This book was originally published in 2 volumes in 1982. A
second edition was released in 4 volumes starting in 2001.
These four volumes (Spades, Hearts, Clubs and Diamonds)
explore the foundations and current research in the field of
recreational games. It is the definitive work on the subject
of mathematical games.
In Volume 1, the authors do the spade-word, presenting
theories and techniques to "dissect" games of varied
structures and formats in order to develop winning
strategies. Their focus is on "adding games," including Nim,
Hackenbush, and Kayles. The inclusion of many examples helps
the reader to put the mathematical analysis to immediate use.
In Volume 2, the authors have a Change of Heart and bend the
rules of Volume 1 to apply them to games that may continue
indefinitely or in which "winning" is not necessarily making
the last move, such as Cut-cake and Loopy Hackenbush.
In Volume 3, the authors examine Games played in Clubs,
giving case studies for coin and paper-and-pencil games, such
as Dots and Boxes and Nimstring.
In Volume 4, the authors present a Diamond of a find,
covering one-player games such as Solitaire and ponder
subjects and questions such as, Purging Pegs Properly,
Pursuing Puzzles Purposefully, and What is Life?
Preface to the Second Edition, Preface, Spade-Work!
1 Whose Game?
Blue-Red Hackenbush / The Tweedledum and Tweedledee
Argument / How Can You Have Half a Move? / ...And Quater
Moves? / Ski-Jumps for Beginners / Don't Just Take the
Average! / What Is a Jump Worth? / Toads-and-Frogs / Do Our
Methods Work? / What is a Game? / When Is a Move Good? /
Figure 8(d) Is Worth 3/4 / References and Further
Reading
2 Finding the Correct Numbers is Simplicity Itself
Which Numbers Are Which? / Simplicity's the Answer! /
Simplest Forms for Numbers / Cutcake / Maundy Cake / A Few
More Applications of the Simplicity Rule / Positive,
Negative, Zero, and Fuzzy Positions / Hackenbush Hotchpotch
/ Sums of Arbitrary Games / The Outcome of a Sum / The
Negative of a Game / Vancelling a Game with its Negative /
Comparing Two Games / Comparing Hackenbush Positions / The
Game of Col / A Star is Born! / Col Contains Such Values /
Game Trees / Green Hackenbush, The Game of Nim, and Nimbers
/ Get Nimble with Nimbers / Childish Hackenbush / Seating
Couples / Winning Strategies / The Sum of Two Finite Games
Can Last Forever / A Theorem about Col / Col-lections and
Collapsings / Maundy Cake / Another Cutcake Variant / How
Childish Can You Get? / References and Further
Reading
3 Some Harder Games and How to Make Them Easier
Poker-Nim / Northcott's Game / Bogus Nim-Heaps and the Mex
Rule / The Sprague-Grundy Theory for Impartial Games / The
White Knight / Adding Nimbers / Wyt Queens / Reversible
Moves in General Games / Deleting Dominated Options /
Toads-and-Frogs with Ups and Downs / Game Tracking and
Identification / What Are Flowers Worth? / A Gallimaufry of
Games / Who Wins Sums of Ups, Downs, Stars, and Numbers? /
A Closer Look at the Stars / The Values {Up|Up} and {0|Up}
/ The Upstart Equality / Gift Horses / The Nim-Addition
Rule in Several Variations / Wyt Queens and Wythoff's Game
/ Answers to Figures 8, 9, and 11 / Toad Versus Frog / Two
Theorems on Simplifying Games / Berlekamp's Rule for
Hackenbush Strings / References and Further Reading
4 Taking and Breaking
Kayles / Games With Heaps / P-Positions and N-Positions /
Subtraction Games / Ferguson's Pairing Property / Grundy
Scales / Other Take-Away Games / Dawson's Chess / The
Periodicity of Kayles / Other Take-and-Break Games /
Dawson's Kayles / Variations / Guiles / Treblecross /
Officers / Grundy's Game / Prim and Dim / Replication of
Nim-Values / Double and Quadruple Kayles / Lasker's Nim /
Some Remarks on Periodicity / Standard Form / A Compendium
of Octal Games / Additional Remarks / Sparse Spaces and
Common Cosets / Will Grundy's Game Be Ultimately Periodic?
