CONTENTS
The page numbers were last updated on 6 Apr 2000.
Nature of This Work 1
Similar Works 2
Coverage 3
Status of the Project 4
Technical 6
New Sections in This Edition 6
Acknowledgements 6
CONTENTS 10
Diacritical Marks and Notation 18
Abbreviations of Journals and Series 19
Abbreviations of Publishers 19
Abbreviations of Months 180
Publishers' Locations 19
SOME OTHER RECURRING REFERENCES 65
1. BIOGRAPHICAL MATERIAL in Chronological Order 68
Alcuin, Fibonacci, Bachet, Leurechon/van Etten, Ozanam, Montucla, Carroll, Hoffmann, Loyd & Loyd Jr, Lucas, Schubert, Ball, Dudeney, Ahrens, Phillips.
2. GENERAL PUZZLE COLLECTIONS AND SURVEYS 74
3. GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL 74
3.A. General Historical Material 74
3.B. Bibliographical Material 74
4. MATHEMATICAL GAMES 80
4.A. General Theory and Nim‑like Games 80
4.A.1. One Pile Game 80
4.A.1.a. The 31 Game 83
4.A.2. Symmetry Arguments 84
4.A.3. Kayles 84
4.A.4. Nim 85
4.A.5. General Theory 87
4.B. Particular Games 88
4.B.1. Tic‑Tac‑Toe = Noughts and Crosses 89
4.B.1.a. In Higher Dimensions 96
4.B.2. Hex 97
4.B.3. Dots and Boxes 98
4.B.4. Sprouts 99
4.B.5. Ovid's Game and Nine Men's Morris 99
4.B.6. Phutball 105
4.B.7. Bridg‑It 105
4.B.8. Chomp 105
4.B.9. Snakes and Ladders 106
4.B.10. Mu Torere 108
4.B.11. Mastermind, etc. 108
4.B.12. Rithmomachia = The Philosophers' Game 108
4.B.13. Mancala Games 109
4.B.14. Dominoes, etc. 110.
5. COMBINATORIAL RECREATIONS 111
5.A. The 15 Puzzle, etc. 111
General 111
Early Alphabetic Versions 111
Loyd 111
The 15 Puzzle 112
5.A.1. Non‑square Pieces 118
5.A.2. Three Dimensional Versions 118
5.A.3. Rolling Piece Puzzles 119
5.A.4. Panex Puzzle 120
5.B. Crossing Problems 120
5.B.1. Lowering from Tower Problem 127
5.C. False Coins with a Balance 128
5.C.1. Ranking Coins with a Balance 130
5.D. Measuring Problems 131
5.D.1. Jugs & Bottles 131
5.D.2. Ruler with Minimal Number of Marks 136
5.D.3. False Coins with a Weighing Scale 137
5.D.4. Timing with Hourglasses 137
5.D.5. Measure Half a Barrel 137
5.E. Euler Circuits and Mazes 137
5.E.1. Mazes 141
5.E.2. Memory Wheels = Chain Codes 144
5.E.2.a Pantactic Squares 145
5.F. Hamiltonian Circuits 146
5.F.1. Knight's Tours and Paths 146
5.F.2. Other Hamiltonian Circuits 152
5.F.3. Knight's Tours in Higher Dimensions 153
5.F.4. Other Circuits In and On a Cube 153
5.G. Connection Problems 154
5.G.1. Gas, Water and Electricity 154
5.H. Coloured Squares and Cubes, etc. 154
5.H.1. Instant Insanity = The Tantalizer 154
5.H.2. MacMahon Pieces 155
5.H.3. Path Forming Puzzles 156
5.H.4. Other and General 157
5.I. Latin Squares and Euler Squares 158
5.I.1. Eight Queens Problem 160
5.I.2. Colouring Chessboard with No Repeats in a Line 163
5.J. Squared Squares, etc. 163
5.J.1. Mrs Perkins's Quilt 165
5.J.2. Cubing the Cube 166
5.J.3. Tiling a Square of Side 70 with Squares of Sides 1, 2, ..., 24 166
5.K. Derangements 166
5.K.1 Deranged Boxes of A, B and A & B 167
5.K.2 Other Logic Puzzles Based on Derangements 167
5.K.3 Cayley's Mousetrap 167
