Contents

    Preface to Second Edition
    Preface
    Spade-Work!
  1. Whose Game?
      Blue-Red Hackenbush
      The Tweedledum and Tweedledee Argument
      How Can You Have Half a Move?
      And Quarter 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 Number is Simplicity Itself
      Which Numbers Are Which?
      Simplicity's the Answer!
      Simplest Forms for Numbers
      Cutcake
      Maundy Cake
      A Few More Applications of the Simplicty Rule
      Positive, Negative, Zero, and Fuzzy Positions
      Hackenbush Hotchpotch
      Sums of Arbitrary Games
      The Outcome of a Sum
      The Negative of a Game
      Cancelling 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 Col-lapsings
      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| 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 Speed
      Games Displaying Arithmetic Periodicity
      A Non-Arithmetic-Periodicity Theorem
      Some Hexadecimal Games
      Further References and 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 Snort Dictionary
      Proof of the Number Avoidance Theorem
      Why Thermostrat Works
      Blockbusting
      ... And 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
      At omic 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 World
      Uppitiness 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

    Index

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