# Contents

Preface to Second Edition
Preface
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?
Do Our Methods Work?
What is a Game?
When Is a Move Good?
Figure 8(d) Is Worth 3/4

2. Finding the Correct Number is Simplicity Itself
Which Numbers Are Which?
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
Col-lections and Col-lapsings
Maundy Cake
Another Cutcake Variant
How Childish Can You Get?

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
Wyt Queens
Reversible Moves in General Games
Deleting Dominated Options
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
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

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
Some Remarks on Periodicity
Standard Form
A Compendium of Octal Games
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

5. Numbers, Nimbers and Numberless Wonders
Domineering
Switch Games
Cashing Cheques
Some Simple Hot Games
The Tiniest Games
Modern Management of Cash Flow
Seating Boys and Girls
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

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

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
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
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