three, in 'aha' or 'heap' problems, in the Pythagorean theorem (though I have now included 6.BF) or in two linear equations in two unknowns, though these often have fanciful settings which are intended to make them amusing and some of these are included -- see  7.R,  7.X,  7.AX.  I also leave out most divination (or 'think of a number') techniques (but a little is covered in 7.M.4.b) and most arithmetic fallacies.  I also leave out Conway's approach to mathematical games -- this is extensively covered by Winning Ways and Frankel's Bibliography.

          The classification of topics is still ad‑hoc and will eventually get rationalised -- but it is hard to sort things until you know what they are!  At present I have only grouped them under the general headings: Biography, General, History & Bibliography, Games, Combinatorics, Geometry, Arithmetic, Probability, Logic, Physics, Topology.  Even the order of these should be amended.  The General section should be subsumed under the History & Bibliography.  Geometry and Arithmetic need to be subdivided.

          I have recently realised that some general topics are spread over several sections in different parts.  E.g. fallacies are covered in 6.P, 6.R, 6.AD, 6.AW.1, 6.AY, 7.F, 7.Y, 7.Z, 7.AD, 7.AI, 7.AL, 7.AN, most of 8, 10.D, 10.E, 10.O.  Perhaps I will produce an index to such topics.  I try to make appropriate cross-references.

          Some topics are so extensive that I include introductory or classifactory material at the beginning.  I often give a notation for the problems being considered.  I give brief explanations of those problems which are not well known or are not described in the notation or the early references.  There may be a section index.  I have started to include references to comprehensive surveys of a given topic -- these are sometimes given at the beginning.

          Recreational problems are repeated so often that it is impossible to include all their occurrences.  I try to be exhaustive with early material, but once a problem passes into mathematical and general circulation, I only include references which show new aspects of the problem or show how the problem is transmitted in time and/or space.  However, the point at which I start leaving out items may vary with time and generally slowly increases as I learn more about a topic.  I include numerous variants and developments on problems, especially when the actual origin is obscure.

          When I began, I made minimal annotations, often nothing at all.  In rereading sections, particularly when adding more material, I have often added annotations, but I have not done this for all the early entries yet.

          Recently added topics often may exist in standard sources that I have not reread recently, so the references for such topics often have gaps -- I constantly discover that Loyd or Dudeney or Ahrens or Lucas or Fibonacci has covered such a topic but I have forgotten this -- e.g. looking through Dudeney recently, I added about 15 entries.  New sections are often so noted to indicate that they may not be as complete as other sections.

          Some of the sources cited are lengthy and I originally added notes as to which parts might be usable in a book of readings -- these notes have now been mostly deleted, but I may have missed a few.

 

                    STATUS OF THE PROJECT

 

          I would like to think that I am about 75% of the way through the relevant material.  However, I recently did a rough measurement of the material in my study -- there is about 8 feet of read but unprocessed material and about 35 feet of unread material, not counting several boxes of unread Rubik Cube material and several feet of semi-read material on my desk and table.  I recently bought two bookshelves just to hold unread material.  Perhaps half of this material is relevant to this work.

          In particular, the unread material includes several works of Folkerts and Sesiano on medieval MSS, a substantial amount of photocopies from Schott, Schwenter and Dudeney (400 columns), some 2000 pages of photocopies recently made at Keele, some 500 pages of photocopies from Martin Gardner's files, as well as a number of letters.  Marcel Gillen has made extracts of all US, German and EURO patents and German registered designs on puzzles -- 26 volumes, occupying about two feet on my shelves.  I have recently acquired an almost complete set of Scripta Mathematica (but I have previously read about half of it), Schwenter-Harsdörffer's Deliciæ Physico‑Mathematicae, Schott's Joco-Seriorum and Murray's History of Board Games Other Than Chess.  I have recently acquired the early issues of Eureka, but there are later issues that I have not yet read and they persist in not sending the current copies I have paid for!

          I have not yet seen some of the earlier 19C material which I have seen referred to and I suspect there is much more to be found.  I have examined some 18C & 19C arithmetic and algebra books looking for problem sections -- these are often given the pleasant name of Promiscuous Problems.  There are so many of these that a reference to one of them probably indicates that the problem appears in many other similar books that I have not examined.  My examination is primarily based on those books which I happen to have acquired.  There are a few 15-17C books which I have not yet examined, notably those included at the end of the last paragraph.

          In working on this material, it has become clear that there were two particularly interesting and productive eras in the 19C.  In the fifteen years from 1857, there appeared about a dozen books in the US and the UK: The Magician's Own Book (1857);  Parlour Pastime, by "Uncle George" (1857);  The Sociable (1858);  The Boy's Own Toymaker, by Landells (1858);  The Book of 500 Curious Puzzles (1859);  The Secret Out (1859);  Indoor and Outdoor Games for Boys and Girls (c1859);  The Boy's Own Conjuring Book (1860);  The Illustrated Boy's Own Treasury (1860, but see below);  The Parlor Magician (1863);  The Art of Amusing, by Bellew (1866);  Parlour Pastimes (1868);  Hanky Panky (1872);  Within Doors, by Elliott (1872);  Magic No Mystery (1876), just to name those that I know.  Most of these are of uncertain authorship and went through several editions and versions.  The Magician's Own Book, The Book of 500 Curious Puzzles, The Secret Out, The Sociable, The Parlor Magician, Hanky Panky, and Magic No Mystery seem to be by the same author(s).  I have recently had a chance to look at a number of previously unseen versions at Sotheby's and at Edward Hordern's and I find that sometimes two editions of the same title are essentially completely different!  This is particularly true for US and UK editions.  Many of the later UK editions say 'By the author of Magician's Own Book etc., translated and edited by W. H. Cremer Jr.'  From the TPs, it appears that they were written by Wiljalba Frikell (1818‑1903) and then translated into English.  However, BMC and NUC generally attribute the earlier US editions to George Arnold (1834-1865), and some catalogue entries explicitly say the Frikell versions are later editions, so it may be that Frikell produced later editions in some other language (French or German ??) and these were translated by Cremer.  On the other hand, the UK ed of The Secret Out says it is based on Le Magicien des Salons.  This is probably Le Magicien des Salons ou le Diable Couleur de Rose, for which I have several references, with different authors! --  J. M. Gassier, 1814;  M. [Louis Apollinarie Christien Emmanuel] Comte, 1829;  Richard (pseud. of A. O. Delarue), 1857 and earlier.  There was a German translation of this.  Some of these are at HPL but ??NYS.  Items with similar names are:  Le Magicien de Société, Delarue, Paris, c1860?  and  Le Manuel des Sorciers (various Paris editions from 178?-1825, cf in Common References).  It seems that this era was inspired by these earlier French books.  To add to the confusion, an advertisement for the UK ed. of Magician's Own Book (1871?) says it is translated from Le Magicien des Salons which has long been a standard in France and Germany.  Toole Stott opines that Frikell had nothing to do with these books -- as a celebrated conjuror of the times, his name was simply attached to the books.  Toole Stott also doubts whether Le Magicien des Salons exists -- but it now seems pretty clear that it does, though it may not have been the direct source for any of these works, but see below.

          Christopher 242 cites the following article on this series.

          Charles L. Rulfs.  Origins of some conjuring works.  Magicol 24 (May 1971) 3-5.  He discusses the various books, saying that Cremer essentially pirated the Dick & Fitzgerald productions.  He says The Magician's Own Book draws on Wyman's Handbook (1850, ??NYS), Endless Amusement, Parlour Magic (by W. Clarke?, 1830s, ??NYS), Brewster's Natural Magic (??NYS).  He says The Secret Out is largely taken, illustrations and all, from Blismon de Douai's Manuel du Magicien (1849, ??NYS) and Richard & Delion's Magicien des salons ou le diable couleur de rose (1857 and earlier, ??NYS).

          Christopher 622 says Harold Adrian Smith [Dick and Fitzgerald Publishers; Books at Brown 34 (1987) 108-114] has studied this series and concludes that Williams was the author of Magician's Own Book, assisted by Wyman.  Actually Smith simply asserts: "The book was undoubtedly [sic] written by H. L. Williams, a "hack writer" of the period, assisted by John Wyman in the technical details."  He gives no explanation for his assertion.  He later says he doubts whether Cremer ever wrote anything.  He suggests The Secret Out book is taken from DeLion.  He states that The Boy's Own Conjuring Book is a London pirate edition.

          Several of the other items are anonymous and there was a tremendous amount of copying going on -- problems are often reproduced verbatim with the same diagram or sometimes with minor changes.  In some cases, the same error is repeated in five different books!  I have just discovered some earlier appearances of the same material in The Family Friend, a periodical which ran in six series from 1849 to 1921 and which I have not yet tracked down further.  However, vol. 1-3 of 1849‑1850 and the volume for Jul‑Dec 1859 contain a number of the problems which appear repeatedly and identically in the above cited books.  Toole Stott 407 is an edition of The Illustrated Boy's Own Treasury of c1847 but the BM copy was destroyed in the war and the other two copies cited are in the US.  If this date is correct, then this book is a forerunner of all the others and a major connection between Boy's Own Book and Magician's Own Book.  I would be most grateful to anyone who can help sort out this material -- e.g. with photocopies of these or similar books or magazines.

          The other interesting era was about 1900.  In English, this was largely created or inspired by Sam Loyd and Henry Dudeney.  Much of this material first appeared in magazines and newspapers.  I have seen much less than half of Loyd's and Dudeney's work and very little of similar earlier material (but see below).  Consequently problems due to Loyd or Dudeney may seem to first appear in the works of Ball (1892, et seq.), Hoffmann (1893) and Pearson (1907).  Further examination of Loyd's and Dudeney's material will be needed to clarify the origin and development of many problems.  Though both started puzzle columns about 1896, they must have been producing material for a decade or more previously which does not seem to be known.  I have just obtained photocopies of 401 columns by Dudeney in the Weekly Dispatch of 1897-1903, but have not had time to study them.  Will Shortz and Angela Newing have been studying Loyd and Dudeney respectively and turning up their material.

          The works of Lucas (1882‑1895), Schubert (1890s) and Ahrens (1900‑1918) were the main items on the Continent and they interacted with the English language writers.  Ahrens was the most historical of these and his book is one of the foundations of the present work.  All of these also wrote in newspapers and magazines and I have not seen all their material.

 

          I would be happy to hear from anyone with ideas or suggestions for this bibliography.  I would be delighted to hear from anyone who can locate missing information or who can provide copies of awkward material.  I am particularly short of information about recreations in the Arabic period.  I prepared a separate file, 'Queries and Problems in the History of Recreational Mathematics', which is about 23 pages, and has recently been updated.  I have also prepared three smaller letters of queries about Middle Eastern, Oriental and Russian sources and these are generally more up-to-date.

 

                    TECHNICAL NOTES

 

          I have prepared a CD containing this and much else of my material.  I divided Sources into four files when I used floppy discs as it was too big to fit on one disc, and I have not yet changed this.  The files are:  1: Introductory material and list of abbreviations/references;  2: Sections 1 - 6;  3: Section 7;  4: Sections 8 - 11.  It is convenient to have the first file separate from the main material, but I might combine the other three files.  (I have tried to send it by email in the past, but this document is very large (currently c4.1MB and the Word version will be longer) and most people who requested it by email found that it overflowed their mailbox and created chaos in their system -- this situation has changed a bit with larger memories and improved transmission speeds.)

          This file started on a DEC-10, then was transferred to a VAX.  It is now on my PC using Script Professional, the development of LocoScript on the Amstrad.  Even in its earliest forms, this provided an easy and comprehensive set of diacritical marks, which are still not all available nor easy to use in WordPerfect or Word (except perhaps by using macros and/or overstriking??).  It also provides multiple cut and paste buffers and easy formatting, though I have learned how to overcome these deficiencies in Word.

          Script provides an ASCII output, but this uses IBM extended ASCII which has 8-bit codes.  Not all computers will accept or print such characters and sometimes they are converted into printer control codes causing considerable confusion.  I have a program that converts these codes to 7-bits -- e.g. accents and umlauts are removed.  However, ASCII loses a great deal of the information, such as sub- and superscripts, so this is not a terribly useful format. 

          Script also provides WordStar and "Revisable-Form-Text DCA" output, but neither of these seems to be very successful (DCA is better than WordStar).  Script later added a WordPerfect exporting facility.  This works well, though some (fairly rare) characters and diacritical marks are lost and the output requires some reformatting.  (Nob Yoshigahara reports that Japanese WordPerfect turns all the extended ASCII characters into Kanji characters!) 

          Reading the WordPerfect output in Word (you may need to install this facility) gives a good approximation to my text, but in  Courier 10pt.  Selecting All  and changing to  Times New Roman 12pt  gives an better approximation.  (Some files use a smaller font of 10pt and I may have done some into 9pt.)  You have to change this in the Header separately, using  View  Header and Footer.  The page layout is awkward as my page numbering header gets put into the text, leaving a large gap at the top.  I go into  Page Setup  and set the  Paper Size  to A4 and the  Top, Bottom and Header Margins  to 15mm and the  Left and Right Margins  to 25mm.  (It has taken me some time to work this out and some earlier files may have other settings.)  However, I find that lines are a bit too close together and underlines and some diacritical marks are lost, so one needs to also go into  Format  Paragraph  Spacing -- Line Spacing  and choose  At least  and  12pt  (or 10pt).  I use hanging indentation in most of the main material and this feature is not preserved in this conversion.  By selecting a relevant section and going into  Format  Paragraph  Indentation -- Special  and selecting  Hanging,  it should automatically select  10.6mm  which corresponds to my automatic spacing of five characters in 12pt.  Further, I use second level hanging indentation in quite a number of places.  You need to create a style which is the basic style with the left hand margin at  10.6mm (or 10 or 11 mm).  When second level indenting is needed, select the desired section and apply this style to it. 

          However, this still leaves some problems.  I use em dashes a bit, i.e.  –,  which gets converted into an underline,  _.  In Word, this is obtained by use of  CTRL  and the  -  sign on the numeric keypad.  One can use the find and replace feature, EXCEPT that a number of other characters are also converted into underlines.  In particular, Cyrillic characters are all converted into underlines.  This is not insuperable as I always(?) give a transliteration of Cyrillic (using the current Mathematical Reviews system) and one can reconstruct the original Cyrillic from it.  I notice that the Cyrillic characters are larger than roman characters and hence may overlap.  One can amend this by selecting the Cyrillic text and going into  Format  Font  Character Spacing  Spacing  and choosing  Expanded  By  2 pt (or thereabout).  But a number of characters with unusual diacritical marks are also converted to underlines or converted to the unmarked character and not all of these are available in Word.  E.g.  ĭ,  which is the transliteration of  й  becomes just  i.  I am slowly forming a Word file containing the Word versions of entries having the Cyrillic or other odd characters, and I will include this file on my CD, named  CYRILLIC.DOC.  For diacritical marks not supported by Word, I use an approximation and/or an explanation. 

          It is very tedious to convert the underlines back to em dashes, so I will convert every em dash to a double hyphen --. 

