Eighth preliminary edition

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.

CONTENTS
INTRODUCTION 1

Nature of This Work 1

Similar Works 2

Coverage 3

Status of the Project 4

Technical Notes 6

New Sections in This Edition 7

Acknowledgements 7

CONTENTS 10
ABBREVIATIONS 20

Diacritical Marks and Notation 20

Abbreviations of Journals and Series 21

Abbreviations of Publishers 21

Abbreviations of Months 21

Publishers' Locations 21

COMMON REFERENCES 22
SOME OTHER RECURRING REFERENCES 79
1. BIOGRAPHICAL MATERIAL in Chronological Order 82
Alcuin, Fibonacci, Pacioli, Bachet, Leurechon/van Etten, Ozanam, Montucla, Carroll, Hoffmann, Loyd & Loyd Jr, Lucas, Schubert, Ball, Dudeney, Ahrens, Perelman, Phillips.
2. GENERAL PUZZLE COLLECTIONS AND SURVEYS 90
3. GENERAL HISTORICAL AND BIBLIOGRAPHICAL MATERIAL 91
3.A. General Historical Material 91

3.B. Bibliographical Material 91

4. MATHEMATICAL GAMES 97
4.A. General Theory and Nim like Games 97

4.A.1. One Pile Game 97

4.A.1.a. The 31 Game 100

4.A.2. Symmetry Arguments 102

4.A.3. Kayles 102

4.A.4. Nim 103

4.A.5. General Theory 105

4.B. Particular Games 106

4.B.1. Tic Tac Toe = Noughts and Crosses 106

4.B.1.a. In Higher Dimensions 114

4.B.2. Hex 115

4.B.3. Dots and Boxes 116

4.B.4. Sprouts 117

4.B.5. Ovid's Game and Nine Men's Morris 118

4.B.6. Phutball 124

4.B.7. Bridg It 124

4.B.8. Chomp 124

4.B.9. Snakes and Ladders 125

4.B.10. Mu Torere 127

4.B.11. Mastermind, etc. 127

4.B.12. Rithmomachia = The Philosophers' Game 128

4.B.13. Mancala Games 129

4.B.14. Dominoes, etc. 130

4.B.15. Svoyi Kosiri 130

5. COMBINATORIAL RECREATIONS 131
5.A. The 15 Puzzle, etc. 131

General 131

Early Alphabetic Versions 131

Loyd 132

The 15 Puzzle 132

5.A.1. Non square Pieces 139

5.A.2. Three Dimensional Versions 140

5.A.3. Rolling Piece Puzzles 141

5.A.4. Panex Puzzle 142

5.B. Crossing Problems 142

5.B.1. Lowering from Tower Problem 151

5.B.2. Crossing a Bridge with a Torch 152

5.C. False Coins with a Balance 152

5.C.1. Ranking Coins with a Balance 155

5.D. Measuring Problems 156

5.D.1. Jugs & Bottles 156

5.D.2. Ruler with Minimal Number of Marks 161

5.D.3. False Coins with a Weighing Scale 162

5.D.4. Timing with Hourglasses 162

5.D.5. Measure Half a Barrel 162

5.E. Euler Circuits and Mazes 163

5.E.1. Mazes 167

5.E.2. Memory Wheels = Chain Codes 172

5.E.2.a Pantactic Squares 173

5.F. Hamiltonian Circuits 174

5.F.1. Knight's Tours and Paths 174

5.F.2. Other Hamiltonian Circuits 181

5.F.3. Knight's Tours in Higher Dimensions 182

5.F.4. Other Circuits In and On a Cube 183

5.G. Connection Problems 183

5.G.1. Gas, Water and Electricity 183

5.H. Coloured Squares and Cubes, etc. 184

5.H.1. Instant Insanity = The Tantalizer 184

5.H.2. MacMahon Pieces 185

5.H.3. Path Forming Puzzles 187

5.H.4. Other and General 187

5.I. Latin Squares and Euler Squares 190

5.I.1. Eight Queens Problem 192

5.I.2. Colouring Chessboard with No Repeats in a Line 195

5.J. Squared Squares, etc. 195

5.J.1. Mrs Perkins's Quilt 197

5.J.2. Cubing the Cube 198

5.J.3. Tiling a Square of Side 70 with Squares of Sides 1, 2, ..., 24 198

5.K. Derangements 198

5.K.1 Deranged Boxes of A, B and A & B 199

5.K.2 Other Logic Puzzles Based on Derangements 199

5.K.3 Cayley's Mousetrap 200

5.L. Ménage Problem 200

5.M. Six People at a Party -- Ramsey Theory 201

5.N. Jeep or Explorer's Problem 201

5.O. Tait's Counter Puzzle: BBBBWWWW to WBWBWBWB 204

5.P. General Moving Piece Puzzles 207

5.P.1. Shunting Puzzles 207

5.P.2. Taquin 209

5.Q. Number of Regions Determined by N Lines or Planes 209