/ Sparse Space Spells Spedd / Games Displaying Arithmetic
Periodicity / A Non-Arithmetic-Periodicity Theorem / Some
Hexadecimal Games / References and Further Reading
5 Numbers, Nimbers and Numberless Wonders
Domineering / Switch Games / Cashing Cheques / Some Simple
Hot Games / The Tiniest Games / Modern Management of Cash
Flow / Tiny Toads-and-Frogs / The Opening Dissection of
Toads-and-Frogs / Seating Boys and Girls / Toads-and-Frogs
Completely Dissected / Toads-and-Frogs with Two Spaces /
More Domineering Values / References
6 The Heat of Battle
Snort / A Graphic Picture of Farm Life / Don't Move In A
Number Unless There's Nothing Else To Do! / What's in it
for Me? / The Left and Right Stops / Cooling--and the
Thermograph / Cooling Settles the Mean Value / How to Draw
Thermographs / When A Player Has Several Options /
Foundations for Thermographs / Examples of Thermographs /
Who Is To Move From The Final Stop? / A Four-Stop Example /
The Cheque-Market Exchange / Equitable Games / Excitable
Games / The Extended Thermograph / Getting the Right Slant
/ The Thermostatic Strategy / Thermostrat's Not Often
Wrong! / Heating / Does The Excitement Show? / Selling
Infinitesimal Values To Your Profit-Conscious Friends /
Nim, Remoteness and Suspense in Hot Games / Overheating /
Cooling the Children's Party / But How Do You Cool A Party
By One Degree? / Three Snort Lemmas / A Short Dictionary /
Proof of the Number Avoidance Theorem / Why Thermostrat
Works / Blockbusting / ... An On We Go! / Hotstrat,
Thermostrat and Sentestrat / References and Further
Reading
7 Hackenbush
Green Hackenbush / Green Trees / Fusion / Proving The
Fusion Principle / A More Complicated Picture / Impartial
Maundy Cake / Blue-Red Hackenbush / Hackenbush Hotchpotch /
Flower Gardens / The Blue Flower Ploy / Atomic Weights /
Atomic Weights of Jungles / Making Tracks in the Jungle /
Tracking Down an Animal / Amazing Jungle / Smart Game in
the Jungle / Unparted Jungles / Blue-Red Hackenbush Can Be
Hard, Too! / Redwood Furniture / Redwood Beds / How Big Is
A Redwood Bed? / What''s The Bottle / Ordinal Addition, The
Colon Principle, and Norton's Lemma / Both Ways of Adding
Impartial Games / Many-Way Maundy Cake / Solution to Figure
15 / Tracks Cleared Through the Amazing Jungle / How Hard
Was The Bed? / NP-Hardness / The Bottle at the End of
Chapter 7 / References and Further Reading
8 It's a Small Small Small Small Word
Uppitness and Uncertainty / Computing Atomic weights /
Eatcake / Splitting The Atom / Turn-and-Eatcake / All You
Need To Know About Atomic Weights But Were Afraid To Ask /
Childish Hackenbush Hotchpotch / Atomic Weights of
Lollipops / Proving Things About Atomic Weights / Playing
Among the Flowers / When is g as uppity as h?