          Finally, I have made a number of diagrams by simple typing without proportional spacing and Word does not permit changing font spacing in mid-line and ignores spaces before a right-alignment instruction.  The latter problem can be overcome by using hard spaces and the former problem is less of a problem, and I think it can be overcome.

 

          Later versions of Script support Hewlett-Packard DeskJets and I am now on my second generation of these, so the 7th and future editions will be better printed (if they ever are!).  However, this required considerable reformatting as the text looks best in proportional spacing (PS) and I found I had to check every table and every mathematical formula and diagram.  Also, to set off letters used as mathematical symbols within text, I find PS requires two spaces on each side of the letter -- i.e.  I refer to  x  rather than to x.  (I find this easier to do than to convert to italics.)  I also sometimes set off numbers with two spaces, though I wasn't consistent in doing this at the beginning of my reformatting.  The conversion to proportional spacing reduced the total length from  587  to  488  pages, a reduction of  16.87%  which is conveniently estimated as  1/6.  The percentage of reduction was fairly consistent throughout the conversion process.

          The printing of Greek characters went amiss in the second part of the 6th Preliminary Edition, apparently due to the printer setting having been changed without my noticing -- this happens if an odd character gets sent to the printer, usually in DOS when trying to use or print a corrupted file, and there is no indication of it.  I was never able to reproduce the effect!

          The conversion to (Loco)Script provided many improved features compared to my earlier DEC versions.  I am using an A4 page  (8¼  by  11⅔  inches) rather than an  8½  by  11 inch page, which gives  60  lines of text per page, four more or  7%  more than when using the DEC or VAX.

 

                    NEW SECTIONS IN THIS EDITION.

 

[SIXTH EDITION: 1: Fibonacci, 1: Montucla;  3.B;  4.A.1.a, 4.B.9, 4.B.10, 4.B.11, 4.B.12;  5.R.1.a, 5.W.1, 5.AA, 5.AB;  6.AS.1.b, 6.AS.2.a, 6.AS.5, 6.AW.4, 6.BP, 6.BQ, 6.BR;  7.I.1, 7.Y.2, 7.AY, 7.AZ; 7.BA;  8.I, 8.J;  9.E.2, 9.K;  10.A.4, 10.A.5, 10.U, 10.V, 10.W;  11.K.6, 11.K.7, 11.K.8.]

 

In the last edition, I had 8.K instead of 8.J in the list of New Sections and in the Contents.

 

1: Pacioli, Carroll, Perelman;  4.B.13, 4.B.14, 4.B.15;  5.B.2, 5.H.3 (the previous 5.H.3 has been renumbered 5.H.4), 5.K.3, 5.R.1.b, 5.X.4, 5.AC, 5.AD, 5.AE, 5.AF, 5.AG.1, 5.AG.2;  6.AJ.4, 6.AJ.5, 6.AS.3.a, 6.AT.8, 6.AT.9, 6.AY.2, 6.BF.4, 6.BF.5, 6.BS, 6.BT, 6.BU, 6.BV, 6.BW;  7.H.6, 7.H.7 (formerly part of 7.H.5), 7.M.4.a, 7.M.4.b, 7.M.6, 7.R.4, 7.AC.3.a, 7.AC.7, 7.AH.1, 7.AJ.1, 7.BB, 7.BC;  8.K, 8.L;  10.A.6, 10.A.7, 10.A.8, 10.D has become 10.D.1, 10.D.2, 10.D.3, 10.E.4, 10.X, 10.Y, 10.Z, 10.AA, 10.AB, 10.AC, 10.AD, 10.AE;  11.N, 11.O, 11.P, 11.Q, 11.R, 11.S.  (65 new sections)

 

 

                    ACKNOWLEDGMENTS

 

          I am immensely indebted to many mathematicians, historians, puzzlers, bookdealers and others who have studied particular topics, as will be apparent.

          I have had assistance from so many sources that I have probably forgotten some, but I would like to give thanks here to the following, and beg forgiveness from anyone inadvertently omitted -- if you remind me, I will make amendment.  In some cases, I simply haven't got to your letter yet!  Also I have had letters from people whose only identification is an undecipherable signature and phone messages from people whose name and phone number are unintelligible. 

          Sadly, a few of these have died since I corresponded with them and I have indicated those known to me with †.

 

André Allard,  Eric J. Aiton†,  Sue Andrew,  Hugh Ap Simon,  Gino Arrighi,  Marcia Ascher,  Mohammad Bagheri,  Banca Commerciale Italiana,  Gerd Baron,  Chris Base,  Rainier [Ray] Bathke,  John Beasley,  Michael Behrend,  Jörg Bewersdorff,  Norman L. Biggs,  C. [Chris] J. Bouwkamp,  Jean Brette,  John Brillhart,  Paul J. Campbell,  Cassa di Risparmio di Firenze,  Henry Cattan,  Marianna Clark,  Stewart Coffin,  Alan & Philippa Collins,  John H. Conway,  H. S. M. Coxeter,  James Dalgety,  Ann E. L. Davis,  Yvonne Dold,  Underwood Dudley,  Anthony W. F. Edwards,  John Ergatoudis,  John Fauvel†,  Sandro Ferace,  Judith V. Field,  Irving Finkel,  Graham Flegg,  Menso Folkerts,  David Fowler,  Aviezri S. Fraenkel,  Raffaella Franci,  Gregory N. Frederickson,  Michael Freude,  Walter W. Funkenbusch,  Nora Gädeke,  Martin Gardner,  Marcel Gillen,  Leonard J. Gordon,  Ron Gow,  Ivor Grattan‑Guinness,  Christine Insley Green,  Jennifer Greenleaves (Manco),  Tom Greeves,  H. [Rik] J. M. van Grol,  Branko Grünbaum,  Richard K. Guy,  John Hadley,  Peter Hajek,  Diana Hall,  Joan Hammontree,  Anton Hanegraaf†,  Martin Hansen,  Jacques Haubrich,  Cynthia Hay,  Takao Hayashi,  Robert L. Helmbold,  Hanno Hentrich,  Richard I. Hess,  Christopher Holtom,  Edward Hordern†,  Peter Hosek,  Konrad Jacobs,  Anatoli Kalinin,  Bill Kalush,  Michael Keller,  Edward S. Kennedy,  Sarah Key (The Haunted Bookshop),  Eberhard Knobloch,  Don Knuth,  Bob Koeppel,  Joseph D. E. Konhauser,  David E. Kullman,  Mogens Esrom Larsen,  Jim Lavis (Doxa (Oxford)),  John  Leech†,  Elisabeth Lefevre,  C. Legel,  Derrick [Dick] H. Lehmer†,  Emma Lehmer,  Leisure Dynamics,  Hendrik W. Lenstra,  Alan L. Mackay,  Andrzej Makowski,  John Malkevitch,  Giovanni Manco,  Tatiana Matveeva,  Ann Maury,  Max Maven,  Jim McArdle,  Patricia McCulloch,  Peter McMullen,  Leroy F. Meyers†,  D. P. Miles,  Marvin Miller,  Nobuo Miura,  William O. J. Moser,  Barbara Moss,  Angela Newing,  Jennie Newman,  Tom and Greta O'Beirne††,  Owen O'Shea,  Parker Brothers,  Alan Parr,  Jean J. Pedersen,  Luigi Pepe,  William Poundstone,  Helen Powlesland,  Oliver Pretzel,  Walter Purkert,  Robert A. Rankin†,  Eleanor Robson,  David J. A. Ross,  Lee Sallows,  Christopher Sansbury,  Sol Saul,  William L. Schaaf,  Doris Schattschneider,  Jaap Scherphuis,  Heribert Schmitz,  Š. Schwabik,  Eileen Scott†,  Al Seckel,  Jacques Sesiano,  Claude E. Shannon†,  John Sheehan,  A. Sherratt,  Will Shortz,  Kripa Shankar Shukla,  George L. Sicherman,  Deborah Singmaster,  Man‑Kit Siu,  Gerald [Jerry] K. Slocum,  Cedric A. B. Smith† (and Sue Povey & Jim Mallet at the Galton Laboratory for letting me have some of Cedric's books),  Jurgen Stigter,  Arthur H. Stone,  Mel Stover†,  Michael Stueben,  Shigeo Takagi†,  Michael Tanoff,  Gary J. Tee,  Andrew Topsfield,  George Tyson†,  Dario Uri,  Warren Van Egmond,  Carlo Viola,  Kurt Vogel†,  Anthony Watkinson,  Chris Weeks,  Maurice Wilkes,  John Winterbottom,  John Withers,  Nob. Yoshigahara,  Claudia Zaslavsky.

 

 

          I would also like to thank the following libraries and museums which I have used:

 

University of Aberdeen;  University of Bristol;   Buckleys Shop Museum, Battle, East Sussex;  University of Calgary;  University of Cambridge;  Marsh's Library, Dublin; 

 

FLORENCE:

          Biblioteca Nazionale;  Biblioteca Riccardiana;

 

University of Keele -- The Turner Collection(†) and its librarian Martin Phillips;

Karl‑Marx‑Universität, Leipzig: Universität Bibliothek and Sektion Mathematik Bibliothek,

          especially Frau Letzel at the latter;

 

LONDON:

          Birkbeck College;  British Library (at Bloomsbury and then at St. Pancras; also at Colindale);  The London Library;  School of Oriental and African Studies, especially Miss Y. Yasumara, the Art Librarian;  Senate House, particularly the Harry Price Library;  South Bank University;  Southwark Public Library;  University College London, especially the Graves Collection and the Rare Book Librarians Jill Furlong, Susan Stead and their staff;  Warburg Institute; 

 

MUNICH:

          Deutsches Museum; Institut für Geschichte der Naturwissenschaften;

 

NEW YORK:

          Brooklyn Public Library;  City College of New York;  Columbia University;

 

Newark Public Library, Newark, New Jersey;

University of Newcastle upon Tyne -- The Wallis Collection and its librarian Lesley Gordon;

 

OXFORD:

          Ashmolean Museum;  The Bodleian Library;  Museum of the History of Science, and its librarian Tony Simcock; 

 

University of Reading;  University of St. Andrews; 

 

SIENA:

          Biblioteca Comunale degli Intronati;  Dipartimento di Matematica, Università di Siena;

 

University of Southampton;  Mathematical Institute, Warsaw.

 

 

          I would like especially to thank the following.

 

          Interlibrary Loans (especially Brenda Spooner) at South Bank University and the British Library Lending Division for obtaining many strange items for me.

          Richard Guy, Bill Sands and the Strens bequest for a most useful week at the Strens/Guy Collection at Calgary in early 1986 and for organizing the Strens Memorial Meeting in summer 1986 and for printing the first preliminary edition of these Sources.

          Gerd Lassner, Uwe Quasthoff and the Naturwissenschaftlich‑Theoretisches Zentrum of the Karl‑Marx‑Universität, for a very useful visit to Leipzig in 1988.

          South Bank University Computer Centre for the computer resources for the early stages of this project, and especially Ann Keen for finding this file when it was lost.

          My School for printing these preliminary editions.

          Martin Gardner for kindly allowing me to excavate through his library and files.

          James Dalgety, Edward Hordern, Bill Kalush, Chris Lewin, Tom Rodgers and Will Shortz for allowing me to rummage through their libraries.

          John Beasley, Edward Hordern, Bill Kalush, Will Shortz and Jerry Slocum for numerous photocopies and copies from their collections.

          Menso Folkerts, Richard Lorch, Michael Segre and the Institut für Geschichte der Naturwissenschaft, Munich, for a most useful visit in Sep 1994 and for producing a copy of Catel.

          Raffaella Franci and the Dipartimento di Matematica and the Centro Studi della Matematica Medioevale at Università di Siena for a most useful visit in Sep 1994.

          Takao Hayashi for much material from Japan and India.

          My wife for organizing a joint trip to Newcastle in Sep 1997 where I made use of the Wallis Collection.

 

          Finally, I would like to thank a large number of publishers, distributors, bookdealers and even authors who have provided copies of the books and documents upon which much of this work is based.  Bookdealers have often let me examine books in their shops.  Their help is greatly appreciated.  There are too many of these to record here, but I would like to mention Fred Whitehart (†1999), England's leading dealer in secondhand scientific books for many years who had a real interest in mathematics.

 

 

                             

 

                              ABBREVIATIONS

 

                    DIACRITICAL MARKS AND NOTATION

 

          Before converting to LocoScript, I used various conventions, given below, to represent diacritical marks.  Each symbol (except  ') occurred after the letter it referred to.  I have now converted these and all mathematical conventions into correct symbols, so far as possible, but I may have missed some, so I am keeping this information for the present.

          Common entries using such marks are given later in this section and only the abbreviated or simplified form is used later -- e.g. I use Problemes for Bachet's work rather than Problèmes.  (Though this may change??)

          Initially, I did not record all diacritical marks, so some may be missing though I have checked almost all items.  I may omit diacritical marks which are very peculiar. 

          Transliterations of Arabic, Sanskrit, Chinese, etc. are often given in very different forms.  See Smith, History, vol. 1, pp. xvii-xxii for a discussion of the problems.  The use of  ^  and  ˉ  seems interchangeable and I have used  ^  when different versions use both  ^  and  ˉ , except when quoting a title or passage when I use the author's form.  [Smith, following Suter, uses  ^  for Arabic, but uses  ˉ  for Indian.  Murray uses  ˉ  for both.  Wieber uses  ˉ  for Arabic.  Van der Linde uses  ´  for Arabic.  Datta & Singh use  ^  for Indian.]

          There are two breathing marks in Arabic -- ayn    and alif/hamzah    -- but originally I didn't have two forms easily available, so both were represented by  '.  I have now converted almost all of these to    and  ’.  These don't seem to be as distinct in the printing as on my screen.

          French practice in accenting capitals is variable and titles are often in capitals, so expected marks may be missing.  Also, older printing may differ from modern usage -- e.g. I have seen: Liège and Liége;  Problèmes, Problêmes and Problémes.  When available, I have transcribed the material as printed without trying to insert marks, but many places insert the marks according to modern French spelling.

          Greek and Cyrillic titles are now given in the original with an English transliteration (using the Amer. Math. Soc. transliteration for Cyrillic).

          I usually ignore the older usage of  v  for  u  and  i  for  j,  so that I give  mathematiqve  as  mathematique  and  xiij  as  xiii.

          I used  a1, a2, ..., ai,  etc. for subscripted variables, though I also sometimes used  a(1), a(2), ..., a(i),  etc.  Superscripts or exponents were indicated by use of  ^,  e.g.  2^3  is  8.  These have been converted to ordinary sub- and superscript usage, but  ^  may be used when the superscript is complicated -- e.g. for  2^ai  or  9^(99). 

          Greek letters were generally spelled out in capitals or marked with square brackets, e.g. PI, [pi], PHI, but these have probably all been converted. 

          My word processor does not produce binomial coefficients easily, so I use  BC(n, k)  for  n!/k!(n‑k)! 