5.L. Ménage Problem 167
5.M. Six People at a Party _ Ramsey Theory 168
5.N. Jeep or Explorer's Problem 169
5.O. Tait's Counter Puzzle: BBBBWWWW to WBWBWBWB 170
5.P. General Moving Piece Puzzles 172
5.P.1. Shunting Puzzles 172
5.P.2. Taquin 174
5.Q. Number of Regions Determined by N Lines or Planes 17
5.Q.1. Number of Intersections Determined by N Lines 175
5.R. Jumping Piece Games 175
5.R.1. Peg Solitaire 175
5.R.1.a. Triangular Version 178
5.R.1.b. Other shapes 179
5.R.2. Frogs and Toads: BBB_WWW to WWW_BBB 179
5.R.3. Fore and Aft _ 3 by 3 Squares Meeting at a Corner 181
5.R.4. Reversing Frogs and Toads: _12...n to _n...21 182
5.R.5. Fox and Geese, etc. 182
5.R.6. Octagram Puzzle 185
5.R.7. Passing Over Counters 185
5.S. Chain Cutting and Rejoining 188
5.S.1. Using Chain Links to Pay for a Room 189
5.T. Dividing a Cake Fairly 189
5.U. Pigeonhole Recreations 190
5.V. Think‑A‑Dot, etc. 191
5.W Making Three Pieces of Toast 191
5.W.1. Boiling Eggs 192
5.X Counting Figures in a Pattern 192
5.X.1. Counting Triangles 192
5.X.2. Counting Rectangles or Squares 194
5.X.3. Counting Hexagons 195
5.X.4. Counting Circles 195
5.Y. Number of Routes in a Lattice 194
5.Z. Chessboard Placing Problems 198
5.Z.1. Kings 198
5.Z.2. Queens 199
5.Z.3. Bishops 200
5.Z.4. Knights 200
5.Z.5. Rooks 201
5.Z.6. Mixtures 201
5.AA. Card Shuffling 201
5.AB. Folding a Strip of Stamps 1203
5.AC. Properties of the Seven Bar Digital Display 203
5.AD. Stacking a Deck to Produce a Special Effect 204
5.AE. Reversing Cups 204
6. GEOMETRIC RECREATIONS 205
6.A. Pi 205
6.B. Straight Line Linkages 206
6.C. Curves of Constant Width 207
6.D. Flexagons 208
6.E. Flexatube 209
6.F. Polyominoes, etc. 210
6.F.1. Other Chessboard Dissections 218
6.F.2. Covering Deleted Chessboard with Dominoes 220
6.F.3. Dissecting a Cross into Zs and Ls 220
6.F.4. Quadrisecting an L‑Tromino, etc. 221
6.F.5. Other Dissections into Polyominoes 223
6.G. Soma Cube 224
6.G.1. Other Cube Dissections 224
6.G.2. Dissection of 63 into 33, 43 and 53, etc. 225
6.G.3. Dissection of a Die into Nine 1 x 1 x 3 226
6.G.4. Use of Other Polyhedral Pieces 226
6.H. Pick's Theorem 226
6.I. Sylvester's Problem of Collinear Points 227
6.J. Four Bugs and Other Pursuit Problems 227
6.K. Dudeney's Square to Triangle Dissection 229
6.L. Crossed Ladders 229
6.L.1. Ladder Over Box 231
6.M. Spider & Fly Problems 231
6.N. Dissection of a 1 x 1 x 2 Block to a Cube 232
6.O. Passing a Cube Through an Equal or Smaller Cube _ Prince Rupert's Problem 233
6.P. Geometrical Vanishing 234
6.P.1. Paradoxical Dissections of the Chessboard Based on
Fibonacci Numbers 234
6.P.2. Other Types 235
6.Q. Knotting a Strip to Make a Regular Pentagon 237
6.R. Geometric Fallacies 238
6.R.1. Every Triangle is Isosceles 238
6.R.2. A Right Angle is Obtuse 239
6.R.3. Lines Approaching but not Meeting 239
6.R.4. Others 239
6.S. Tangrams, et al. 239
General Histories 239
Specific Items 240
6.S.1. Loculus of Archimedes 245
6.S.2. Other Sets of Pieces 246
6.T. No Three in a Line Problem 247