5.Q.1. Number of Intersections Determined by N Lines 210

5.R. Jumping Piece Games 210

5.R.1. Peg Solitaire 210

5.R.1.a. Triangular Version 213

5.R.1.b. Other shapes 214

5.R.2. Frogs and Toads: BBB_WWW to WWW_BBB 215

5.R.3. Fore and Aft -- 3 by 3 Squares Meeting at a Corner 217

5.R.4. Reversing Frogs and Toads: _12...n to _n...21 218

5.R.5. Fox and Geese, etc. 219

5.R.6. Octagram Puzzle 222

5.R.7. Passing Over Counters 223

5.S. Chain Cutting and Rejoining 226

5.S.1. Using Chain Links to Pay for a Room 227

5.T. Dividing a Cake Fairly 227

5.U. Pigeonhole Recreations 228

5.V. Think A Dot, etc. 229

5.W Making Three Pieces of Toast 230

5.W.1. Boiling Eggs 230

5.X Counting Figures in a Pattern 231

5.X.1. Counting Triangles 231

5.X.2. Counting Rectangles or Squares 233

5.X.3. Counting Hexagons 235

5.X.4. Counting Circles 235

5.Y. Number of Routes in a Lattice 235

5.Z. Chessboard Placing Problems 238

5.Z.1. Kings 239

5.Z.2. Queens 239

5.Z.3. Bishops 240

5.Z.4. Knights 241

5.Z.5. Rooks 241

5.Z.6. Mixtures 241

5.AA. Card Shuffling 242

5.AB. Folding a Strip of Stamps 244

5.AC. Properties of the Seven Bar Digital Display 244

5.AD. Stacking a Deck to Produce a Special Effect 245

5.AE. Reversing Cups 245

5.AF. Spotting Dice 245

5.AG. Rubik's Cube and Similar Puzzles 246

5.AG.1. Rubik's Cube 246

5.AG.2. Hungarian Rings, etc. 246
6. GEOMETRIC RECREATIONS 248
6.A. Pi 248

6.B. Straight Line Linkages 249

6.C. Curves of Constant Width 250

6.D. Flexagons 251

6.E. Flexatube 253

6.F. Polyominoes, etc. 253

6.F.1. Other Chessboard Dissections 262

6.F.2. Covering Deleted Chessboard with Dominoes 264

6.F.3. Dissecting a Cross into Zs and Ls 264

6.F.4. Quadrisecting an L Tromino, etc. 266

6.F.5. Other Dissections into Polyominoes 268

6.G. Soma Cube 269

6.G.1. Other Cube Dissections 270

6.G.2. Dissection of 6³ into 3³, 4³ and 5³, etc. 271

6.G.3. Dissection of a Die into Nine 1 x 1 x 3 271

6.G.4. Use of Other Polyhedral Pieces 272

6.H. Pick's Theorem 272

6.I. Sylvester's Problem of Collinear Points 273

6.J. Four Bugs and Other Pursuit Problems 273

6.K. Dudeney's Square to Triangle Dissection 275

6.L. Crossed Ladders 275

6.L.1. Ladder Over Box 277

6.M. Spider & Fly Problems 278

6.N. Dissection of a 1 x 1 x 2 Block to a Cube 279

6.O. Passing a Cube Through an Equal or Smaller Cube

-- Prince Rupert's Problem 279

6.P. Geometrical Vanishing 280

6.P.1. Paradoxical Dissections of the Chessboard Based on

Fibonacci Numbers 280

6.P.2. Other Types 282

6.Q. Knotting a Strip to Make a Regular Pentagon 285

6.R. Geometric Fallacies 285

6.R.1. Every Triangle is Isosceles 286

6.R.2. A Right Angle is Obtuse 287

6.R.3. Lines Approaching but not Meeting 287

6.R.4. Others 287

6.S. Tangrams, et al. 287

General Histories 287

Specific Items 289

6.S.1. Loculus of Archimedes 299

6.S.2. Other Sets of Pieces 300

6.T. No Three in a Line Problem 301

6.U. Tiling 302

6.U.1. Penrose Pieces 302

6.U.2. Packing Bricks in Boxes 302

6.V. Silhouette and Viewing Puzzles 303

6.W. Burr Puzzles 306

6.W.1. Three Piece Burr 306

6.W.2. Six Piece Burr = Chinese Cross 307

6.W.3. Three Piece Burr with Identical Pieces 309

6.W.4. Diagonal Six Piece Burr = Trick Star 309

6.W.5. Six Piece Burr with Identical Pieces 310

6.W.6. Altekruse Puzzle 310

6.W.7. Other Burrs 310

6.X. Rotating Rings of Polyhedra 311

6.Y. Rope Round the Earth 312

6.Z. Langley's Adventitious Angles 314

6.AA. Nets of Polyhedra 314

6.AB. Self Rising Polyhedra 316

6.AC. Conway's Life 316

6.AD. Isoperimetric Problems 316

6.AD.1. Largest Parcel One Can Post 318

