Contents

Preface to Second Edition
Preface
SpadeWork!
 Whose Game?

BlueRed Hackenbush
The Tweedledum and Tweedledee Argument
How Can You Have Half a Move?
And Quarter Moves?
SkiJumps for Beginners
Don't Just Take the Average!
What Is a Jump Worth?
ToadsandFrogs
Do Our Methods Work?
What is a Game?
When Is a Move Good?
Figure 8(d) Is Worth 3/4
References and Further Reading  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
Collections and Collapsings
Maundy Cake
Another Cutcake Variant
How Childish Can You Get?
References and Further Reading
 Some Harder Games and How to Make Them Easier

PokerNim
Northcott's Game
Bogus NimHeaps and the Mex Rule
The SpragueGrundy Theory for Impartial Games
The White Knight
Adding Nimbers
Wyt Queens
Reversible Moves in General Games
Deleting Dominated Options
ToadsandFrogs 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 NimAddition 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
 Taking and Breaking

Kayles
Games With Heaps
PPositions and NPositions
Subtraction Games
Ferguson's Pairing Property
Grundy Scales
Other TakeAway Games
Dawson's Chess
The Periodicity of Kayles
Other TakeandBreak Games
Dawson's Kayles
Variations
Guiles
Treblecross
Officers
Grundy's Game
Prim and Dim
Replication of NimValues
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 NonArithmeticPeriodicity Theorem
Some Hexadecimal Games
Further References and Reading
 Numbers, Nimbers and Numberless Wonders

Domineering
Switch Games
Cashing Cheques
Some Simple Hot Games
The Tiniest Games
Modern Management of Cash Flow
Tiny ToadsandFrogs
The Opening Dissection of ToadsandFrogs
Seating Boys and Girls
ToadsandFrogs Completely Dissected
ToadsandFrogs with Two Spaces
More Domineering Values
References
 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
Coolingand 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 FourStop Example
The ChequeMarket 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 ProfitConscious 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
 Hackenbush

Green Hackenbush
Green Trees
Fusion
Proving The Fusion Principle
A More Complicated Picture
Impartial Maundy Cake
BlueRed 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
BlueRed 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
ManyWay Maundy Cake
Solution to Figure 15
Tracks Cleared Through the Amazing Jungle
How Hard Was The Bed?
NPHardness
The Bottle at the End of Chapter 7
References and Further Reading
 It's a Small Small Small Small World

Uppitiness and Uncertainty
Computing Atomic Weights
Eatcake
Splitting The Atom
TurnandEatcake
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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Copyright © 19962018 Alexander Bogomolny
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