6.U. Tiling 248
6.U.1. Penrose Pieces 248
6.U.2. Packing Bricks in Boxes 248
6.V. Silhouette and Viewing Puzzles 248
6.W. Burr Puzzles 251
6.W.1. Three Piece Burr 251
6.W.2. Six Piece Burr = Chinese Cross 251
6.W.3. Three Piece Burr with Identical Pieces 253
6.W.4. Diagonal Six Piece Burr = Trick Star 253
6.W.5. Six Piece Burr with Identical Pieces 254
6.W.6. Altekruse Puzzle 254
6.W.7. Other Burrs 254
6.X. Rotating Rings of Polyhedra 255
6.Y. Rope Round the Earth 256
6.Z. Langley's Adventitious Angles 257
6.AA. Nets of Polyhedra 258
6.AB. Self‑Rising Polyhedra 259
6.AC. Conway's Life 259
6.AD. Isoperimetric Problems 259
6.AD.1. Largest Parcel One Can Post 262
6.AE. 6" Hole Through Sphere Leaves Constant Volume 261
6.AF. What Colour Was The Bear? 262
6.AG. Moving Around a Corner 264
6.AH. Tethered Goat 265
6.AI. Trick Joints 266
6.AJ. Geometric Illusions 267
6.AJ.1. Two Pronged Trident 269
6.AJ.2. Tribar and Impossible Staircase 270
6.AJ.3. Café Wall Illusion 271
6.AK. Polygonal Path Covering N x N Lattice of Points, Queen's Tours, etc. 271
6.AL. Steiner‑Lehmus Theorem 273
6.AM. Morley's Theorem 274
6.AN. Volume of the Intersection of Two Cylinders 274
6.AO. Configuration Problems 275
6.AO.1. Place Four Points Equidistantly = Make Four Triangles with Six
Matchsticks 282
6.AO.2. Place an Even Number in Each Line 282
6.AP. Dissections of a Tetrahedron 283
6.AP.1. Two Pieces 283
6.AP.2. Four Pieces 283
6.AQ. Dissections of a Cross, T or H 284
6.AR. Quadrisected Square Puzzle 285
6.AS. Dissection of Squares into a Square 285
6.AS.1. Twenty 1, 2, Ö5 Triangles Make a Square, i.e. Five Equal Squares to a
Square 286
6.AS.1.a. Greek Cross to a Square 288
6.AS.1.b. Other Greek Cross Dissections 289
6.AS.2. Two (Adjacent) Squares to a Square 289
6.AS.2.a. Two Equal Squares to a Square 291
6.AS.3. Three Equal Squares to a Square 291
6.AS.3.a. Three Equal 'Squares' to a Hexagon 291
6.AS.4. Eight Equal Squares to a Square 292
6.AS.5. Rectangle to a Square or Other Rectangle 292
6.AT. Polyhedra and Tessellations 293
6.AT.1. Regular Polyhedra 293
6.AT.2. Star and Stellated Polyhedra 294
6.AT.3. Archimedean Polyhedra 296
6.AT.4. Uniform Polyhedra 297
6.AT.5. Regular‑Faced Polyhedra 298
6.AT.6. Tessellations 298
6.AT.6.a. Tessellating with Congruent Figures 298
6.AT.7. Plaiting of Polyhedra 299
6.AT.8. Dürer's Octahedron 299
6.AT.9. Other Polyhedra 300
6.AU. Three Rabbits, Dead Dogs and Trick Ponies 300
China 301
China/Paderborn 301
Europe 302
Modern Versions of the Three Rabbits Puzzle 305
Dead Dogs 306
Trick Ponies 308
6.AV. Cutting Up in Fewest Cuts 308
6.AW. Division into Congruent Pieces 308
6.AW.1. Mitre Puzzle 308
6.AW.2. Rep‑Tiles 309
6.AW.3. Dividing a Square into Congruent Parts 310
6.AW.4. Dividing an L-Tromino into Congruent Parts 311
6.AX. The Packer's Secret 311
6.AY. Dissect 3A x 2B to Make 2A x 3B, etc. 311
6.AY.1. O'Beirne's Steps 313
6.AY.2. Swiss Flag Puzzle 314
6.AZ. Ball Pyramid Puzzles 314
6.BA. Cutting a Card so One Can Pass Through It 315