6.AE. 6" Hole Through Sphere Leaves Constant Volume 319

6.AF. What Colour Was The Bear? 319

6.AG. Moving Around a Corner 322

6.AH. Tethered Goat 323

6.AI. Trick Joints 324

6.AJ. Geometric Illusions 325

6.AJ.1. Two Pronged Trident 328

6.AJ.2. Tribar and Impossible Staircase 329

6.AJ.3. Café Wall Illusion 330

6.AJ.4. Stereograms 331

6.AJ.5. Impossible Crate 331

6.AK. Polygonal Path Covering N x N Lattice of Points, Queen's Tours, etc. 331

6.AL. Steiner Lehmus Theorem 334

6.AM. Morley's Theorem 335

6.AN. Volume of the Intersection of Two Cylinders 335

6.AO. Configuration Problems 336

6.AO.1. Place Four Points Equidistantly = Make Four Triangles with Six

Matchsticks 343

6.AO.2. Place an Even Number in Each Line 345

6.AP. Dissections of a Tetrahedron 346

6.AP.1. Two Pieces 346

6.AP.2. Four Pieces 346

6.AQ. Dissections of a Cross, T or H 347

6.AR. Quadrisected Square Puzzle 348

6.AS. Dissection of Squares into a Square 349

6.AS.1. Twenty 1, 2, 5 Triangles Make a Square, i.e. Five Equal Squares to a

Square 349

6.AS.1.a. Greek Cross to a Square 352

6.AS.1.b. Other Greek Cross Dissections 353

6.AS.2. Two (Adjacent) Squares to a Square 353

6.AS.2.a. Two Equal Squares to a Square 356

6.AS.3. Three Equal Squares to a Square 356

6.AS.3.a. Three Equal 'Squares' to a Hexagon 356

6.AS.4. Eight Equal Squares to a Square 357

6.AS.5. Rectangle to a Square or Other Rectangle 357

6.AT. Polyhedra and Tessellations 358

6.AT.1. Regular Polyhedra 358

6.AT.2. Star and Stellated Polyhedra 362

6.AT.3. Archimedean Polyhedra 364

6.AT.4. Uniform Polyhedra 369

6.AT.5. Regular Faced Polyhedra 369

6.AT.6. Tessellations 369

6.AT.6.a. Tessellating with Congruent Figures 369

6.AT.7. Plaiting of Polyhedra 371

6.AT.8. Dürer's Octahedron 371

6.AT.9. Other Polyhedra 371

6.AU. Three Rabbits, Dead Dogs and Trick Ponies 372

China 373

Other Asia 375

Paderborn 378

Medieval Europe 380

Modern Versions of the Three Rabbits Puzzle 388

Dead Dogs 389

Trick Mules 394

6.AV. Cutting Up in Fewest Cuts 394

6.AW. Division into Congruent Pieces 394

6.AW.1. Mitre Puzzle 394

6.AW.2. Rep Tiles 395

6.AW.3. Dividing a Square into Congruent Parts 396

6.AW.4. Dividing an L-Tromino into Congruent Parts 397

6.AX. The Packer's Secret 397

6.AY. Dissect 3A x 2B to Make 2A x 3B, etc. 398

6.AY.1. O'Beirne's Steps 400

6.AY.2. Swiss Flag Puzzle 400

6.AZ. Ball Pyramid Puzzles 401

6.BA. Cutting a Card so One Can Pass Through It 402

6.BB. Doubling an Area Without Changing Its Height or Width 402

6.BC. Hoffmann's Cube 403

6.BD. Bridge a Moat with Planks 403

6.BE. Reverse a Triangular Array of Ten Circles 405

6.BF. Pythagorean Recreations 405

6.BF.1. The Broken Bamboo 406

6.BF.2. Sliding Spear = Leaning Reed 407

6.BF.3. Well Between Two Towers 408

6.BF.4. Rail Buckling 3411

6.BF.5. Travelling on Sides of a Right Triangle 412

6.BG. Quadrisect a Paper Square with One Cut 412

6.BH. Moiré Patterns 412

6.BI. Venn Diagrams for n Sets 413

6.BJ. 3D Dissection Puzzles 415

6.BK. Superellipse 415

6.BL. Tan^-1 ⅓ + Tan^-1 ½ = Tan^-1 1, etc. 416

6.BM. Dissect Circle into Two Hollow Ovals 417

6.BN. Round Peg in Square Hole or Vice Versa 417

6.BO. Butterfly Problem 418

6.BP. Early Matchstick Puzzles 418

6.BQ. Covering a Disc with Discs 419

6.BR. What is a General Triangle? 420

6.BS. Form Six Coins into a Hexagon 420

6.BT. Placing Objects in Contact 421

6.BU. Construction of n-gons 421

6.BV. Geometric Constructions 423

6.BW. Distances to Corners of a Square 424
7. ARITHMETIC & NUMBER THEORETIC RECREATIONS 426
7.A. Fibonacci Numbers 426