          Many problems have solutions which are sets of fractions with the same denominator and I abbreviate  a/z,  b/z,  c/z  as  (a, b, c)/z.  Notations for particular problems are explained at the beginning of the topic. 

          Rather than attempting to italicise letters used as symbols, I generally set them off by double-spaces on each side -- see examples above.  Other mathematical notations may be improvised as necessary and should be obvious.

          Recall that the symbols below occurred after the letter they referred to, except for  ' .

 

"         denoted umlaut or diaeresis in general, e.g.:  ä, ë, ï, ö, ü.

/         was used after a letter for accent acute,  ́, after  l  for  ł  in Polish, and after  o  for  ø  in Scandinavian.

\         denoted accent grave,  ̀.

^        denoted the circumflex,  ^,  in Czech, etc.; the overbar (macron)  ˉ   or  ^  for a long vowel in Sanskrit, Hindu, etc.; and the overbar used to indicate omission in medieval MSS.

@       denoted the cedilla (French  ç  and Arabic  ş)  and the ogonek or Polish hook (Polish  ą).

.         denoted the underdot in  ḥ, ḳ, ṇ, ṛ, ṣ,  ,  in Sanskrit, Hindu, Arabic.  These are sometimes written with a following  h  -- e.g.  k  may also be written  kh  and I may sometimes have used this.  (It is difficult to search for  . , etc., so not all of these may be converted.)  This mark vanishes when converted to WordPerfect.

*        denoted the overdot for  ġ, , ṅ,  in Sanskrit, Hindu, Arabic.  This vanishes over  m  and  n  in WordPerfect.

~        denoted the Spanish tilde  ~  and the caron or hachek  ˇ,  in  ğ, š.  The breve is a curved version,  ˘,  of the same symbol and is essentially indistinguishable from the caron.  It occurs in Russian  й,  which is translitereated as  ĭ.

_        denoted the underbar in  , j, ṯ (I cannot find a  j  with an underbar in Arial).  This mark vanishes in WordPerfect.

'         denotes breathing marks in Arabic, etc.  There are actually two forms of this -- ayn    and alif/hamzah    -- but I didn't have two forms easily available and originally entered both as apostrophe  '  .  These normally occur between letters and I placed the  '  in the same space.  I have converted most of these.

 

          Commonly occurring words with diacritical marks are: Académie, arithmétique, bibliothèque, Birkhäuser, café, carré, école, Erdös, für, géomètre, géométrie, Göttingen, Hanoï -- in French only, ‑ième, littéraire, mathématique, mémoire, ménage, misère, Möbius, moiré, numérique, Pétersbourg, probabilités, problème (I have seen problêmes??), Rätsel, récréation, Sändig, siècle, société, Thébault, théorie, über, umfüllung.

 

          I have used  ??  to indicate uncertainty and points where further work needs to be done.  The following symbols after  ??  indicate the action to be done.

          *                  check for diacritical marks, etc.

          NX               no Xerox or other copy

          NYS             not yet seen

          NYR  not yet read

          o/o               on order

          SP                check spelling

          Other comments may be given.

 

 

          ABBREVIATIONS OF JOURNALS AND SERIES. 

See:  AMM,  CFF,  CM,  CMJ,  Family Friend,  G&P,  G&PJ,  HM,  JRM,  MG,  MiS,  MM,  MS, MTg,  MTr,  M500,  OPM,  RMM,  SA,  SM,  SSM  in Common References below.

 

          ABBREVIATIONS OF PUBLISHERS. 

See:  AMS,  C&W,  CUP,  Loeb Classical Library,  MA,  MAA,  NCTM,  OUP  in Common References below.

 

          ABBREVIATIONS OF MONTHS.  All months are given by their first three letters in English:  Jan,  Feb,  ....

 

          PUBLISHERS' LOCATIONS.  The following publisher's locations will not be cited each time.  Other examples may occur and can be found in the file PUBLOC.

 

AMS (American Mathematical Society),  Providence, Rhode Island, USA.

Chelsea Publishing,  NY, USA. 

CUP (Cambridge University Press),  Cambridge, UK. 

Dover,  NY, USA.

Freeman,  San Francisco, then NY, USA.

Harvard University Press,  Cambridge, Massachusetts, USA.

MA (Mathematical Association),  Leicester, UK.

MAA (Mathematical Association of America),  Washington, DC, USA.

NCTM (National Council of Teachers of Mathematics),  Reston, Virginia, USA.

Nelson,  London, UK. 

OUP (Oxford University Press),  Oxford, UK (and also NY, USA).

Penguin,  Harmondsworth, UK. 

Simon & Schuster,  NY, USA.

 

 

 

                              COMMON REFERENCES.

 

          NOTES.  When referring to items below, I will usually include the earliest reasonable date, even though the citation may be to a much later edition.  For example, I would say "Canterbury Puzzles, 1907", even though I am citing problem numbers or pages from the 1958 Dover reprint of the 1919 edition.  Sometimes the earlier editions are hard to come by and I have sometimes found that the earlier edition has different pagination -- in that case I will (eventually) make the necessary changes.

          Edition information in parentheses indicates items or editions that I have not seen, though I don't always do this when the later version is a reprint or facsimile.

 

 

Abbaco.                        See:  Pseudo-dell'Abbaco.

Abbot Albert. Abbot Albert von Stade.  Annales Stadenses.  c1240.  Ed. by J. M. Lappenberg.  In: Monumenta Germaniae Historica, ed. G. H. Pertz, Scriptorum t. XVI, Imp. Bibliopolii Aulici Hahniani, Hannover, 1859 (= Hiersemann, Leipzig, 1925), pp. 271‑359.  (There are  13  recreational problems on pp. 332‑335.)  [Vogel, on p. 22 of his edition of the Columbia Algorism, dates this as 1179, but Tropfke gives 1240, which is more in line with Lappenberg's notes on variants of the text.  The material of interest, and several other miscellaneous sections, is inserted at the year 1152 of the Annales, so perhaps Vogel was misled by this.]  I have prepared an annotated translation of this: The problems of Abbot Albert (c1240).  I have numbered the problems and will cite this problem number.

Abraham.                      R. M. Abraham.  Diversions and Pastimes.  Constable, London, 1933  = Dover, 1964 (slightly amended and with different pagination, later retitled: Tricks and Amusements with Coins, Cards, String, Paper and Matches).  I will cite the Constable pages (and the Dover pages in parentheses).

Ackermann.                  Alfred S. E. Ackermann.  Scientific Paradoxes and Problems and Their Solutions.  The Old Westminster Press, London, 1925.

D. Adams.  New Arithmetic.  1835. 

                                    Daniel Adams (1773-1864).  ADAMS NEW ARITHMETIC.  Arithmetic, in which the principles of operating by numbers are analytically explained, and synthetically applied; thus combining the advantages to be derived both from the inductive and synthetic mode of instructing:  The whole made familiar by a great variety of useful and interesting examples, calculated at once to engage the pupil in the study, and to give him a full knowledge of figures in their application to all the practical purposes of life.  Designed for the use of schools and academies in the United States.  J. Prentiss, Keene, New Hampshire, 1836, boarded.  1-262 pp + 2pp publisher's ads, apparently inserted backward.  [Halwas 1-6 lists 1st ed as 1835, then has 1837, 1838, 1839, 1842, c1850.]  This is a reworking of The Scholar's Arithmetic of 1801.

D. Adams.  Scholar's Arithmetic.  1801.

                                    Daniel Adams (1773-1864).  The Scholar's Arithmetic;  or,  Federal Accountant:  Containing.  I. Common arithmetic, ....  II. Examples and Answers with Blank Spaces, ....  III. To each Rule, a Supplement, comprehending, 1. Questions ....  2. Exercises.  IV. Federal Money, ....  V. Interest cast in Federal Money, ....  VI. Demonstration by engravings ....  VII. Forms of Notes, ....  The Whole in a Form and Method altogether New, for the Ease of the Master and the greater Progress of the Scholar.  Adams & Wilder, Leominster, Massachusetts, 1801; 2nd ed, 1802.  3rd ed ??.  4th ed, by Prentiss, 1807; 6th ed, 1810; 10th ed, 1816; Stereotype Edition, Revised and Corrected, with Additions, 1819, 1820, 1824; John Prentiss, Keene, New Hampshire, 1825.  [Halwas 8-14.]  I have the 1825, whose Preface is for the 10th ed of 1816, so is probably identical to that ed.  The Preface says he has now made some revisions.  The only change of interest to us is that he has added answers to some problems.  So I will cite this as 1801 though I will be giving pages from the 1825 ed.  The book was thoroughly reworked as Adams New Arithmetic, 1835.

M. Adams.  Indoor Games.  1912.

                                    Morley Adams, ed.  The Boy's Own Book of Indoor Games and Recreations.  "The Boy's Own Paper" Office, London, 1912; 2nd ptg, The Religious Tract Society, London (same address), 1913.  [This is a major revision of: G. A. Hutchison, ed.; Indoor Games and Recreations; The Boy's Own Bookshelf; New ed., Religious Tract Society, London, 1891 (possibly earlier) -- see 5.A.]

M. Adams.  Puzzle Book.  1939.

                                    Morley Adams.  The Morley Adams Puzzle Book.  Faber & Faber, London, 1939.

M. Adams.  Puzzles That Everyone Can Do.  1931.

                                    Morley Adams.  Puzzles that Everyone Can Do.  Grant Richards, London, 1931, boarded.

AGM.           Abhandlungen zur Geschichte der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen.  Begründet von Moritz Cantor.  Teubner, Leipzig.  The first ten volumes were Supplements to Zeitschrift für Math. u. Physik, had a slightly different title and are often bound in with the journal volume.

Ahrens, Wilhelm Ernst Martin Georg (1872-1927).  See:  A&N,  MUS,  3.B,  7.N.

al‑Karkhi.                      Aboû Beqr Mohammed Ben Alhaçen Alkarkhî [= al‑Karagi  = al‑Karajī].  Untitled MS called Kitāb al-Fakhrī (or just Alfakhrî) (The Book Dedicated to Fakhr al-Din).  c1010.  MS 952, Supp. Arabe de la Bibliothèque Impériale, Paris.  Edited into French by Franz Woepcke as: Extrait du Fakhrî.  L'Imprimerie Impériale, Paris, 1853; reprinted by Georg Olms Verlag, Hildesheim, 1982.  My page citations will be to Woepcke.  Woepcke often refers to Diophantos, but his numbering gets ahead of Heath's.

Alberti.  1747.               Giuseppe Antonio (or Giusepp-Antonio) Alberti (1715-1768).  I Giochi Numerici Fatti Arcani Palesati da Giuseppe Antonio Alberti.  Bartolomeo Borghi, Bologna, 1747, 1749.  Venice, 1780, 1788(?).  4th ed., adornata di figure, Giuseppe Orlandelli for Francesco di Niccolo' Pezzana, Venice, 1795  (reprinted: Arnaud, Florence, 1979), 1813.  Adornata di 16 figure, Michele Morelli, Naples, 1814.  As: Li Giuochi Numerici Manifestati, Edizione adorna di Figure in rame, Giuseppe Molinari, Venice, 1815.

                    The editions have almost identical content, but different paginations.  I have compared several editions and seen little difference.  The 1747 ed. has a dedication which is dropped in the 2nd ed. which also omits the last paragraph of the Prefazione.  I only saw one other point where a few words were changed.  I will give pages of 1747 (followed by 1795 in parenthesis).  Much of Alberti, including almost all the material of interest to us and many of the plates, is translated from vol. 4 of the 1723 ed. of Ozanam. 

                    (Serge Plantureux's 1993 catalogue describes a 1747-1749 ed. with Appendice al Trattato de' Giochi Numerici (1749, 72 pp) & Osservazioni all'Appendice de' Giochi Numerici (38 pp), ??NYS.  The copy in the Honeyman Collection had the Appendice.  Christopher 3 has the Osservazioni.  The Appendice is described by Riccardi as a severe criticism of Alberti, attributed to Giovanni Antonio Andrea Castelvetri and published by Lelio dall Volpe, Bologna, 1749.  The Osservazioni are Alberti's response.)

 

Alcuin (c735-804).

                                    Propositiones Alcuini doctoris Caroli Magni Imperatoris ad acuendos juvenes.  9C. 

                    IN: B. Flacci Albini seu Alcuini, Abbatis et Caroli Magni Imperatoris Magistri.  Opera Omnia: Operum pars octava: Opera dubia.  Ed. D. Frobenius, Ratisbon, 1777, Tomus secundus, volumen secundum, pp. 440‑448.  ??NYS.  Revised and republished by J.‑P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 101, Paris, 1863, columns 1143‑1160.

                    A different version appears in: Venerabilis Bedae, Anglo‑Saxonis Presbyteri.  Opera Omnia: Pars Prima, Sectio II -- Dubia et Spuria: De Arithmeticus propositionibus.  Tomus 1, Basel, 1563.  (Rara, 131, says there were earlier editions: Paris, 1521 (part), 1544‑1545 (all), 1554, all ??NYS.)  Revised and republished by J.‑P. Migne as: Patrologiae Cursus Completus: Patrologiae Latinae, Tomus 90, Paris, 1904, columns 665‑672.  Incipiunt aliae propositiones ad acuendos juvenes is col. 667‑672.  A version of this occurs in Ens' Thaumaturgus Mathematicus of 1636 -- cf under Etten.

                    The Alcuin has 53 numbered problems with answers.  The Bede has 3 extra problems, but the problems are not numbered, there are only 31 1/2 answers and there are several transcription errors.  The editor has used the Bede to rectify the Alcuin.

                    There is a recent critical edition of the text by Folkerts -- Die älteste mathematische Aufgabensammlung in lateinischer Spräche: Die Alkuin zugeschriebenen Propositiones ad Acuendos Iuvenes; Denkschriften der Österreichischen Akademie der Wissenschaften, Mathematische‑naturwissenschaftliche Klasse 116:6 (1978) 13‑80.  (Also separately published by Springer, Vienna, 1978.  The critical part is somewhat revised as: Die Alkuin zugeschriebenen "Propositiones ad Acuendos Iuvenes"; IN: Science in Western and Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; Birkhäuser, Basel, 1993, pp. 273-281.)  He finds that the earliest text is late 9C and is quite close to the first edition cited above.  He uses the same numbers for the problems as above and numbers the extra Bede problems as 11a, 11b, 33a.  I use Folkerts for the numbering and the titles of problems.

                    John Hadley kindly translated Alcuin for me some years ago and made some amendments when Folkerts' edition appeared.  I annotated it and it appeared as: Problems to Sharpen the Young, MG 76 (No. 475) (Mar 1992) 102-126.  A slightly corrected and updated edition, containing some material omitted from the MG version, is available as Technical Report SBU-CISM-95-18, School of Computing, Information Systems, and Mathematics, South Bank University, Oct 1995, 28pp.

                    Menso Folkerts and Helmuth Gericke have produced a German edition: Die Alkuin zugeschriebenen Propositiones ad Acuendos Juvenes (Aufgabe zur Schärfung des Geistes der Jugend); IN: Science in Western and Eastern Civilization in Carolingian Times, ed. by P. L. Butzer & D. Lohrmann; Birkhäuser, Basel, 1993, pp. 283-362.