6.BB. Doubling a Square Without Changing Its Height or Width 316
6.BC. Hoffmann's Cube 316
6.BD. Bridge a Moat with Planks 316
6.BE. Reverse a Triangular Array of Ten Circles 317
6.BF. Pythagorean Recreations 318
6.BF.1. The Broken Bamboo 318
6.BF.2. Sliding Spear = Leaning Reed 319
6.BF.3. Well Between Two Towers 321
6.BF.4. Rail Buckling 323
6.BF.5. Travelling on Sides of a Right Triangle 323
6.BG. Quadrisect a Paper Square with One Cut 324
6.BH. Moiré Patterns 324
6.BI. Venn Diagrams for n Sets 325
6.BJ. 3D Dissection Puzzles 326
6.BK. Superellipse 327
6.BL. Tan-1 _ + Tan-1 ˝ = Tan-1 1, etc. 327
6.BM. Dissect Circle into Two Hollow Ovals 328
6.BN. Round Peg in Square Hole or Vice Versa 329
6.BO. Butterfly Problem 329
6.BP. Early Matchstick Puzzles 329
6.BQ. Covering a Disc with Discs 330
6.BR. What is a General Triangle? 331
6.BS. Form Six Coins into a Hexagon 331
6.BT. Placing Objects in Contact 332
6.BU. Construction of n-gons 332
6.BV. Geometric Constructions 334
7. ARITHMETIC & NUMBER‑THEORETIC RECREATIONS 335
7.A. Fibonacci Numbers 335
7.B. Josephus or Survivor Problem 337
7.C. Egyptian Fractions 349
7.D. The First Digit Problem 350
7.E. Monkey and Coconuts Problems 350
7.E.1. Versions with All Getting the Same 361
7.F. Illegal Operations Giving Correct Result 363
7.G. Inheritance Problems 363
7.G.1. Half + Third + Ninth, etc. 363
7.G.2. Posthumous Twins, etc. 368
7.H. Division and Sharing Problems _ Cistern Problems 370
7.H.1. With Growth _ Newton's Cattle Problem 385
7.H.2. Division of Casks 386
7.H.3. Sharing Unequal Resources _ Problem of the Pandects 388
7.H.4. Each Doubles Other's Money to Make All Equal, etc. 390
7.H.5. Sharing Cost of Stairs, etc. 392
7.H.6. Sharing a Grindstone 393
7.H.7. Digging Part of a Well 394
7.I. Four Fours, etc. 396
7.I.1. Largest Number Using Four Ones, etc. 401
7.J. Salary Puzzle 402
7.K. Congruences 404
7.K.1. Casting Out Nines 404
7.L. Geometric Progressions 406
7.L.1. 1 + 7 + 49 + ... & St. Ives 408
7.L.2. 1 + 2 + 4 + .... 409
7.L.2.a. Chessboard Problem 410
7.L.2.b. Horseshoe Nails Problem 413
7.L.2.c. Use of 1, 2, 4, ... as Weights, etc. 414
7.L.3. 1 + 3 + 9 + ... and Other Systems of Weights 415
7.M. Binary System and Binary Recreations 417
7.M.1. Chinese Rings 418
7.M.2. Tower of Hanoi 420
7.M.2.a. Tower of Hanoi with More Pegs 423
7.M.3. Gray Code 424
7.M.4. Binary Divination 424
7.M.4.a. Ternary Divination 426
7.M.4.b. Other Divinations Using Binary or Ternary 426
7.M.5. Loony Loop = Gordian Knot 428
7.M.6. Binary Button Games 429
7.N. Magic Squares 431
7.N.1. Magic Cubes 442
7.N.2. Magic Triangles 445
7.N.3. Anti‑Magic Squares and Triangles 446
7.N.4. Magic Knight's Tour 447
7.N.5. Other Magic Shapes 448
7.O. Magic Hexagon 450
7.O.1 Other Magic Hexagons 451
7.P. Diophantine Recreations 452
7.P.1. Hundred Fowls and Other Linear Problems 453
7.P.2. Chinese Remainder Theorem 467
7.P.3. Archimedes' Cattle Problem 472
7.P.4. Present of Gems 473
7.P.5. Selling Different Amounts 'At Same Prices' Yielding the Same 474