7.B. Josephus or Survivor Problem 429

7.C. Egyptian Fractions 443

7.D. The First Digit Problem 443

7.E. Monkey and Coconuts Problems 444

7.E.1. Versions with All Getting the Same 457

7.F. Illegal Operations Giving Correct Result 458

7.G. Inheritance Problems 459

7.G.1. Half + Third + Ninth, etc. 459

7.G.2. Posthumous Twins, etc. 464

7.H. Division and Sharing Problems -- Cistern Problems 467

7.H.1. With Growth -- Newton's Cattle Problem 484

7.H.2. Division of Casks 486

7.H.3. Sharing Unequal Resources -- Problem of the Pandects 487

7.H.4. Each Doubles Other's Money to Make All Equal, etc. 490

7.H.5. Sharing Cost of Stairs, etc. 492

7.H.6. Sharing a Grindstone 494

7.H.7. Digging Part of a Well 494

7.I. Four Fours, etc. 496

7.I.1. Largest Number Using Four Ones, etc. 503

7.J. Salary Puzzle 504

7.K. Congruences 505

7.K.1. Casting Out Nines 506

7.L. Geometric Progressions 508

7.L.1. 1 + 7 + 49 + ... & St. Ives 510

7.L.2. 1 + 2 + 4 + .... 513

7.L.2.a. Chessboard Problem 513

7.L.2.b. Horseshoe Nails Problem 517

7.L.2.c. Use of 1, 2, 4, ... as Weights, etc. 518

7.L.3. 1 + 3 + 9 + ... and Other Systems of Weights 519

7.M. Binary System and Binary Recreations 521

7.M.1. Chinese Rings 523

7.M.2. Tower of Hanoi 527

7.M.2.a. Tower of Hanoi with More Pegs 530

7.M.3. Gray Code 531

7.M.4. Binary Divination 531

7.M.4.a. Ternary Divination 533

7.M.4.b. Other Divinations Using Binary or Ternary 533

7.M.5. Loony Loop = Gordian Knot 537

7.M.6. Binary Button Games 537

7.N. Magic Squares 540

Surveys 540

Possible Early References 541

7.N.1. Magic Cubes 554

7.N.2. Magic Triangles 556

7.N.3. Anti Magic Squares and Triangles 557

7.N.4. Magic Knight's Tour 558

7.N.5. Other Magic Shapes 559

7.O. Magic Hexagon 562

7.O.1 Other Magic Hexagons 563

7.P. Diophantine Recreations 565

7.P.1. Hundred Fowls and Other Linear Problems 565

7.P.2. Chinese Remainder Theorem 582

7.P.3. Archimedes' Cattle Problem 588

7.P.4. Present of Gems 589

7.P.5. Selling Different Amounts 'At Same Prices' Yielding the Same 589

7.P.6. Conjunction of Planets, etc. 594

7.P.7. Robbing and Restoring 595

7.Q. Blind Abbess and her Nuns -- Rearrangement Along Sides of a 3 x 3 Square

Conserving Side Totals 597

7.Q.1. Rearrangement on a Cross 600

7.Q.2. Rearrange a Cross of Six to Make Two Lines of Four 601

7.R. "If I Had One From You, I'd Have Twice You" 602