                    See also:  David Singmaster.  The history of some of Alcuin's Propositiones.  IN: Charlemagne and his Heritage  1200 Years of Civilization and Science in Europe: Vol. 2  Mathematical Arts; ed. by P. L. Butzer, H. Th. Jongen & W. Oberschelp; Brepols, Turnhout, 1998, pp. 11‑29.

 

AM.  1917.                   H. E. Dudeney.  Amusements in Mathematics.  Nelson, 1917.  (There were reprintings in 1919, 1920, 1924, 1925, 1927, 1928, 1930, 1932, 1935, 1938, 1939, 1941, 1943, 1946, 1947, 1949, 1951, but it seems that the date wasn't given before 1941?)  = Dover, 1958.

AMM.                          American Mathematical Monthly.

AMS.                           American Mathematical Society.

Les Amusemens.  1749.

                                    Les Amusemens Mathématiques Precedés Des Elémens d'Arithmétique, d'Algébre & de Géométrie nécessaires pour l'intelligence des Problêmes.  André‑Joseph Panckoucke, Lille, 1749.  Often listed with Panckoucke as author (e.g. by the NUC, the BNC and Poggendorff), but the book gives no such indication.  Sometimes spelled Amusements.  There were 1769 and 1799 editions.

Apianus.  Kauffmanss Rechnung.  1527. 

                                    Petrus Apianus (= Peter Apian or Bienewitz or Bennewitz) (1495‑1552).  Eyn Newe Unnd wolgegründte underweysung aller Kauffmanss Rechnung in dreyen Büchern / mit schönen Regeln uň [NOTE:  ň  denotes an  n  with an overbar.] fragstucken begriffen.  Sunderlich was fortl unnd behendigkait in der Welschē Practica uň Tolletn gebraucht wirdt / des gleychen fürmalss wider in Teützscher noch in Welscher sprach nie gedrückt.  durch Petrum Apianū von Leyssnick / d Astronomei zů Ingolstat Ordinariū / verfertiget.  Georgius Apianus, Ingolstadt, (1527), facsimile, with the TP of the 1544 ed. and 2pp of publication details added at the end, Polygon-Verlag, Buxheim-Eichstätt, 1995, with 8pp commentary leaflet by Wolfgang Kaunzner.  (The TP of this has the first known printed version of Pascal's Triangle.)  Smith, Rara, pp. 155-157.  (The  d  is an odd symbol, a bit like a  δ  or an  8,  which is used regularly for  der  both as a single word and as the ending of a word, e.g.  and  for  ander.)  Smith notes that Apianus follows Rudolff (1526) very closely.

AR.  c1450.                  Frater Friedrich Gerhart (attrib.).  Latin & German MSS, c1450, known as Algorismus Ratisbonensis.  Transcribed and edited from 6 MSS by Kurt Vogel as: Die Practica des Algorismus Ratisbonensis; C. H. Beck'sche Verlagsbuchhandlung, Munich, 1954.  (Kindly sent by Prof. Vogel.)  Vogel classifies the problems and gives general comments on the mathematics on pp. 155‑189.  He gives detailed historical notes on pp. 203‑232.  When appropriate, I will cite these pages before the specific problems.  He says (on p. 206) that almost all of Munich 14684 (see below) is included in AR.

Arnold, George.  See:  Book of 500 Puzzles,  Boy's Own Conjuring Book,  Hanky Panky.

Arrighi, Gino.  See:  Benedetto da Firenze,  Calandri,  Pseudo-Dell'Abbaco,  della Francesca,  Gherardi,  Lucca 1754,  P. M. Calandri.

Aryabhata.                    Āryabhata (I)) [NOTE:  ţ  denotes a  t  with a dot under it and  ş  denotes an  s  with a dot under it.]  (476-  ). Āryabhatīya.  499.  Critically edited and translated into English by Kripa Shankar Shukla, with K. V. Sarma.  Indian National Science Academy, New Delhi, 1976.  (Volume 1 of a three volume series -- the other two volumes are commentaries, of which Vol. 2 includes the commentary Āryabhatīya-Bhāşya, written by Bhaskara I in 629.  Aryabhata rarely gives numerical examples, so Bhaskara I provided them and these were later used by other Indian writers such as Chaturveda, 860.  The other commentaries are later and of less interest to us.  Prof. Shukla has sent a photocopy of an introductory booklet, which is an abbreviated version of the introductory material of Vol. 1, with some extensions relating Aryabhata to other writers.)  The material is organized into verses.  There is an older translation by Walter Eugene Clark as:  The  Âryabhaţîya  of  Âryabhaţa; Univ. of Chicago Press, Chicago, 1930.  (There was an Aryabhata II, c950, but he only occurs in 7.K.1.)

A&N.                           Wilhelm Ahrens.  Altes und Neues aus der Unterhaltungsmathematik.  Springer, Berlin, 1918.

 

Bachet, Claude‑Gaspar (1581-1638).  See:  Problemes.

Bachet-Labosne.  See:  Problemes.

Badcock.  Philosophical Recreations, or, Winter Amusements.  [1820].

                                    Philosophical Recreations, or, Winter Amusements.  Thomas Hughes, London, nd [1820].  [BCB 18-19; OCB, pp. 180 & 197.  Heyl 22-23.  Toole Stott 75-77.  Christopher 54-56.  Wallis 34 BAD, 35 BAD.  These give dates of 1820, 1822, 1828.]  HPL [Badcock] RBC has three versions with slightly different imprints on the title pages, possibly the three dates mentioned.

                    Wallis 34 BAD has this bound after the copy of:  John Badcock; Domestic Amusements, or Philosophical Recreations ...; T. Hughes, London, nd [1823], and it is lacking its Frontispiece and TP -- cf in 6.BH.  HPL [Badcock] has both books, including the folding Frontispieces.  The earlier does not give an author, but its Preface is signed J. B. and the later book does give his name and calls itself a sequel to the earlier.  Toole Stott 75-80 clearly describes both works.  Some of the material is used in Endless Amusement II. 

Baker.  Well Spring of Sciences.  1562?

                                    Humfrey Baker (fl. 1557-1587).  The Well Sprynge of Sciences Which teacheth the perfect worke and practise of Arithmeticke both in whole numbers and fractions, with such easye and compendious instruction into the sayde arte, ....  Rouland Hall for James Rowbotham, London, 1562.  [Smith, Rara, p. 327, says it was written in 1562 but wasn't actually printed until 1568, but a dealer says the 1st ed. was 1564 and there was a 4th ed. in 1574, which I have examined.]  Apparently much revised and extended, (1580).  Reprinted, with title: The Wel [sic] Spring of Sciences: Which teacheth the perfect worke and practise of Arithmetike; Thomas Purfoote, London, 1591.  I have seen Thomas Purfoot, London, 1612, which is essentially identical to 1591.  I have also seen:  Christopher Meredith, London, 1646;  Christopher Meredith, London, 1650;  R. & W. L. for Andrew Kemb, London, 1655;  which are all the same, but differently paged than the 1591.  I have also seen Baker's Arithmetick, ed. by Henry Phillippes, Edward Thomas, London, 1670, which has different pagination and some additional problems compared to the 1646/1655 ed.  [Smith, Rara, 327-330 & 537, says it was rewritten in 1580, but there is little difference between the 1580 and the many later editions, so the 1591 ed. is probably close to the 1580 ed.  The copy of the 1562 in the Graves collection ends on f. 160r, but an owner has written a query as to whether the book is complete.  Neither Smith nor De Morgan seems to have seen a 1562 so they don't give a number of pages for it.  (STC records no copies of the 1562, 1564, 1576, 1584, 1607 editions, but there was a 1576 by [T. Purfoote], apparently the 5th ed., of c500pp, in the Honeyman Collection.)  Almost all the problems of interest occur on ff. 189r-198r of the 1591 ed. and hence are not in the Graves copy of the 1562 ed., but H&S 61 refers to one of these problems as being in Baker, 1568.  The 1574 ends at fol. 200 (misprinted as 19?, where the ? is an undecipherable blob) and Chapter 16, which is headed:  The 16 Chapter treateth of sportes and pastime, done by number,  is on ff. 189r-200v, and contains just a few recreations, as in Recorde.  So I will date the book as 1562?, but most of the later material as 1580?.  The problems of 7.AF.1 and 10.A may be in Graves copy of the 1562 ed. -- ??check.  I will cite the 1580?, 1646 and 1670 editions, e.g.  1580?: ff. 192r 193r; 1646: pp. 302-304; 1670: pp. 344-345.]  Bill Kalush has recently sent a CD with 1574, 1580, 1591, 1598, 1602, 1607, 1612, 1617, 1650, 1655 on it -- ??NYR.

Bakhshali MS.               The Bakhshālī Manuscript, c7C.  This MS was found in May 1881 near the village of Bakhshālī, in the Yusufzāī district of the Peshawer division, then at the northwestern frontier of India, but apparently now in Pakistan.  This is discussed in several places, such as the following, but a complete translation has only recently appeared.  David Pingree says it is 10C, but his student Hayashi opts for 7C which seems pretty reasonable and I will adopt c7C.

                    1.             A. F. Rudolf Hoernle.  Extract of his report in some journal of the previous year.  The Indian Antiquary 12 (Mar 1883) 89-90.  A preliminary report, saying it was found near Bakhshâlî in the Yusufzai District in the Panjâb.

                    2.             A. F. Rudolf Hoernle.  On the Bakhshālī Manuscript.  Berichte des VII. Internationalen Orientalisten‑Congresses, Wien, 1886.  Alfred Hölder, Vienna, 1889.  Arische Section, p. 127-147 plus three folding plates.  Cf next item.  I will cite this as Hoernle, 1886.

                    3.             A. F. Rudolf Hoernle.  The Bakhshali manuscript.  The Indian Antiquary 17 (Feb 1888) 33‑48 & Plate I opp. p. 46;  275‑279 & Plates II & III opp. pp. 276 & 277.  This is essentially a reprint of the previous item, with a few changes or corrections, but considerable additional material.  He dates it c4C.  I will cite this as Hoernle, 1888.

                    4.             G. R. Kaye.  The Bakhshālī Manuscript – A Study in Medieval Mathematics.  Archæeological Survey of India – New Imperial Series XLIII: I-III, with parts I & II as one volume, (1927‑1933).  (Facsimile reprint in two volumes, Cosmo Publications, New Delhi, 1981 – this is a rather poor facsimile, but all the text is preserved.  I have a letter detailing the changes between the original and this 'facsimile'.)  I will only cite Part I – Introduction, which includes a discussion of the text.  Part II is a discussion of the script, transliteration of the text and pictures of the entire MS.  Part III apparently was intended to deal with the language used, but Kaye died before completing this and the published Part III consists of only a rearranged version of the MS with footnotes explaining the mathematics.  Gupta, below, cites part III, as Kaye III and I will reproduce these citations.  He dates it c12C.

                    5.             B. Datta.  The Bakhshâlî mathematics.  Bull. Calcutta Math. Soc. 21 (1929) 1‑60.  This is largely devoted to dating of the MS and of its contents.  He asserts that the MS is a copy of a commentary on some lost work of 4C or 5C (?).

                    6.             R. C. Gupta.  Some equalization problems from the Bakhshālī manuscript.  Indian Journal of the History of Science 21 (1986) 51-61.  Notes that Hoernle gave the MS to the Bodleian Library in 1902, where it remains, with shelf mark MS. Sansk. d.14.  He follows Datta in believing that this is a commentary on a early work, though the MS is 9C, as stated by Hoernle.  He gives many problems from Kaye III, sometimes restoring them, and he discusses them in more detail than the previous works.

                    7.             Takao Hayashi.  The Bakhshālī Manuscript  An ancient Indian mathematical treatise.  Egbert Forsten, Groningen, Netherlands, 1995.  (Based on his PhD Dissertation in History of Mathematics, Brown University, May 1985, 774pp.)  A complete edition and translation with extensive discussion of the context of the problems.  He dates it as 7C. 

Ball, Walter William Rouse (1850-1925).  See:  Ball‑FitzPatrick;  MRE.

Ball‑FitzPatrick.

                                    French translation of MRE by J. Fitz‑Patrick, published by Hermann, Paris.

                    1st ed., Récréations et Problèmes Mathématiques des Temps Anciens & Modernes.  From the 3rd ed, 1896, of MRE, 'Revue et augmentée par l'auteur'.  1898.  The Note says 'M. Ball ... a bien voulu apporter à la troisième édition anglaise des additions et des modifications importantes.'  352pp.

                    2nd ed., Récréations et Problèmes Mathématiques des Temps Anciens et Modernes.  From the 4th ed, 1905, of MRE, 'et enrichie de nombreuses additions'.

              As three volumes, 1907‑09.  [I have vol. 1, 1907, which is 356pp.  Pp. 327‑355 is a note by A. Hermann, Comptabilité d'une persone qui dépense plus que son revenu.  I have not yet seen the other volumes to compare with the 1926 reprint, but Strens's notes in his copy indicate that they are identical.]

              Reprinted in one vol., Gabay, Paris, 1992, 544pp.

              Reprinted, 1926-1927.  The only copies that I have seen are bound as one volume, but with separate pagination.  My copy has left out the title pages of vols. 2 & 3.  The copy in the Strens Collection has these title pages, but its vol. II is 1908.  The 1926 vol. 1 says Nouvelle édition française, but the 1927 vol. 3 says Deuxième édition française. 

                    [Vol. 1 is 326pp, omitting the note by Hermann.  Vol. 2 is 363pp (pp. 322‑355 is a historical note on the cubic, based on Cossali (1797)).  Vol. 3 is 363pp including: Notes diverses de M. Aubry, pp. 137‑206 (or 340? -- the Table des Matières and the page set up do not make it clear if Aubry's Notes end on p. 206);  Note de M. Fitz‑Patrick, La géométrie par le pliage et découpage du papier, pp. 341‑360;  A. Margossian, De l'ordonnance des nombres dans les carrés magiques impairs, pp. 1‑60 (pp. 61-64 is a Note on the same subject, presumably part of Margossian's material);  Capt. Reinhart, some geometric notes, pp. 130-136.]

Barnard.  50 Observer Brain-Twisters.  1962.

                                    Douglas St. Paul Barnard.  Fifty Observer Brain‑Twisters  A Book of Mathematical and Reasoning Problems.  Faber, 1962.  US ed.:  A Book of Mathematical and Reasoning Problems: Fifty Brain Twisters;  Van Nostrand, 1962.  The editions have identical pagination.

Bartl.  c1920.                János Bartl.  Preis-Verzeichnis von Bartl's Akademie für moderne magische Kunst.  Hamburg, c1920.  Reprinted by Olms Verlag, Zürich, 1983, as: Zauberkatalog Bartl.  References are to the section: Vexier- und Geduldspiele, pp. 305‑312.

Bartoli.  Memoriale.  c1420.