/ Go Fly A Kite! / All Remote Stars Agree / Large and Small
Flowerbeds / Playing Under a Lucky Star / General Multiples
of Up / Proof of the Remote Star Rules / Proof That Atomic
Weight = Uppitiness / The Wholeness of Hackenbush
Hotchpotch / Proper Care of the Eccentric / Galvinized
Games / Trading Triangles / Multiples of Positive Games /
Multiples Work! / First for the "With" Rule / Now for the
"Without" Rule / Shifting Multiples Of Up By Stars / A
Theorem on Incentives / Seating Families of Five /
references
Preface to the Second Edition, Preface, Change of Heart
9 If You Can't Beat 'Em, Join 'Em!
All the King's Horses / We Can Join Any Games / How Remote
Is a Horse? / What If the First Horse to Get Stuck Wins? /
A Slightly Slower Join / Moving Horses Impartially /
Cutting Every Cake / Eatcakes / When to Put Your Money on
the Last Horse / Slow Horses Join the Also-Rans / Let Them
Eat Cake / All the King's Horses on a Quater-Infinite Board
/ Cutting Your Cakes and Eating Them / References and
Further Reading
10 Hot Battles Followed by Cold Wars
Cold Games - Numbers Are Still Numbers / Hot Games - The
Battle Is Joined! / Tolls, Timers and Tallies / Which Is
the Best Option? / Hot Positions / Cold Positions / Tepid
Positions / A Tepid Game / Select Boys and Girls / Mrs.
Grundy / How to Play Mise`re Unions of Partizan Games /
Urgent Unions (Shotgun Weddings?) / Predeciders -
Overriders and Suiciders / Falada / Two More Falada Games /
Baked Alaska / A Felicitous Falada Field / The Rules for
Tallies on Infinite Tolls / Time May Be Shorther than You
Think!
11 Games Infinite and Indefinite
Infinite Enders / The Infinite Ordinal Numbers / Other
Numbers / Infinite Nim / The Infinite Sprague-Grundy and
Smith Theories / Some Superheavy Atoms / Fixed, Mixed and
Free / Onsides and Offsides, Upsums and Downsums / Stoppers
/ on, off and dud / How Big Is
on? / It's Bigger than All of Them! / Sidling
Towards a Game / Sidling Picks Sides / Stoppers Have Only
One Side / 'Tis!-'Tis n!-'Tis!-'Tis n!-... / Loopy
Hackenbush / Disentangling Loopy Hackenbush / Loopily
Infinite Hackenbush / Sisyphus / Living with Loops /
Comparing Loopy Games / The Swivel Chair Strategy /
Stoppers Are Nice / Plumtrees Are Nicer! / Taking Care of
Plumtrees / Working With Upsums and Downsums / on,
off and hot / A Summary of Some Sum
Properties / The House of Cards / The Degree of Loopiness /
Classes and Varieties / No Highway / Backsliding
Toads-and-Frogs / Bach's Carousel / Getting on in
Checkers / Proof of the Sidling Theorem / Answer to
Exercise One / tis and tisn / upon /
Backsliding Toads-and-Frogs / KOs in Go / References and
Further Reading
12 Games Eternal - Games Entailed
Fair Shares and Varied Pairs / How Soon Can You Win? /
There May Be Open Positions (O-Positions) /
Adders-and-Ladders / Just How Loopy Can You Get? / Corrall
Automotive Betterment Scheme / Sharing Out Other Kinds of
Nut / Fair Shares and Unequal Partners / Sweets and Nuts,
and Maybe a Date? / The Additional Subtraction Games /
Horsefly / Selective and Subselective Compounds of
Impartial Games / Entailing Moves / Sunny and Loony
Positions / Calculating with Entailed Values / Nim with
Entailing Moves / Goldbach's Nim / Wyt Queens with Trains /
Adding Tails to Prim and Dim / Complimenting Moves /
On-the-Rails / De Bono's L-Games / Proving the Outcome
Rules for Loopy Positions / Fair Shares and Unequal
Partners / Were Your Ways Winning Enough? / Did You Moves
First in Horsefly? / References and Further Reading
13 Survival in the Lost World
Mise`re Nim / Reversible Moves / The Endgame Proviso / The
Awful Truth / What's Left of the Old Rules? / As Easy as
Two and Two? / The Mise`re Form of Grundy's Game / Animals
and Their Genus / What Can We With the Genus? / Firm,
Fickle and Tame / Which Animals are Tame... / ...and Which
are Restive? / Some Tame Animals in the Good Child's Zoo /
Mise`re Wyt Queens / Jelly Beans and Lemon Drops / Stalking
Adders and Taking Squares / "But What if They're Wild?"