7.R.1. Men Find a Purse and 'Bloom of Thymaridas' 608

7.R.2. "If I Had 1/3 of Your Money, I Could Buy the Horse" 614

7.R.3. Sisters and Brothers 623

7.R.4. "If I Sold Your Eggs at my Price, I'd Get ...." 623

7.S. Dilution and Mixing Problems 624

7.S.1. Dishonest Butler Drinking Some and Replacing with Water 624

7.S.2. Water in Wine Versus Wine in Water 625

7.T. Four Number Game 626

7.U. Postage Stamp Problem 627

7.V. x^y = y^x and Iterated Exponentials 627

7.W. Card Piling over a Cliff 628

7.X. How Old is Ann? and Other Age Problems 629

7.Y. Combining Amounts and Prices Incoherently 638

7.Y.1. Reversal of Averages Paradox 640

7.Y.2. Unfair Division 641

7.Z. Missing Dollar and Other Erroneous Accounting 642

7.AA. Negative Digits 643

7.AA.1. Negative Bases, etc. 644

7.AB. Perfect Numbers, etc. 645

7.AC. Cryptarithms, Alphametics and Skeleton Arithmetic 647

7.AC.1. Cryptarithms: SEND + MORE = MONEY, etc. 647

7.AC.2. Skeleton Arithmetic: Solitary Seven, etc. 650

7.AC.3. Pan Digital Sums 652

7.AC.3.a Insertion of Signs to Make 100, etc. 653

7.AC.4. Pan Digital Products 654

7.AC.5. Pan Digital Fractions 656

7.AC.6. Other Pan Digital and Similar Problems 657

7.AC.7. Self-descriptive Numbers, Pangrams, etc. 660

7.AD. Selling, Buying and Selling Same Item 661

7.AD.1. Pawning Money 662

7.AE. Use of Counterfeit Bill or Forged Cheque 662

7.AF. Arithmetic Progressions 664

7.AF.1. Collecting Stones 664

7.AF.2. Clock Striking 666

7.AG. 2592 667

7.AH. Multiplying by Reversing 668

7.AH.1. Other Reversal Problems 669

7.AI. Impossible Exchange Rates 669

7.AJ. Multiplying by Shifting 669

7.AJ.1. Multiplying by Appending Digits 671

7.AK. Lazy Worker 671

7.AL. If A is B, What is C? 674

7.AM. Crossnumber Puzzles 676

7.AN. Three Odds Make an Even, etc. 678

7.AO. Divination of a Permutation 680

7.AP. Knowing Sum vs Knowing Product 684

7.AQ. Numbers in Alphabetic Order 687

7.AR. 1089 687

7.AS. Cigarette Butts 690

7.AT. Bookworm's Distance 690

7.AU. Number of Cuts to Make n Pieces 691

7.AV. How Long to Strike Twelve? 692

7.AW. 28/7 = 13 692

7.AX. Sum = Product, etc. 692

7.AY. Sum of Powers of Digits 693

7.AZ. Divination of a Pair of Cards from its Rows 694

7.BA. Cycle of Numbers with Each Closer to Ten than the Previous 696

7.BB. Iterated Functions of Integers 696

7.BC. Unusual Difficulty Making Change 697