                                    Francesco Bartoli (  -1425).  Memoriale (= Notebook) containing some 30 mathematical problems copied during 1400?-1425.  Ms 1 F 54 of the Archives départementales du Vaucluse, France.  ??NYS -- described and quoted in:  Jacques Sesiano; Les problèmes mathématiques du Memoriale de F. Bartoli; Physis 26:1 (1984) 129-150.

BC.                              Binomial Coefficient, i.e.  BC(n, k) = n!/k!(n-k)!.

BCB.  See: Hall, BCB.

BDM.  See under DSB.

Bede, The Venerable (c672-735).  (Now St. Bede.)  See:  Alcuin.

Benedetto da Firenze.  c1465.

                                    Benedetto da Firenze.  Trattato d'Abacho.  c1465.  This was a popular treatise and Van Egmond's Catalog 356 lists 18 copies under Benedetto.  Six show  B  as author, one has Benedetto, one has Benedetto da Firenze, one has  Po Ma  and one has Filipo Chalandri, so it seems Benedetto is the most likely author.  The MSS date from c1465 to c1525 and contain 9 to 25 chapters. 

                    The version in Cod. Acq. e doni 154, Biblioteca Medicea Laurenziana, Florence, c1480.  has been transcribed and edited by Gino Arrighi as:  Pier Maria Calandri; Tractato d'Abbacho; Domus Galilaeana, Pisa, 1974.  The incipit names  Po Ma  as author.  Cf Van Egmond's Catalog 96.  This version has 23 chapters.

Benson.  1904.              J. K. Benson.  The Book of Indoor Games for Young People of All Ages.  C. Arthur Pearson, London, 1904.  [This copies a lot from Hoffmann (or a common ancestor?).] 

                    Much of the material of Indoor Games is repeated in: J. K. Benson, ed.; The Pearson Puzzle Book; C. Arthur Pearson, London, nd [1921 -- BMC].  This is not in BMC or NUC under Benson -- but I have seen an ad listing this as by Mr. X and it is listed under Mr. X in BMC.  Puzzle Book pp. 1-96  =  Indoor Games pp. 189-257;  Puzzle Book pp. 109-114  =  Indoor Games pp. 258-262.  The only different material in Puzzle Book is pp. 97-108.  Neither book refers to the other.  Cf Mr. X in Section 4.A.1

Berkeley & Rowland.  Card Tricks & Puzzles.  1892.

                                    "Berkeley" [Peel, Walter H.]  &  Rowland, T. B.  Card Tricks and Puzzles.  The Club  Series, George Bell & Sons, London, 1892 -- according to BMC, but my copy is 1897.  Card Puzzles, etc., pp. 1-74 is by Berkeley; Arithmetical Puzzles, pp. 75-120 is by Rowland.

Berlekamp, Elwyn R. (1940-  )  See:  Winning Ways.

Bestelmeier.  1801-1803.

                                    G. H. [Georg Hieronimus] Bestelmeier.  Magazin von verschiedenen Kunst‑ und andern nützlichen Sachen ....  [Toy catalogues.]  Nuremberg, 1801‑1803.

                    Eight issues and cumulative classified index reprinted by Olms, Zurich, 1979.  Issue VII is 1801; the others are 'neue verbesserte Auflage', 1803.  This includes items numbered 1 through 1111.

                    Selections, with English translations.  Daniel S. Jacoby, ed.  The Amazing Catalogue of the Esteemed Firm of George Hieronimus Bestelmeier.  Selected Excerpts from Editions of 1793 and 1807.  [A comment inside makes me wonder if 1793-1807 is meant??]  Merrimack Publishing Corp., NY, 1971, 82pp.  The numeration is the same as in the Olms edition, but the Jacoby continues to item 1321.  Obviously these later items come from the 1807 edition, but we cannot tell if they might date from 1805, say, nor whether all the earlier items go back to 1793.  Jerry Slocum uses Jacoby in his Compendium and has kindly provided photocopies of Jacoby's pp. 70-82 containing all the items after 1111 and some examples of the earlier items.  Jacoby does not translate the texts, but just provides English labels for each picture and these labels are sometimes unconnected with the text. 

                    Many of Bestelmeier's items are taken from Catel; Kunst-Cabinet; 1790.  Sometimes the figure is identical (often reversed) or is a poor copy.  Texts are often copied verbatim, or slightly modified, but often abbreviated.  E.g. Catel often explains the puzzle and this part is frequently omitted in Bestelmeier.  Bestelmeier was the successor to Catel, qv.  The booklet by Slocum & Gebhardt (qv under Catel) gives precise datings for the various parts of these catalogues, but I have not yet entered these details.

Bhaskara I.  629.

                                    Bhāskara I.  Āryabhaţīya-Bhāşya.  [NOTE:  ţ  denotes a  t  with a dot under it and  ş  denotes an  s  with a dot under it.]   629.  Critically edited, including an English Appendix of the numerical examples used, by Kripa Shankar Shukla.  Indian National Science Academy, New Delhi, 1976.  (Vol. 2 of a three volume series devoted to the Āryabhaţīya (499) of Aryabhata (476-  ), qv.)  Bhaskara I repeats and exposits Aryabhata verse by verse, but Aryabhata rarely gives numerical examples, so Bhaskara I provided them and these were later used by other Indian writers such as Chaturveda, 860.  His earlier Maha-Bhaskariya (Mahā‑Bhāskarīya) of c629 is cited in 7.P.2.  Shukla's Appendix is sometimes brief, but sometimes very detailed, e.g. on the 26 examples of Chinese remainder problems.

Bhaskara II (1114-c1185).

                                    Bhâskara II (1114-c1185, see Colebrooke).

Biggs, Norman L.  See:  BLW.

Bijaganita.                     Bîjaganita of Bhaskara II, 1150 (see Colebrooke).

The Bile Beans Puzzle Book.  1933. 

                                    Bile Beans (C. E. Fulford, Ltd., Leeds, England).  The Bile Beans Puzzle Book.  1933.

Birtwistle.  Math. Puzzles & Perplexities.  1971.

                                    Claude Birtwistle.  Mathematical Puzzles and Perplexities  How to Make the Most of Them.  George Allen & Unwin, London, 1971.

Birtwistle.  Calculator Puzzle Book.  1978.

                                    Claude Birtwistle.  The Calculator Puzzle Book.  Paperfronts (Elliot Right Way Books), Kingswood, Surrey, 1978.  (There is a US ed. by Bell, NY, 1978.)

BL(LD).                       British Library (Lending Division).

Blasius.  1513.               Johannis (or Joannes) Martinus Blasius (later denoted Sileceus or Sciliceus).  Liber Arithmetice Practice Astrologis Phisicis et Calculatioribus admodum utilis.  Thomas Kees for Joannis Parui & Joannis Lambert (in colophon; TP has Jehanlambert), Paris, 1513.  Facsimile by Heffer Scientific Reprint, Cambridge, 1960.  See Smith, Rara, pp. 95-97.  The Glaisher article in 7.P.5 [Messenger of Mathematics 53 (1923-24) 1‑131] discusses this book and says he only knows one example of it, which he has in front of him, so I suspect this facsimile is from that copy.  See Rara 95-97.  The Honeyman Collection had a copy, saying it was printed for J. Petit and J. Lambert and that copy had Petit's device on the TP while the TP shown in Rara has Lambert's device, which is as in this facsimile.  There was a reprinting in 1514 and extended editions in 1519 (ed. by Oronce Finé) and 1526 (ed. by T. Rhaetus) [Honeyman Collection, nos. 350-352].

BLC.                            British Library Catalogue, replacing BMC, in progress since 1970s. 

BLC-Ø                         Indicates that I could not find the item in the BLC.

BLW.  1976. Norman L. Biggs,  E. Keith Lloyd  &  Robin J. Wilson.  Graph Theory 1736‑1936.  OUP, 1976.

Blyth.  Match-Stick Magic.  1921.

                                    Will Blyth.  Match-Stick Magic.  C. Arthur Pearson, London, 1921, reprinted 1923, 1939.

BM(C).                         British Museum (Catalogue (of books) to 1955.  c1963). 

BMC65.                       Supplement to the above Catalogue for 1956‑1965.  c1968.

BN(C).                         Bibliothèque National, Paris.  (Catalogue, 1897-1981.)

Bodleian.                      The Bodleian Library, University of Oxford, or its catalogue.

Bonnycastle.  Algebra.  1782

                                    John Bonnycastle (??-1821).  An Introduction to Algebra, with Notes and Observations; designed for the Use of Schools, and Other Places of Public Education.  1782.  The first nine editions appeared "without any material alterations".  In 1815, he produced a 10th ed., "an entire revision of the work" which "may be considered as a concise abridgment" of his two volume Treatise on Algebra, 1813, (2nd ed. in 1820).  The 1815 ed. had an Appendix: On the application of Algebra to Geometry.  I have a copy of the 7th ed., 1805, printed for J. Johnson, London, and it is identical to the 2nd ed. of 1788 except for a problem in the final section of Miscellaneous Questions.  However, the 9th ed. of 1812 has page numbers advanced by 10 except toward the end of the book.  I also have the 13th ed. of 1824, printed for J. Nunn and 11 other publishers, London, 1824.  This version has an Addenda: A New Method of resolving Numerical Equations, by his son Charles Bonnycastle (1797-1840), but is otherwise identical to the 10th ed. of 1815.  The earlier text was expanded by about 10% in 1815, so a number of problems only occur in later editions.  I will cite these later problems as 1815 and will cite the earlier problems as 1782.  [Halwas 36-38 gives some US editions.]

 

Book of 500 Puzzles.  1859.

                                    The Book of 500 Curious Puzzles: Containing a Large Collection of Entertaining Paradoxes, Perplexing Deceptions in Numbers, and Amusing Tricks in Geometry.  By the author of "The Sociable," "The Secret Out," "The Magician's Own Book," "Parlor Games," and " Parlor Theatricals," etc.  Illustrated with a great Variety of Engravings.  Dick & Fitzgerald, NY, 1859.  Compiled from The Sociable (qv) and Magician's Own Book.  Pp. 1-2 are the TP and its reverse.  Pp. 3‑36, are identical to pp. 285-318 of The Sociable; pp. 37-54 are identical to pp. 199-216 of Magician's Own Book and pp. 55-116 are identical to pp. 241-302 of Magician's Own Book.  [Toole Stott 103 lists it as anonymous.  NUC, under Frikell, says to see title.  NUC, under Book, also has an 1882 ed, compiled by William B. Dick.  Christopher 129.  C&B lists it under Cremer.]

 

                    The authorship of this and the other books cited -- The Sociable, The Secret Out, The Magician's Own Book, Parlor Games, and Parlor Theatricals, etc. -- is confused.  BMC & NUC generally assign them to George Arnold (1834-1865) or Wiljalba (or Gustave) Frikell (1818 (or 1816) - 1903), sometimes with Frikell as UK editor of Arnold's US version -- but several UK versions say they are translated and edited by W. H. Cremer Jr, and one even cites an earlier French book (though the given title may not exist!, but cf Manuel des Sorciers, 1825) -- see the discussion under Status of The Project, in the Introduction, above.  The names of Frank Cahill, Henry Llewellyn Williams and Gustave Frikell (Jr.) are sometimes associated with versions of these as authors or coauthors.  The Preface of The Sociable says that most of the Parlor Theatricals are by Frank Cahill and George Arnold -- this may indicate they had little to do with the parts that interest us.  Toole Stott 640 opines that this reference led Harry Price to ascribe these books to these authors.

                    A publisher's ad in the back says: "The above five books are compiled from the "Sociable" and "Magician's Own."", referring to: The Parlor Magician [Toole Stott 543, 544]; Book of Riddles and Five Hundred Home Amusements [Toole Stott 107, 951]; Book of Fireside Games [possibly Toole Stott 300??]; Parlor Theatricals; The Book of 500 Curious Puzzles.  However, [Toole Stott 951] is another version of The Book of Riddles and Five Hundred Home Amusements "by the author of "Fireside Games" [Toole Stott 300], "The Parlor Magic" [perhaps Toole Stott 543, 544], "Parlor Tricks with Cards" [Toole Stott 1056 lists this as by Frikell, "abridged from The Secret Out" (see also 547, 1142)], ..."; Dick & Fitzgerald, 1986 [sic, but must mean 1886??]. 

                    See Magician's Own Book for more about the authorship.

                    See also:  Boy's Own Book,  Boy's Own Conjuring Book,  Illustrated Boy's Own Treasury,  Indoor and Outdoor,  Landells: Boy's Own Toy-Maker,  The Secret Out,  Hanky Panky,  The Sociable.

 

Book of Merry Riddles.  1629?

                                    The Book of Merry Riddles.  London, 1629.  [Santi 235.] 

          Several reprints.  Also known as Prettie Riddles.

                    A Booke of Merry Riddles; Robert Bird, London, 1631.  [Mark Bryant; Dictionary of Riddles; Routledge, 1990, p. 100.]

                    Booke of Merry Riddles; John Stafford & W. G., London, 1660.

                    Reprint of the 1629 in: J. O. Halliwell; The literature of the sixteenth and seventeenth centuries; London, 1851, pp. 67‑102.  [Santi 235.]

                    Reprint of the 1660 in: J. O. Halliwell; The Booke of Merry Riddles, together with proper questions, and witty proverbs, to make pleasant pastime.  Now first reprinted from the unique edition printed at London in the year 1660.  For the author, London, 1866.  This was a printing of 25 copies.  There is a copy at UCL and a MS note at the end says 15 copies were destroyed on 9 Apr 1866, signed: J. O. H., with Number 9 written below.  [Santi 307.]  I have seen this, but some of the riddles are quoted by other authors and I will date all items as 1629? until I examine other material.

                    Reprint of the 1629 in: Alois Brandl; Shakespeares Book of Merry Riddles und die anderen Räthselbücher seiner Zeit; Jahrbuch der deutschen Shakespeare-Gesellschaft 42 (1906) 1-64 (with the 1631 ed on pp. 53-63).  ??NYR.  [Santi 235 & 237.]

Borghi.  Arithmetica.  1484.

                                    Pietro Borghi  = Piero Borgo or Borgi (?? - ³1494).  Qui comenza la nobel opera de arithmethica ne la qual se tracta tute cosse amercantia pertinente facta & compilata p Piero borgi da veniesia.  Erhard Ratdolt, Venice, 1484.  2 + 116 numbered ff.  This is the second commercial arithmetic printed in Italy and was reprinted many times.  See Rara 16-22.  This edition was reproduced in facsimile, with notes by Kurt Elfering, as: Piero Borghi; Arithmetica  Venedig 1484; Graphos, Munich, 1964;  in: Veröffentlichungen des Forschungsinstituts des Deutschen Museums für die Geschichte der Naturwissenschaften und der Technik, Reihe C -- Quellentexte und Übersetzunge, Nr. 2, 1965.