Asks the Bad Child / Mise`re Kayles / The Noah's Ark Theorem
/ The Half-Tame Theorem / Guiles / Dividing Rulers /
Dawson, Officers, Grundy / All Subtraction Games Reduce to
Nim / Prim and Dim / Proof of the Noah's Ark Theorem /
Mise`re Octal Games / Notes / Stop Press: Even More Games
are Tameable!
Preface to the Second Edition, Preface to the Original
Edition, Games in Clubs!
14 Turn and Turn About
Turning Turtles / Mock Turtles / Odius and Evil Numbers /
Moebius, Mogul and Gold Moidores / The Mock Turtle Theorem
/ Why Moebius? / Mogul / Motley / Twins, Triplets, Etc. /
The Ruler Game / Circumscribed Games / Turnips (or Ternups)
/ Grunt / Sym / Two-Dimensional Turning Games / Acrostic
Twins / Turning Corners / Nim-Multiplication / Swirling
Tartans / The Tartan Theorem / Rugs, Carpets, Windows and
Doors / Acrostic Games / Stripping and Streaking /
Uglification and Derision / Unlocking Doors / Sparring,
Boxing and Fencing / "Coins" (or Heaps) with Infinitely
Many (or 2^(2^{N}) "Sides" / References and Further
Reading
15 Chips and Strips
The Silver Dollar Game - Profit from Gaming Tables /
Antonim / Synoim / Simonim / Staircase Fives / Twopins
(pronounced "Tuppins") / Cram / Welter's Game / Four-Coin
Welter is Just Nim / And So's Three-Coin Welter! / The
Congruence Modulo 16 / Frieze Patterns / Inverting the
Welter Function / The Abacus Positions / The Abacus
Strategy / The Mis`ere Form of Welter's Game /
Kotzig's Nim / Fibonacci Nim / More Generally Bounded Nim /
Epstein's Put-or-Take-a-Square Game / Tribulations and
Fibulations / Third One Lucky / Hickory, Dickory, Dock /
D.U.D.E.N.E.Y / Strings of Pearls / Schuhstrings / The
Princess and the Roses / One-Step, Two-Step / More on
Subtraction Games / Smallest Nim, Largest Nim / Moore's
Nim_{k} / The More the Merrier / Moore and More /
Not with a Bang but a Whim / Did You Win the Silver Dollar?
/ How Was Your Arithmetic? / In Put-or-Take-a-Square, 92 Is
an N-Position / Tribulations and Fibulations / Our
Code of Behavior for Princes / References and Further
Reading
16 Dots-and-Boxes
Double-Dealing Leads to Double-Crosses / How Long Is
"Long"? / The 4-Box Game / The 9-Box Game / The 16-Box Game
/ Other Shapes of Board / Dots-and-Boxes and
Strings-and-Coins / Nimstring / Why Long Is Long / To Take
or Not to Take a Coin in Nimstring / Sprague-Grundy Theory
for Nimstring Graphs / All Long Chains Are the Same / Which
Mutations Are Harmless / Chopping and Changing / Vines /
Dots + Doublecrosses = Turns / How Dodie Can Win the 4-Box
Game / When Is it Best to Lose Control? / Computing the
Values of Vines / Loony Endgames Are NP-Hard / Solutions to
Dots-and-Boxes Problems / Some More Nimstring Values /
Nimbers for Nimstring Arrays / References and Further
Reading
17 Spots and Sprouts
Rims / Rails / Loops-and-Branches / Contours / Lucasta / A
Child's Guide to Normal Lucasta / The Mis`ere
Form of Lucasta / The Positions (7,3,1) and (11,1,1) /
Cabbages; or Bugs, Caterpillars and Cocoons / Jocasta /
Sprouts / Brussels Sprouts / Stars-and-Stripes / Bushenhack
/ Genetic Codes for Nim / Bushenhack Positions Have Genetic
Codes! / Von Neumann Hackenbush / The Joke in Jocasta / The
Worm in Brussels Sprouts / Bushenhack / References and
Further Reading
18 The Emperor and His Money