7.P.6. Conjunction of Planets, etc. 478
7.P.7. Robbing and Restoring 479
7.Q. Blind Abbess and her Nuns _ Rearrangement Along Sides of a 3 x 3 Square
Conserving Side Totals 481
7.Q.1. Rearrangement on a Cross 483
7.Q.2. Rearrange a Cross of Six to Make Two Lines of Four 484
7.R. "If I Had One From You, I'd Have Twice You" 484
7.R.1. Men Find a Purse and 'Bloom' of Thymarides 489
7.R.2. "If I Had 1/3 of Your Money, I Could Buy the Horse" 494
7.R.3. Sisters and Brothers 501
7.R.4. "If I Sold Your Eggs at my Price, I'd Get ...." 501
7.S. Dilution and Mixing Problems 501
7.S.1. Dishonest Butler Drinking Some and Replacing with Water 502
7.S.2. Water in Wine Versus Wine in Water 502
7.T. Four Number Game 503
7.U. Postage Stamp Problem 504
7.V. xy = yx and Iterated Exponentials 504
7.W. Card Piling over a Cliff 505
7.X. How Old is Ann? and Other Age Problems 506
7.Y. Combining Amounts and Prices Incoherently 513
7.Y.1. Reversal of Averages Paradox 515
7.Y.2. Unfair Division 516
7.Z. Missing Dollar and Other Erroneous Accounting 516
7.AA. Negative Digits 517
7.AA.1. Negative Bases, etc. 517
7.AB. Perfect Numbers, etc. 518
7.AC. Cryptarithms, Alphametics and Skeleton Arithmetic 520
7.AC.1. Cryptarithms: SEND + MORE = MONEY, etc. 520
7.AC.2. Skeleton Arithmetic: Solitary Seven, etc. 522
7.AC.3. Pan‑Digital Sums 523
7.AC.4. Pan‑Digital Products 525
7.AC.5. Pan‑Digital Fractions 527
7.AC.6. Other Pan‑Digital Expressions 527
7.AC.7. Self-descriptive Numbers, Pangrams, etc. 530
7.AD. Selling, Buying and Selling Same Item 531
7.AD.1. Pawning Money 532
7.AE. Use of Counterfeit Bill or Forged Cheque 532
7.AF. Arithmetic Progressions 533
7.AF.1. Collecting Stones 534
7.AF.2. Clock Striking 536
7.AG. 2592 537
7.AH. Multiplying by Reversing 537
7.AH.1. Other Reversal Problems 538
7.AI. Impossible Exchange Rates 538
7.AJ. Multiplying by Shifting 538
7.AJ.1. Multiplying by Appending Digits 540
7.AK. Lazy Worker 540
7.AL. If A is B, What is C? 542
7.AM. Crossnumber Puzzles 544
7.AN. Three Odds Make an Even, etc. 545
7.AO. Divination of a Permutation 547
7.AP. Knowing Sum vs Knowing Product 550
7.AQ. Numbers in Alphabetic Order 552
7.AR. 1089 552
7.AS. Cigarette Butts 554
7.AT. Bookworm's Distance 555
7.AU. Number of Cuts to Make n Pieces 555
7.AV. How Long to Strike Twelve? 556
7.AW. 28/7 = 13 556
7.AX. Sum = Product, etc. 557
7.AY. Sum of Powers of Digits 557
7.AZ. Divination of a Pair of Cards from its Rows 558
7.BA. Cycle of Numbers with Each Closer to Ten than the Previous 560
7.BB. Iterated Functions of Integers 560
7.BC. Unusual Difficulty Making Change 561
8. PROBABILITY RECREATIONS 562
8.A. Buffon's Needle Problem 562
8.B. Birthday Problem 562
8.C. Probability that a Triangle is Acute 564
8.D. Attempts to Modify Boy‑Girl Ratio 565
8.E. St. Petersburg Paradox 565
8.F. Problem of Points 565
8.G. Probability that Three Lengths Form a Triangle 57
8.H. Probability Paradoxes 567
8.H.1. Bertrand's Box Paradox 567
8.H.2. Bertrand's Chord Paradox 568
8.I. Taking the Next Train 568