                      The 3rd ed of 1491 had a title: Libro dabacho.  From the 4th ed of 1501, the title was Libro de Abacho, so this is sometimes used as the title for the first editions also.  Rara indicates that the printing was revised to 100 numbered ff by the 4th ed. of 1491.  I have examined a 1509 ed. by Jacomo Pentio, Venice, ??NX.  This has 100 numbered ff, but the last three ff contain additional material, though Rara doesn't mention this until the 11th ed of 1540.  H&S discusses a problem and the folio in the 1540 ed is the same as in the 1509 ed.  The locations of interest in the 1509 ed. are c18ff before the corresponding locations of the 1484.  Van Egmond's Catalog 293-297 lists 13 Venetian editions from 1484 to 1567.

                    It has been conjectured that this was a pseudonym of Luca Pacioli, but there is no evidence for this [R. Emmett Taylor; No Royal Road  Luca Pacioli and His Times; Univ. of North Carolina Press, Chapel Hill, 1942, pp. 60 & 349].

                    See also:  D. E. Smith; The first great commercial arithmetic; Isis 8 (1926) 41-49.

Bourdon.  Algèbre.  7th ed., 1834.

                                    Louis Pierre Marie Bourdon (1779-1854).  Élémens d'Algèbre.  7th ed., Bachelier, Paris, 1834.  (1st ed, 1817; 5th, 1828; 6th, 1831; 8th, 1837; 1840.  Undated preface in the 7th ed. describes many changes, so I will cite this as 1834, though much of the material would have occurred earlier.)

 

Boy's Own Book.  1828.

                                    William Clarke, ed.  The Boy's Own Book.  The bibliography of this book is extremely complex -- by 1880, it was described as having gone through scores of editions.  My  The Bibliography of Some Recreational Mathematics Books  has 11 pages listing 76 English (40 UK, 37 US, 1 Paris) versions and a Danish version, implying 88 English (50 UK, 37 US, 1 Paris) versions, and 10 (or 11) related versions, and giving a detailed comparison of the versions that I have seen.  Because of the multiplicity of versions, I have cited it by title rather than by the original editor's name, which is not in any of the books (except the modern facsimile) though this attribution seems to be generally accepted.  I have examined the following versions, sometimes in partial photocopies or imperfect copies. 

                    Vizetelly, Branston and Co., London, 1828, 448pp.;  2nd ed., 1828, 462pp.;  3rd ed., 1829, 464pp (has an inserted advertisement sheet);  6th ed??, c1830, 462pp?? (my copy lacks TP, pp. 417-418, 431-436, 461-462);  9th ed., 1834, 462pp.  Longman, Brown & Co., London, 24th ed., 1846, 462pp.  [The latter five are identical, except for a bit in the Prelude (and the extra sheet in 3rd ed), so I will just cite the first of these as 1828‑2.  It seems that all editions from the 2nd of 1828 through the 29th of 1848, 462pp. are actually identical except for a bit of the Prelude (and the advertisement sheet in the 3rd ed.)]

                    First American Edition.  Munroe & Francis, Boston  &  Charles S. Francis, NY, 1829, 316pp.  Facsimile by Applewood Books, Bedford, Massachusetts, nd [1998?].  This is essentially an abridgement of the 2nd ed of 1828, copying the Prelude and adding  "So far the London Preface.  The American publishers have omitted a few articles, entirely useless on this side of the Atlantic, ...."  The type is reset, giving some reduction in pages.  A number of the woodcuts have been omitted.  The section title pages are omitted.  Singing Birds, Silkworms, White Mice, Bantams, Magnetism, Aerostatics, Chess and Artificial Fireworks are omitted.  Angling, Rabbits, Pigeons, Optics are reduced.  Rosamond's Bower is omitted from Paradoxes and Puzzles.  Surprisingly, The Riddler is increased in size.  The 2pp Contents is omitted and an 8pp Index is added.

                    Baudry's European Library  &  Stassin & Xavier,  Paris, 1843, 448pp.  [The existence of a Paris edition was previously unknown to the vendor and myself, but it is Heyl 354 and he cites Library of Congress.  It is very different than the English and US editions, listing J. L. Williams as author.  Even when the topic is the same, the text, and often the topic's name, are completely rewritten.  See my  The Bibliography of Some Recreational Mathematics Books  for details -- in it I have found it generally necessary to treat this book separately from all other editions.  I will cite it as 1843 (Paris).  Much of this, including almost all of the material of interest is copied exactly in  Anon: Boy's Treasury, 1844,  qv, and in translated form in  de Savigny, Livre des Écoliers, 1846,  qv.  The problem of finding the number of permutations of the letters of the alphabet assumes 24 letters, which makes me wonder if these books are based on some earlier French work.  Heyl 355 is probably the same book, with slight variations in the title, by Dean and Munday, London, c1845.]

                    David Bogue, London, 1855, 611pp.  [It seems that this version first appears in 1849 and continues through about 1859, when two sections were appended.]

                    [W. Kent (late D. Bogue), London, 1859, 624pp??.  For almost all material of interest, this is identical to the 1855 ed, so I will rarely (if ever?) cite it.]

                    [Lockwood & Co., London, 1861, 624pp.  Identical to the 1859 ed., so I will not cite it.]

                    Lockwood & Co., London, 1868, 696pp.

                    [Lockwood & Co., London, 1870, 716pp.  Identical to 1868 with 20pp of Appendices, so page numbers for material of interest are the same as in 1868, so I will not cite it.]

                    [Crosby Lockwood & Co., London, 1880, 726pp.  Identical to 1870, but having the Appendices and 20 more pages incorporated into a new section.  For almost all material of interest, the page numbers are 30 ahead of the 1868 & 1870 page numbers, so I will not cite it except when the page numbers are not as expected.]

                    [5th (US?) ed., Worthington, NY, 1881, 362pp.  For almost all material of interest, this is identical to the 1829 (US) ed., so I will rarely (if ever?) cite it.]

          I will cite pages with edition dates and edition numbers or locations if needed (e.g.  1828-2: 410  or  1829 (US): 216).  See also:  Book of 500 Puzzles,  Boy's Own Conjuring Book,  Illustrated Boy's Own Treasury.

                    Anonymous.  The Riddler; A Collection of Puzzles, Charades, Rebusses, Conundrums, Enigmas, Anagrams, &c. for the Amusement of Little Folks.  S. Babcock, New Haven, Connecticut, 1835.  22pp.  My copy has leaf 11/12 half missing and leaf 17/18 missing; NUC & Toole Stott 1392 say it should be 24pp, so presumably leaf 23/24 is also missing here.  [Toole Stott 1392 has The Riddler: or, Fire-Side Recreations; a collection ..., 1838, also listed in NUC.]  Paradoxes and Puzzles section consists of the introduction and 11 puzzles copied almost exactly from the Paradoxes and Puzzles section of Boy's Own Book, 2nd ed. of 1828 and this material is all in the first American edition of 1829.  Other material is charades, etc. and is all in both these versions of Boy's Own Book.  Shortz states that this is the first American book with puzzles -- but there were at least five American versions of Boy's Own Book before this and all the material in The Riddler, except some woodcuts, is taken from Boy's Own Book, so this pamphlet seems to be a pirate version.  NUC also lists a 1838 version.

 

Boy's Own Conjuring Book.  1860.

                                    The Boy's Own Conjuring Book: Being a Complete Hand-book of Parlour Magic; and Containing over One Thousand Optical, Chemical, Mechanical, Magnetical, and Magical Experiments, Amusing Transmutations, Astonishing Sleights and Subtleties, Celebrated Card Deceptions, Ingenious Tricks with Numbers, Curious and Entertaining Puzzles, Charades, Enigmas, Rebuses, etc., etc., etc.  Illustrated with nearly two hundred engravings.  Intended as a source of amusement for one thousand and one evenings.  Dick and Fitzgerald, NY, 1860.  384pp.  [Toole Stott 115, corrected, lists this as (1859), and under 114, describes it as an extended edition of The Magician's Own Book -- indeed the running head of the book is The Magician's Own Book! -- but see below.  Toole Stott 481 cites a 1910 letter from Harris B. Dick, of the publishers Dick & Fitzgerald.  He describes The Boy's Own Conjuring Book as a reprint of Magician's Own Book "evidently gotten up and printed in London, but singularly enough it had printed in the book on the title-page -- New York, Dick & Fitzgerald."  Indeed, all the monetary terms are converted into British.  Harold Adrian Smith [Dick and Fitzgerald Publishers; Books at Brown 34 (1987) 108-114] states that this is a London pirate edition.  BMC has 384pp, c1860.  NUC has a 384pp version, nd.  Christopher 145-149 are five versions from 1859 and 1860, though none has the blue cover of my copy.  Christopher 145 says the 1859 versions were printed by Milner & Sowerby, Halifax, and describes it as an extraction from Magician's Own Book, but see below.  Christopher 148 cites Smith's article.]  I also have a slightly different version with identical contents except omitting the date and frontispiece, but with a quite different binding, probably Christopher 149.  [NUC lists 334pp, nd; 416pp, nd and 416pp, 1860.  Toole Stott 114 is a 416pp version, 1861.  Toole Stott 959 is a 534pp version, 1861.  C&B cite a New York, 1859 with 416pp, a New York, nd, 334pp and London, c1850 (surely too early?).]

                    I have now compared this with The Magician's Own Book of 1857 and it is essentially a minor reworking of that book.  The Magician's Own Book has 17 chapters and an answers chapter and a miscellaneous chapter of items which are almost all also listed in the Contents under earlier sections.  All together, there are some 635 items.  The Boy's Own Conjuring Book copies about 455 of these items essentially directly, completely omitting the chapters on Electricity, Galvanism, Magnetism, Geometry, Art, Secret Writing and Strength, and almost completely omitting the chapter on Acoustics.  Of the 488 items in the other chapters, 453 are copied into the Boy's Own Conjuring Book, and this has in addition two of the acoustic problems, 125 new miscellaneous problems and 38pp of charades, riddles, etc.  (The later UK edition of Magician's Own Book is very different from the US edition.)  Many of the problems are identical to the Boy's Own Book or the Illustrated Boy's Own Treasury.   See also:  Book of 500 Puzzles,  Boy's Own Book,  Illustrated Boy's Own Treasury,  Landells: Boy's Own Toy‑Maker.

 

Boy's Treasury.  1844.

                                    Anonymous.  The Boy's Treasury of Sports, Pastimes, and Recreations.  With four hundred engravings.  By Samuel Williams.  [The phrasing on the TP could be read as saying Williams is the author, but the NUC entry shows he was clearly listed as the designer in later editions and his name appears on the Frontispiece.]  D. Bogue, London, 1844.  Despite the similarity of title, this is quite different from Illustrated Boy's Own Treasury and the similar books of c1860.  [Toole Stott 116.  Toole Stott 117 is another ed., 1847, 'considerably extended'.  Toole Stott gives US editions: 959; 960; 118; 199 & 961-965 are 1st, 1847; 2nd, 1847; 3rd, 1848; 6 versions of the 4th, 1850, 1848, 1849, 1852, 1854, 1848.  Hall, BCB 37 is a US ed. of 1850 = Toole Stott 119.  Christopher 151 is a US version of 1850?  NUC lists 9 versions, all included in Toole Stott.  Toole Stott cites some BM copies, but I haven't found this in the BMC.  A section of this, with some additional material, was reissued as Games of Skill and Conjuring: ..., in 1860, 1861, 1862, 1865, 1870 -- see Toole Stott 312-317.]

                    I have now found that much of this, including all the material of interest, is taken directly from the 1843 Paris edition of  Boy's Own Book,  qv, by J. L. Williams, including many of the illustrations - indeed they have the same Frontispiece, with S. Williams' name on it.

BR.  c1305.                  Greek MS, c1305, Codex Par. Suppl. Gr. 387, fol. 118v‑140v.  Transcribed, translated and annotated by Kurt Vogel as: Ein Byzantinisches Rechenbuch des frühen 14.Jahrhunderts; Wiener Byzantinistische Studien, Band VI; Hermann Böhlaus Nachf., Wien, 1968.  I will cite problem numbers and pages from this -- Vogel gives analysis of the methods on pp. 149‑153 and historical comments on pp. 154‑160, but I will not cite these.

Brahmagupta, c628.  See:  Brahma‑sphuta‑siddhanta;  Colebrooke.

Brahma‑sphuta‑siddhanta.

                                    Bráhma‑sphuta‑siddhânta of Brahmagupta, 628 (see Colebrooke).  He only states rules, which are sometimes obscure.  It appears from Colebrooke, p. v, and Datta (op. cit. under Bakhshali, p. 10), that almost all the illustrative examples and all the solutions are due to Chaturveda Prthudakasvâmî in 860.  Brahmagupta's rules are sometimes so general that one would not recognise their relevance to these examples and I have often not cited Brahmagupta.  E.g. cistern problems are given as examples to Brahmagupta's verse on how to add and subtract fractions.  (See also Datta & Singh, I, p. 248.)  Some of these comments are taken from Bhaskara I in 629.

Brush.          Hubert Phillips.  Brush Up Your Wits.  Dent, London, 1936.

BSHM.                         British Society for the History of Mathematics.  The produce a useful Newsletter.

Buteo.  Logistica.  1559.

                                    Johannes Buteo (= Jean Borrel, c1485-c1560 or c1492-1572).  Ioan. Buteonis Logistica, quæ & Arithmetica vulgò dicitur in libros quinque digesta: quorum index summatim habetur in tergo.  Gulielmus Rovillius, Lyons, 1559.  Most of the material is in books IV and V.  H&S cites some problems in the 1560 ed with the same pages as in the 1559 ed, so these editions are presumably identical.  See Rara 292-294.

 

c.                                 circa, e.g. c1300.  Also  c=  means "approximately equal", though  @  will be used in mathematical contexts.

C.                                 Century, e.g. 13C, -5C.

Calandri.  Arimethrica.  1491.

                                    Philippo Calandri.  Untitled.  Frontispiece is labelled "Pictagoras arithmetrice introductor".  Text begins: "Philippi Calandri ad nobilem et studiosus Julianum Laurentii Medicē de arimethrica opusculū."  Lorenzo de Morgiani & Giovanni Thedesco da Maganza, Florence, 1491.  Van Egmond's Catalog 298-299.  The Graves collection has two copies dated 1491, one with the folio number  c iiii  misprinted as  b iiii - cf Van Egmond for other differences in this unique variant.  There was a reprint by Bernardo Zucchetta, Florence, 1518 -- ??NYS but mentioned:  in a handwritten note in one of the Graves copies of the 1491 (giving Bernardo Zucchecta, 1517);  in Smith, Rara, p. 48 (giving Bernardo Zuchetta, 1518);  in Riccardi [I, col. 208-209] (giving Bernardo Zuchecta, 1515)  and in Van Egmond's Catalog 299.  "It is the first printed Italian arithmetic with illustrations accompanying problems, ...." (Smith, Rara, pp. 46‑49).  There are about 50 of these illustrations, which appear to be woodcuts, but they are quite small, about 25mm (1") square, and the same picture is sometimes repeated for a related but inappropriate problem.  Rara reproduces some of these, slightly reduced.  Riccardi [I, col. 208-209] says there may have been a 1490 ed. by Bernardo Zuchecta, but Van Egmond did not find any example.