Sylver Coinage / How Long Will It Last? / Some Openings Are
Bad / Are All Openings Bad? / Not All Openings Are Bad /
Strategy Stealing / Quiet Ends / Doubling and Tripling? /
Halving and Thiding? / Finding the Right Combinations /
What Shall I do When g is Two / The Great Unknown /
Are Outcomes Computable? / The Etiquette of Sylver Cinage /
Chomp / Zig-Zag / More Cliques for Sylver Coinage / 5-Pairs
/ Positions Containing 6 / Sylver Coinage Has Infinite
Nim-Values / A Few Final Questions / References and Further
Reading
19 The King and the Consumer
Chessgo, Kinggo and Dukego / Quadraphage / The Angel and
the Square-Eater / Strategy and Tactics / Dukego / The Edge
Attack / The Edge Defence / A Memoryless Edge Defence / The
Edge-Corner Attack / Strategic and Tactical Stones / Corner
Tactics / Defence on Large Square Boards / The 33 x 33
Board / The Centred King / Leaving the Central Region / The
Cornered King / The Sidelined King / How Chas. Can Win on a
34 x 34 Board / Rectangular Boards / Many-Dimensional
Angels / Games of Encirclement / Wolves-and-Sheep / Tablut
/ Saxon Hnefatafl / King and Rook Versus King / References
and Further Reading
20 Fox and Geese
Some Properties of Our Strategy / What Is the Value of
Fox-and-Geese? / Fox-Flocks-Fox / Towards Greater Precision
/ The Indefinite Board / How the Geese Survive Triumphantly
/ Eight Exciting Escapades / Tactics and Startegy for the
Fox / Fox's Play on Definite Boards / The Scrimmage
Sequence / Values of the Initial Positions / Sophisticated
New Software / Values of FOXTAC Positions / High Region /
Welton's Region of Value 2over / Lower "Delta" Region /
Initial Values / Parity in the High Scrimmage Region /
Early Values / The abc's to Trifurcate the Two-Ish
Transition / Extending the Backbone to Lower Altitudes /
Early Values of Other Formations / Resolving the Exceptions
/ Museums / Solutions to Problems / Open Problems /
Maharajah and Sepoys / References and Further Reading
21 Hare and Hounds
The French Military Hunt / Two Trial Games / History / The
Different Kinds to Place / The Opposition / When Has the
Hare Escaped? / Losing the Opposition / A Strategy for the
Hare / On the Small Board / On the Medium and Larger Boards
/ Answers to Questions / A Sound Bound for a Hound? / All
Is Found for the Small Board Hound / Proof of the
Thirty-One Theorem / References and Further Reading
22 Lines and Squares
Tit-Tat-Toe, My First Go, Three Jolly Butcher Boys All in a
Row / Magic Fifteen / Spit Not So, Fat Fop, as if in Pan! /
Jam / How Long Can You Fool Your Friends? / Analysis of
Tic-Tac-Toe / Ovid's Game, Hopscotch, Les Pendus / Six
Men's Morris / Nine Men's Morris / Three Up / Four-in-a-Row
/ Five-in-a-Row / G0-Moku / Six, Seven, Eight, Nine,..., in
a Row / n-Dimensional k-in-a-Row / Strategy
Stealing in Tic-Tac-Toe Games / Hex / Bridgit / How Does
the First Player Win? / The Shannon Switching Game / The
Black Path Game / Lewthwaite's Game / Meander / Winners and
Losers / Dodgem / Dodgerydoo / Philosopher's Football /
Count foxy words And stay awake Using lively wit / Amazons
/ Checkers / Chess / Cherries / Clobber / Go / Konane /
Reversi-Othello / Scrabble / Shogi / Sowing Games / Answer
to Problems / References and Further Reading
Preface to the Second Edition, Preface to the Original
Edition, Solotaire Diamonds!