8.J. Clock Patience or Solitaire 568
8.K. Sucker Bets 569
8.L. Nontransitive Games 569
9. LOGICAL RECREATIONS 571
9.A. All Cretans are Liars, etc. 571
9.B. Smith _ Jones _ Robinson Problem 572
9.C. Forty Unfaithful Wives 573
9.D. Spots on Foreheads 573
9.E. Strange Families 574
9.E.1. That Man's Father is My Father's Son, etc. 580
9.E.2. Identical Siblings who are not Twins 583
9.F. The Unexpected Hanging 583
9.G. Truthtellers and Liars 583
9.H. Prisoner's Dilemma 585
9.I. Hempel's Raven Paradox 585
9.J. Use of a Fallen Signpost 585
9.K. Carroll's Barber Paradox 585
10. PHYSICAL RECREATIONS 587
10.A. Overtaking and Meeting Problems 587
10.A.1. Circling an Army 598
10.A.2. Number of Buses Met 599
10.A.3. Times from Meeting to Finish Given 600
10.A.4. The Early Commuter 601
10.A.5. Head Start Problems 601
10.A.6. Double Crossing Problems 602
10.B. Fly Between Trains 602
10.C. Lewis Carroll's Monkey Problem 603
10.D. Mirror Problems 604
10.D.1. Mirror Reversal Paradox 604
10.D.2. Other Mirror Problems 605
10.D.3. Magic Mirrors 605
10.E. Wheel Paradoxes 606
10.E.1. Aristotle's Wheel Paradox 606
10.E.2. One Wheel Rolling Around Another 606
10.E.3. Hunter and Squirrel 607
10.F. Floating Body Problems 607
10.G. Motion in a Current or Wind 608
10.H. Snail Climbing out of Well 610
10.I. Limited Means of Transport _ Two Men and a Bike, etc. 614
10.J. Resistor Networks 614
10.K. Problem of the Date Line 615
10.L. Falling Down a Hole Through the Earth 617
10.M. Celts = Rattlebacks 618
10.M.1. Tippee Tops 620
10.N. Ship's Ladder in Rising Tide 620
10.O. Erroneous Averaging of Velocities 620
10.P. False Balance 621
10.Q. Push a Bicycle Pedal 622
10.R. Clock Hand Problems 622
10.S. Walking in the Rain 624
10.T. Centrifugal Puzzles 624
10.U. Shortest Route Via a Wall 625
10.V. Pick Up Puzzles = Pluck It 625
10.W. Puzzle Vessels 625
10.X. How Far Does a Phonograph Needle Travel? 629
10.Y. Double Cone Rolls Uphill 629
10.Z. The Wobbler 630
10.AA. Non-Regular Dice 630
10.AB. Bicycle Track Problems 632
10.AC. Roberval's Balance 634
11. TOPOLOGICAL RECREATIONS 635
11.A. Scissors on String 635
11.B. Two People Joined by Ropes at Wrists 636
11.C. Two Balls on String Through Leather Hole and Strap = Cherries Puzzle 637
11.D. Solomon's Seal 638
11.E. Loyd's Pencil Puzzle 639
11.F. The Imperial Scale 640
11.G. Trick Purses 640
11.H. Removing Waistcoat Without Removing Coat 641
11.H.1. Removing Loop from Arm 642
11.I. Heart and Ball Puzzle and Other Loop Puzzles 642
11.J. Möbius Strip 644
11.K. Wire Puzzles 646
11.K.1. Ring and Spring Puzzle 647
11.K.2. String and Spring Puzzle 648
11.K.3. Magic Chain = Tumble Rings 648
11.K.4. Puzzle Rings 648
11.K.5. Ring Mazes 649
11.K.6. Interlocked Nails, Hooks, Horns, etc. 649
11.K.7. Horseshoes Puzzle 650
11.K.8. The Caught Heart 650
11.L. Jacob's Ladder and Other Hinging Devices 650
11.M. Puzzle Boxes 651
11.N. Three Knives Make a Support 652
11.O. Borromean Rings 654
11.P. The Lonely Monk 654
11.Q. Turning an Inner Tube Inside Out 655
11.R String Figures 655