Calandri.  Aritmetica.  c1485.

                                    Filippo Calandri.  Aritmetica.  c1485 [according to Van Egmond's Catalog 158-159].  Italian MS in Codex 2669, Biblioteca Riccardiana di Firenze.  Edited by Gino Arrighi, Edizioni Cassa di Risparmio di Firenze, Florence, 1969.  2 vol.: colour facsimile; transcription of the text.  Copies of the facsimile were exhausted about 1980 and repeated requests to the Cassa di Risparmio have not produced a reprint, though they usually send a copy of the text volume every time I write!  I have now (1996) acquired a example of the 2 vol. set and I find that copies of the text volume which are not part of a set have 8 colour plates inserted, but these are not in the copy in the set. 

                    I cite folios from the facsimile volume and pages from the text volume.  These are in direct correspondence with the original except for those pages with full page illustrations.  The original begins with a blank side with a Frontispiece verso, then 9 sheets (18 pp.) of full page tables, then two blank sheets.  The numbered folios then begin and go through 110.  Ff. 1r - 32r  are  pp. 3 - 65  of the text.  F. 32v is a full page calculation which is not in the text.  Then  ff. 33r - 110r  are  pp. 66 - 220  of the text.  F. 110v is a full page illustration omitted in the text.  The first 80 folio numbers are in elaborate Roman numerals centred at the head of the page.  (These are sometimes unusually written -- e.g. XXIIIIII.)  The later folios were not originally numbered and were later numbered in the top right corner using Hindu-Arabic numerals.

                    In Sep 1994, I examined the original MS, though it is on restricted access.  The original colours are rather more luminous than in the facsimile, but the facsimile is a first class job.  The history of this codex is obscure.  It is said to have belonged to Piero di Lorenzo dei Medici and it may be the book catalogued in the library of Francesco Pandolfini, c1515, as 'uno libretto ... di Filippo Calandri in arithmetica'.  The Riccardi family collected continuously from their rise in the mid 15C until the library was acquired by the city in 1813.  A number of items from the Pandolfini catalogue can be identified as being in the Riccardiana.  Van Egmond's dating may be early as some claim this was produced for Giuliano de' Medici, who was born in 1479. 

Calandri.  Raccolta.  c1495.

                                    Filippo Calandri.  Una Raccolta di Ragioni.  In: Cod. L.VI.45, Biblioteca Comunale di Siena.  Ed. by D. Santini.  Quaderni del Centro Studi della Matematica Medioevale, No. 4, Univ. di Siena, 1982.  Van Egmond's Catalog 193 identifies this as ff. 75r-111v of the codex, titles it Ragone Varie and gives a date of c1495.

Calandri.  See also:  Benedetto da Firenze,  P. M. Calandri.

Cardan.  Ars Magna.  1545.

                                    Jerome Cardan  = Girolamo Cardano  = Hieronymus Cardanus (1501‑1576).  Artis Magnae sive de Regulis Algebraicis  Liber Unus.  Joh. Petreium, Nuremberg, 1545, ??NYS  Included in Vol. IV of the Opera Omnia, Joannis Antonius Huguetan & Marcus Antonius Ravaud, Lyon, 1663, and often reprinted, e.g. in 1967.  NEVER CITED??

Cardan.  Practica Arithmetice.  1539.

                                    Jerome Cardan  = Girolamo Cardano  = Hieronymus Cardanus (1501‑1576).  Practica Arithmetice, & Mensurandi Singularis.  (Or: Practica Arithmeticae Generalis Omnium Copiosissima & Utilissima, in the 1663 ed.)  Bernardini Calusci, Milan, 1539.  Included in Vol. IV of the Opera Omnia, 1663, see above.  Some of the section numbers are omitted in the Opera Omnia and have to be intuited.  I will give the folios from the 1539 ed. followed by the pages of the 1663 ed., e.g. ff. T.iiii.v-T.v.r (p. 113).

Cardan.  De Rerum Varietate.  1557.

                                    Jerome Cardan  = Girolamo Cardano  = Hieronymus Cardanus (1501‑1576).  De Rerum Varietate.  Henricus Petrus, Basel, 1557; 2nd ed., 1557; 5th ed., 1581, ??NYS.  Included in Vol. III of the Opera Omnia, 1663, see above.

Cardan.  De Subtilitate.  1550.

                                    Jerome Cardan  = Girolamo Cardano  = Hieronymus Cardanus (1501‑1576).  De Subtilitate Libri XXI.  J. Petreium, Nuremberg, 1550; Basel, 1553; 6th ed., 1560; and five other 16C editions, part ??NYS.  Included in Vol. III of the Opera Omnia, 1663, see above.  French ed. by Richard Leblanc, Paris, 1556, 1584, titled: Les Livres d'Hieronymus Cardanus: De la Subtilité et subtiles Inventions, ensemble les causes occultes et les raisons d'icelles; 9th ed., 1611.  I have seen a note that the 1582 ed. by Henricus Petrus, Basel, was augmented by a riposte to attacks by Scaliger with further illustrations.

Carlile.  Collection.  1793.

                                    Richard Carlile.  A Collection of One Hundred and Twenty Useful and Entertaining Arithmetical, Mathematical, Algebraical, and Paradoxical Questions: With the Method of Working Each.  Printed by T. Brice for the author, Exeter, 1793.  Wallis 227 CAR, ??NX.  Includes a number of straightforward problems covered here, but I have only entered the more unusual examples.

Carroll-Collingwood.  1899.

                                    The Lewis Carroll Picture Book.  Stuart Dodgson Collingwood, ed.  T. Fisher Unwin, London, 1899.  = Diversions and Digressions of Lewis Carroll, Dover, 1961.  = The Unknown Lewis Carroll, Dover, 1961(?).  Reprint, in reduced format, Collins, c1910.  The pagination of the main text is the same in the original and in both Dover reprints, but is quite different than the Collins.  I will indicate the Collins pages separately.  The later Dover has 42 additional photographs.

Carroll-Gardner.  c1890?  or  1996

                                    Martin Gardner.  The Universe in a Handkerchief.  Lewis Carroll's Mathematical Recreations, Games, Puzzles and Word Plays.  Copernicus (Springer, NY), 1996.  As with Carroll-Wakeling, Carroll material will be dated as 1890?, but there is much material by Gardner which is dated 1996.

Carroll-Wakeling.  c1890?

                                    Lewis Carroll's Games and Puzzles.  Newly Compiled and Edited by Edward Wakeling.  Dover and the Lewis Carroll Birthplace Trust, 1992.  This is mostly assembled from various manuscript sheets of Carroll's containing problems which he intended to assemble into a puzzle book.  Wakeling has examined a great deal of such material, including a mass of Carroll's notes to Bartholomew Price (1818‑1898) who was Sedleian Professor of Natural Philosophy at Oxford in 1853-1898.  Price was at Pembroke College, becoming the Master, adjacent to Carroll's Christ Church.  He had tutored Carroll (1833‑1898) and they were close friends and in continual contact until their deaths, both in 1898.  However, few of the papers are dated and they are simply loose sheets with no indication of being in order, so there is no way to date the undated sheets and I have given a fairly arbitrary date of c1890? for these, though Carroll was more active before then rather than after.  Some items are taken from Carroll's youthful magazines or his correspondence and hence are more precisely dated.  The correspondence is more fully given in Carroll-Collingwood.

                    In response to an inquiry, Wakeling wrote on 28 May 2003 and said that some of the Carroll-Price notes were typewritten 'probably using Dodgson's Hammond typewriter, purchased in 1888.'  This gives a somewhat more precise dating than my c1890? and I will give:  1888 to 1898  for such items, unless there is other evidence.

Carroll-Wakeling II.  c1890?

                                    Rediscovered Lewis Carroll Puzzles.  Newly Compiled and Edited by Edward Wakeling.  Dover, 1995.  See the notes to Carroll-Wakeling, above.

Cassell's.  1881.

                                    Cassell's Book of In‑Door Amusements, Card Games, and Fireside Fun.  Cassell, Peter, Gilpin & Co., London, 1881; Cassell, London, 1973.  217pp [probably + 1p + 6pp Index] (pp. 1-8 are preliminary matter).  [There was a companion volume: Cassell's Book of Sports and Pastimes.  In 1887, the two were  combined, with the spine titled Cassell's Book of Outdoor Sports and Indoor Amusements.  The front cover says Out Door Sports, the back cover says Indoor Amusements, while the title page says Cassell's Book of Sports and Pastimes.  It contains all the main text of Book of In‑Door Amusements, ..., advanced by 744 pages.  From at least 1896, Card Games and Parlour Magic were completely revised and later there were a few other small changes.  The title varies slightly.  Manson (qv) is a 1911 revision and extension to 340pp of main text.]

Catel.  Kunst-Cabinet.  1790.

                                    Peter Friedrich Catel.  Mathematisches und physikalisches Kunst-Cabinet, dem Unterrichte und der Belustigung der Jugend gewidmet.  Nebst einer zweckmässigen Beschreibung der Stücke, und Anzeige der Preise, für welche sie beim Verfassser dieses Werks P. F. Catel in Berlin zu bekommen sind.  [I.e. this is a catalogue of items for sale by post!]  Lagarde und Friedrich, Berlin & Libau, 1790.  [MUS #113.]  P. iv says he started his business in 1780.

                    There is a smaller Vol. 2, with the same title, except 'beim Verfasser dieses Werkes P. F. Catel' is replaced by 'in der P. F. Catelschen Handlung', and the publisher is F. L. Lagarde, Berlin, 1793. 

                    My thanks to M. Folkerts for getting a copy of the example in the Deutsches Museum made for me.

                    All citations are to vol. 1 unless specified.

                    Many of Bestelmeier's items are taken from Catel.  Sometimes the figure is identical (often reversed) or is a poor copy.  Texts are often copied verbatim, or slightly modified, but usually abbreviated.  E.g. Catel often explains the puzzle and this part is frequently omitted in Bestelmeier.  Bestelmeier was the successor to Catel.  Dieter Gebhardt has searched for the various editions and associated price lists of the Catel and Bestelmeier catalogues in German libraries and he and Jerry Slocum have published the details in:  Jerry Slocum  &  Dieter Gebhardt.  Puzzles from Catel's Cabinet and Bestelmeier's Magazine  1785 to 1823.  English translations of excerpts from the German Catel-Katalog and Bestelmeier-Katalog.  Intro. by David Singmaster.  History of Puzzles Series.  The Slocum Puzzle Foundation, PO Box 1635, Beverly Hills, California, 90213, USA, 1997.  I have not yet made detailed entries from this which gives precise dates for the various parts of these catalogues.

CFF.                             Cubism for Fun.  This is the Newsletter of the Nederlandse Kubus Club (NKC) (Dutch Cubists Club) which has been in English since the mid 1980s. 

Chambers -- see: Fireside Amusements.

Charades, Enigmas, and Riddles.  1859.

                                    Charades, Enigmas, and Riddles.  Collected by A Cantab. [BLC gives no author.  "A Cantab." was a common pseudonym.  One such author of about the right time and nature was George Haslehurst.]  (Cambridge, 1859). 

                    3rd ed., J. Hall and Son, Cambridge, 1860, HB.  Half-title, 6 + 96pp. 

                    4th ed., Bell & Daldy, London, 1862.  8 preliminaries (i = half-title; FP facing iii = TP; v-viii = Introduction; errata slip; two facing plates illustrating a charade for Harrowgate [sic] Waters), 1-160, 32pp publisher's ads, dated Jan 1863; (my copy is lacking pp. 63-64).  The three plates are signed J.R.J.  This is a substantial expansion of the 3rd ed.

                    I also have photocopy of part of the 5th ed., Bell and Daldy, London, 1865, and this shows it was even larger than the 4th ed, but most of the problems of interest have the same or similar problem numbers in the three editions that I have seen.  I will cite them as in the following example.  1860: prob. 28, pp. 59 & 63; 1862: prob. 29, pp. 135 & 141; 1865: prob. 573, pp. 107 & 154.

Chaturveda. Chaturveda Pŗthudakasvâmî [NOTE:  ŗ  denotes an  r  with an underdot.].  Commentator on the Brahma‑sphuta‑siddhanta (qv), 860.  Some of these comments are taken from Bhaskara I in 629.  Shukla calls him Pŗthūdaka, but Colebrooke cites him as Ch.

Chessics.                       Chessics.  The Journal of Generalised Chess.  Produced by G. P. Jelliss, 5 Biddulph Street, Leicester, LE2 1BH.  No. 1 (Mar 1976) -- No. 29 & 30 (1987).  Succeeded by G&PJ.

Child.  Girl's Own Book.

                                    Mrs. L. Maria Child [= Mrs. Child  = Lydia Maria Francis, later Child].  The Girl's Own Book.  The bibliography of this book is confused.  According to the Opies [The Singing Game, p. 481], the first edition was Boston, 1831 and there was a London 4th ed. of 1832, based on the 2nd US ed.  However the earliest edition in the BMC is a 6th ed. of 1833.  I have examined and taken some notes from the 3rd ed., Thomas Tegg, London, 1832 -- unfortunately I didn't have time to go through the entire book so I may have missed some items of interest.  I have also examined the following.

                    Clark Austin & Co., NY, nd [back of original TP says it was copyrighted by Carter, Hendee, & Babcock in Massachusetts in 1833];  facsimile by Applewood Books, Bedford, Massachusetts, nd [new copy bought in 1998 indicates it is 4th ptg, so c1990].  The facsimile is from a copy at Old Sturbridge Village.  The back of the modern TP says the book was first published in 1834 and the Cataloguing-in-Publication data says it was originally published by Carter, Hendee and Babcock in 1834.  However, the earliest version in the NUC is Clark, Austin, 1833.  I am confused but it seems likely that Carter, Hendee and Babcock was the original publisher in Boston in 1831 and that that this facsimile is likely to be from 1833 or an 1834 reprint of the same.  The pagination is different than in the 1832 London edition I have seen.

                    The Tenth Edition, with Great Additions.  By Mrs. Child.  Embellished with 144 Wood Cuts.  Thomas Tegg, London (& three other copublishers), 1839.  12 + 307 pp + 1p publisher's ad.  Has Preface to the Second Edition but no other prefaces.  This Preface is identical to that in the 1833 NY ed, except that it omits the final P.S. of season's greetings.  The 1833 NY essentially has the same text, but they have different settings and different illustrations with some consequent rearrangement of sections.  However the main difference is that the NY ed omits 41pp of stories.  There are a number of minor differences which lead to the NY ed having 9 extra pages of material.

                    The Eleventh Edition, with Great Additions.  By Mrs. Child.  Embellished with 124 Wood Cuts.  Thomas Tegg, London (& three other copublishers), 1842.  12 + 363 pp + 1p publisher's ad.  The Preface is identical to that in the 10th ed, but omits 'to the Second Edition' after Preface.  90 pp of games and 40 pp of enigmas, charades, rebuses, etc. have been added; 56 pp of stories have been dropped.

                    The Girl's Own Book of Amusements, Studies and Employments.  New Edition.  Considerably enlarged and modernized by Mrs. L. Valentine, and others.  William Tegg, London, 1876.  This differs considerably from the previous editions.