23 Purging Pegs Properly
Central Solitaire / Dudeney, Bergholt and Beasley /
Packages and Purges / Packages Provide Perfect Panacea /
The Rule of Two and the Rule of Three / Some Pegs Are More
Equal Than Others / Reiss's 16 Solitaire Position Classes /
The Continental Board / Playing Backwards and Forwards /
Pagoda Functions / The Solitaire Army / Managing Your
Resources / Unproductivity / The Prodigal Son's Opening /
Deficit Accounting and the G.N.P. / Accounting for Two-Peg
Reversal Problems / Forgetting the Order Can Be Useful /
Beasley's Exit Theorems / A Stolid Survivor Problem /
Another Hard Problem / The Spinner / Our Fine Finalist /
Doing the Splits / All Soluble One-Peg Problems on the
Continental Board / The Last Two Moves / A20-Man Solitaire
Army / Fool's Solitaire, ETC. / Beasley Proves Bergholt Is
Best / The Classical Problems / References and Further
Reading
24 Pursuicing Puzzles Purposefully
Soma / Blocks-in-a-Box / Hidden Secrets / The Hidden
Secrets of Soma / Hoffman's Arithmetico-Geometric Puzzle /
Coloring Three-by-Threee-by-Three by Three, Bar Three /
Wire and String Puzzles / The Magic Mirror Method / The
Barmy Braid / The Artful Arrow / The Magic Movie Method /
Party Tricks and Chinese Rings / Chinese Rings and the Gray
Code / The Tower of Hanoi / A Solitaire-Like Puzzle and
Some Coin-Sliding Problems / The Fifteen Puzzle and the
Lucky Seven Puzzle / All Other Courses for Point-to-Point /
Rubik's Hungarian Cube - Buvos Kocka / Just How Chaotic Can
the Cube Get? / Chief Colors and Chief Faces / Curling the
Cube / A: Aloft, Around (Adjust) and About / B: Bottom
Layer Corner Cubelets / C: Central Layer Edge Cubelets / D:
Domiciling the Top Edge Cubelets / E: Exchanging Pairs of
Top Corners / F: Finishing Flips and Fiddles / Explanations
/ Improvements / Elena's Elements / Are You Partial to
Partial Puzzles? / Other "Hungarian" Objects / A Trio of
Sliding Block Puzzles / Tactics for Solving Such Puzzles /
Counting Your Moves / Paradoxical Pennies / Paradoxical
Dice / More on Magic Squares / The Magic Tesseract / Adam's
Amazing Magic Hexagon / Strip-Jack-Naked, or
Beggar-My-Neighbour **1 / The Great Tantalizer /
Polyominoes, Polyiamonds and Searching Policy / Alan
Schoen's Cyclotome / MacMahon's Superdominoes / Quintominal
Dodecahedra / The Doomsday Rule / ... and Easter Easily /
How Old is the Moon? / Jewish New Year (Rosh Hashna) /
Blocks-in-a-Box / The Somap / Solutions to the
Arithmetico-Geometric Puzzle / ... and One for "Three" Too!
/ Hares and Tortoises / The Lucky Seven Puzzle / Top Face
Alterations for the Hungarian Cube / The Century Puzzle /
Adams's Amazing Magic Hexagon / Flags of the Allies
Solution / All Hexiamond Solutions Found / The Three
Quintominal Dodecahedra / Answer to Exercise for Experts /
Where Do the Black Edges of MacMahon Squares Go? / A
Doomsday Answer / References and Further Reading
25 What is Life?
Still Life / Life Cycles / The Glider and Other Space Ships
/ The Unpredictability of Life / Gardens of Eden / Life's
Problems are Hard! / Making a Life Computer / When Glider
Meets Glider / How to Make a NOT Gate / The Eater / Gliders
Can Build Their Own Guns! / The Kickback Reaction /
Thinning a Glider Stream / Building Blocks for Our Computer
/ Auxiliary Storage / How We Move Blocks / A Little
Difficulty / Mission Completed - Will Self-Destruct / Life
is Still Being Lived! / Life Computers Can Reproduce! /
Genetic Engineering / Whither Life? / References and
Further Reading