                    I will cite the above by the dates  1832, 1833, 1839, 1842, 1876.

                    Various sources list:  13th ed., 1844 [BMC, Toole Stott 831];  Clark Austin, NY, 1845 [NUC];  16th ed., 1853 [BMC];  17th ed. by Madame de Chatelain, 1856 [BMC, NUC, Toole Stott 832];  18th ed. by Madame de Chatelain, 1858 [BMC, Toole Stott 833];  1858 [Osborne Collection (at Univ. of Toronto)];  rev. by Mrs. R. Valentine, 1861 [BMC, Osborne Collection];  rev. by Mrs. R. Valentine, 1862 [BMC, NUC];  rev. by Mrs. R. Valentine, 1864 [BMC];  rev. by Mrs. R. Valentine, 1867 [BMC];  enlarged by Mrs. L. Valentine, 1868 [NUC];  enlarged by Mrs. L. Valentine, 1869 [BMC];  enlarged by Mrs. L. Valentine, 1873 [NUC];  enlarged by Mrs. L. Valentine, 1875 [NUC];  enlarged by Mrs. L. Valentine, 1876 [BMC];

                    Heyl gives the following under the title  The Little Girl's Own Book:  Carter, Hendee and Co., Boston, 1834;  American Stationers Co, John B. Russell, Boston, 1837;  Edward Kearney, NY, 1847;  NY, 1849.

                     I think there were at least 33 editions.  See my  The Bibliography of Some Recreational Mathematics Books  for more details.  Cf Fireside Amusements, below, which is largely copied from Child.

Chiu Chang Suan Ching.  c-150?

                                    Chiu Chang Suan Ching (Nine Chapters on the Mathematical Art).  (Also called Chiu Chang Suan Shu and variously transliterated.  The pinyin is Jiŭ Zhāng Suàn Shù.)  c‑150?  German translation by K. Vogel; Neun Bücher arithmetischer Technik; Vieweg, Braunschweig, 1968.  My citations will be to chapter and problem, and to the pages in Vogel.  (Needham said, in 1958, that Wang Ling was translating this, but it doesn't seem to have happened.)  Some of the material dates from the early Han Dynasty or earlier, say c-200, but Chap. 4 & 9, the most original of all, have no indication of so early a date.  A text of c50 describes the contents of all the chapters and Høyrup suggests that Chap. 4 & 9 and the final assembly of the book should be dated to the [early] 1C.

Christopher.  1994.

                                    Maurine Brooks Christopher & George P. Hansen.  The Milbourne Christopher Library.  Magic, Mind Reading, Psychic Research, Spiritualism and the Occult  1589-1900.  Mike Coveney's Magic Words, Pasadena, 1994.  1118 entries.  References are to item numbers.

Christopher II.  1998.

                                    Maurine Brooks Christopher & George P. Hansen.  The Milbourne Christopher Library -- II.  Magic, Mind Reading, Psychic Research, Spiritualism and the Occult  1589-1900.  Mike Coveney's Magic Words, Pasadena, 1998.  3067 entries.  References are to item numbers.  Recently received, ??NYR.

Chuquet.  1484.             Nicolas Chuquet.  Problèmes numériques faisant suite et servant d'application au Triparty en la science des nombres de Nicolas Chuquet Parisien.  MS No. 1346 du Fonds Français de la Bibliothèque Nationale, 1484, ff. 148r-210r.  Published in an abbreviated version as: Aristide Marre; Appendice au Triparty en la science des nombres de Nicolas Chuquet Parisien; Bulletino di bibliografia e di storia delle scienze matematiche e fisiche 14 (1881) 413‑460.  (The first part of the MS was published by Marre; ibid. 13 (1880) 593-814; ??NYS)  Marre generally transcribes the text of the problem, but just gives the answer without any of the text of the solution.  I will cite problems by number.  There are 166 problems.  (Much of this was used in his student's book: Estienne de la Roche; Larismethique novellement composee par maistre Estienne de la roche dict Villefrāche; Lyons, 1520, ??NYS.  (Rara 128‑130).) 

                    FHM        Graham Flegg,  Cynthia Hay  &  Barbara Moss.  Nicolas Chuquet, Renaissance Mathematician.  A study with extensive translation of Chuquet's mathematical manuscript completed in 1484.  Reidel, Dordrecht, 1985.  This studies the entire MS, of which the above Appendice is only the second quarter.  It often gives a full English translation of the text of the problem and the solution, but it may summarize or skip when there are many similar problems.  The problems in the first part of the MS are not numbered in FHM.  I will cite this as FHM xxx, where xxx is the page number, with 'English in FHM xxx' when the problem is explicitly translated.

Clark.  Mental Nuts.  1897, 1904, 1916.

A book of Old Time Catch or Trick Problems  Regular old Puzzlers that kept your Grandad up at night.  Copyright, 1897, by S. E. Clark, Philadelphia.  Flood & Conklin Co.  Makers of Fine Varnishes, Newark, N.J.  100 problems and answers.  32pp + covers.

A book of 100 Catch or Trick Problems  Their simplicity invites attack, while their cunningly contrived relations call forth our best thought and reasoning.  Copyright, 1897, by S. E. Clark, Philadelphia.  Revised 1904 Edition.  Waltham Watches, Waltham, Massachusetts.  This was an promotional item and jewellers would have their address printed on the cover.  My example has:  With the compliments of  J. H. Allen  Jeweler [sic]  Shelbina, Mo.  Thanks to Jerry Slocum for this.   In fact there are only 95 problems; numbers 68, 75, 76, 78, 84 are skipped.  32pp + covers.

Revised Edition 1916, with no specific company mentioned.  Enlarged PHOTOCOPY from Robert L. Helmbold.  100 numbered problems, but some figures inserted after no. 75 are the solutions to a problem in the other editions and I have counted this as a problem (no. 75A), making 101 problems.  28pp + covers.

                     The editions are considerably different.   Only 40 problems occur in all three editions.  There are 50 problems common to 1897 and 1904, 42 common to 1897 and 1916 and 71 common to 1904 and 1916, though this counting is a bit confused by the fact that problems are sometimes combined or expanded or partly omitted, etc.  Solutions are brief.  It includes a number of early examples or distinct variants, which is remarkable for a promotional item.  I have entered 36 of the 1897 problems plus 13 of the 1904 problems not in 1897 and 7 of the 1916 problems not in 1897 or 1904.  Many others are standard examples of topics covered in this work, but are not sufficiently early to be worth entering. 

                     I originally had the 1904 ed and cited the 1904 problems as 1897 on the grounds that editions of this period do not change much, but having now seen the 1897 and 1916 eds, I realise that the editions are very different, so I will cite the actual dates.  Since only the 1897 version is paginated, I will just cite problem numbers; the solutions are at the back.

Clarke, William.  See:  Boy's Own Book.

CM.                             Crux Mathematicorum (originally titled Eureka until 4:3)

CMJ.                            The College Mathematics Journal.  Before the early 1980s, this was the Two Year College Mathematics Journal.

Colebrooke.  1817.

                                    Henry Thomas Colebrooke (1765-1837), trans.  Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bháscara.  John Murray, London, 1817.  Contains Lîlâvatî and Bîjaganita of Bhâskara II (1150) and Chapters XII (Arithmetic) and XIII (Algebra) of the Bráhma‑sphuta‑siddhânta of Brahmagupta (628).  There have been several reprints, including Sändig, Wiesbaden, 1973.  (Edward Strachey produced a version: Bija Ganita: or the Algebra of the Hindus; W. Glendinning, London, 1813; by translating a Persian translation of 1634/5.)

Collins.  Book of Puzzles.  1927.

                                    A. Frederick Collins.  The Book of Puzzles.  D. Appleton and Co., NY, 1927. 

Collins.  Fun with Figures.  1928.

                                    A. Frederick Collins.  Fun with Figures.  D. Appleton and Co., NY, 1928. 

Columbia Algorism.  c1350.

                                    Anonymous Italian MS, c1350 [according to Van Egmond's Catalog 253‑254], Columbia X511 .A1 3.  Transcribed and edited by K. Vogel; Ein italienisches Rechenbuch aus dem 14.Jahrhundert; Veröffentlichungen des Forschungsinstituts des Deutschen Museums für die Geschichte der Naturwissenschaften und der Technik, Reihe C, Quellentexte und Übersetzungen, Nr. 33, Munich, 1977.  My page references will be to this edition.  Van Egmond says it has a title in a later hand: Rascioni de Algorismo.

               The Algorism is discussed at length in Elizabeth B. Cowley; An Italian mathematical manuscript; Vassar Medieval Studies, New Haven, 1923, pp. 379‑405.

Conway, John Horton. (1937-  ).  See:  Winning Ways.

Cowley, Elizabeth B.  See:  Columbia Algorism.

CP.  1907.                    H. E. Dudeney.  Canterbury Puzzles.  (1907);  2nd ed. "with some fuller solutions and additional notes", Nelson, 1919;  4th ed.  = Dover, 1958.  (I have found no difference between the 2nd and 4th editions, except Dover has added an extra note on British coins and stamps.  I now have a 1st ed, which has different page numbers, but I have not yet added them.)

CR                               Comptes Rendus des Séances de l'Académie des Sciences, Paris.

Crambrook.  1843.        W. H. M. Crambrook.  Crambrook's Catalogue of Mathematical & Mechanical Puzzles Deceptions and Magical Curiosities, contained in the Necromantic Tent, Royal Adelaide Gallery, West Strand, London.  ...  To which is added, a Complete Exposé [of] the Baneful Arts by which unwary Youth too often become the prey of professed gamesters.  And ... an extract from The Anatomy of Gambling.  Second Edition, Corrected & Enlarged.  T. C. Savill, 107 St. Martin's Lane, 1843.  23pp.  Photocopy provided by Slocum.  [According to: Edwin A. Dawes; The Great Illusionists; Chartwell Books, Secaucus, New Jersey, 1979, p. 138, this is the first known magical catalogue.  It has a list of about 100 puzzles on pp. 3-5, with the rest devoted to magic tricks.  Unfortunately there are no pictures.  Comparison with Hoffmann helped identify some of the puzzles, but I can not identify many of them.  I have marked almost all these entries with ?? or check??, but the only way one can check is if actual examples or an illustrated catalogue turn up.  Some of the names are so distinctive that it seems certain that the item does fit where I have cited it; others are rather speculative.  There are several names which may turn up with more investigation.  Toole Stott 190 says there should be 48pp, though the later pages may be the added material on gambling.]

Cremer, William Henry, Jr.  See under:  Book of 500 Puzzles,  Hanky Panky,  Magician's Own Book.

CUP.                            Cambridge University Press.

Cyclopedia.  1914.

                                    Sam Loyd's Cyclopedia of 5,000 Puzzles, Tricks and Conundrums (ed. by Sam Loyd Jr).  Lamb Publishing, 1914  = Pinnacle or Corwin, 1976.  This is a reprint of Loyd's "Our Puzzle Magazine", a quarterly which started in June 1907 and ran till 1908.  See OPM for details.

C&B.  1920. Sidney W. Clarke  &  Adolphe Blind.  The Bibliography of Conjuring  And Kindred Deceptions.  George Johnson, London, 1920.  Facsimile by Martino Fine Books, Mansfield Centre, Connecticut, nd [obtained new in 1998].

C&W.          Chatto & Windus, London.

 

Datta & Singh.              Bibhutibhusan Datta & Avadhesh Narayan Singh.  History of Hindu Mathematics.  Combined edition of Parts I (1935) and II (1938), Asia Publishing House, Bombay, 1962.  NOTE: This book makes some contentious assertions.  Readers are referred to the following reviews.

                    O. Neugebauer.  Quellen und Studien zur Geschichte der Mathematik 3B (1936) 263-271.

                    S. Gandz.  Isis 25 (1936) 478-488.

Datta, B.  See:  Bakhshali MS;  Datta & Singh.

De Morgan (1806-1871).  See:  Rara.

De Viribus.  See:  Pacioli.

dell'Abbaco.  See:  Pseudo-dell'Abbaco.

Depew.  Cokesbury Game Book.

                                    Arthur M. Depew.  The Cokesbury Game Book.  Abingdon-Cokesbury Press, NY & Nashville, 1939.  [The back of the TP says it is copyright by Whitmore & Smith -- ??  The Acknowledgements say material has been assembled from various sources and colleagues who have been collecting and writing over the previous thirty years.]

Dickson.                       Leonard Eugene Dickson (1874-1954).  History of the Theory of Numbers, 3 vols.  Carnegie Institution of Washington, Publication 256, 1919-1923;  facsimile reprint by Chelsea, 1952.

Dilworth.  Schoolmaster's Assistant.

                                    Thomas Dilworth.  The Schoolmaster's Assistant:  Being a Compendium of Arithmetic, both Practical and Theoretical.  (1743);  11th ed., Henry Kent, London, 1762 (partly reproduced by Scott, Foresman, 1938.)  20th ed., Richard & Henry Causton, London, 1780.  De Morgan suggests the 1st ed. was 1744 or 1745, but the testimonials are dated as early as Jan 1743, so I will assume 1743.  Comparison of a 1762 ed. (Wallis 321 DIL) with my 1780 ed. shows the 1780 ed. is identical to the 1762 ed., except the section on exchange is much expanded, so the page numbers of all material of interest are increased by 12pp.  I will cite the pages of the 1762 ed., but give the date as 1743.  [Wallis also has:  14th ed., 1767;  15th ed., 1768;  1783;  22nd ed., 1785;  1791;  24th ed., 1792;  1793;  33rd ed., 179-;  1799;  1800;  1804.]  [Halwas 149‑162 are some US editions.]

Diophantos.  c250.

                                    Diophantos.  Arithmetica.  c250.  In: T. L. Heath; Diophantos of Alexandria; 2nd ed., (OUP, 1910); Dover, 1964.  Note: Bachet edited a Greek and Latin version of Diophantos in 1620, which inserted 45 problems from the Greek Anthology at the end of Book V.  (It was in Fermat's copy of this work that Fermat wrote the famous marginal note now called his Last Theorem; Fermat's son published an edition with his father's annotations in 1670, but the original copy was lost in a fire.)

DNB.                           Leslie Stephen, ed.  The Dictionary of National Biography.  Smith, Elder and Co., London, 1885‑1901 in 22 volumes.  OUP took it over in 1917.  Decennial Supplements were added. 

                    Compact Edition, with Supplement amalgamating the six decennial supplements to 1960, OUP, 1975.  The Compact Ed. shows the original volumes and pages so I will cite them in ( ), followed by the pages in the Compact Ed.

Dodson.  Math. Repository.  (1747?); 1775.

                                    James Dodson.  The Mathematical Repository.  Containing Analytical Solutions of near Five Hundred Questions, mostly selected from Scarce and Valuable Authors.  Designed As Examples to Mac-Laurin's and other Elementary Books of Algebra; And To conduct Beginners to the more difficult Properties of Numbers.  2nd ed., J. Nourse, London,