Winning Ways for Your Mathematical Plays, Volume 2

Elwyn R. Berlekamp

• Publisher: A K Peters/CRC Press
• Year: 2018
• ISBN:
  9780429945601
• Edition: 2
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

In the quarter of a century since three mathematicians and game theorists collaborated to create Winning Ways for Your Mathematical Plays, the book has become the definitive work on the subject of mathematical games. Now carefully revised and broken down into four volumes to accommodate new developments, the Second Edition retains the original's wealth of wit and wisdom. The authors' insightful strategies, blended with their witty and irreverent style, make reading a profitable pleasure. In Volume 2, the authors have a Change of Heart, bending the rules established in Volume 1 to apply them to games such as Cut-cake and Loopy Hackenbush. From the Table of Contents: - If You Can't Beat 'Em, Join 'Em! - Hot Bottles Followed by Cold Wars - Games Infinite and Indefinite - Games Eternal--Games Entailed - Survival in the Lost World

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Games Infinite and Indefinite

“I find”, said ’e, “things very much as ’ow I’ve always found,

For mostly they goes up and down, or else goes round and round”.

Patrick Reginald Chalmers, Green Days and Blue Days, Roundabouts and Swings.

For ever and ever when I move.

How dull it is to pause, to make an end.

Alfred Lord Tennyson, Ullysses, I.21.

Infinite Games

Most of our examples in Winning Ways have had only finitely many positions. But a game can have infinitely many positions and still satisfy the ending condition. A few such are described in the first part of this chapter. More interesting are the games obtained by dropping the ending condition. which we call loopy games, since it’s often possible to find oneself returning to the same position over and over again. In Chapter 12 we’ll describe, amongst other things, C.A.B. Smith’s complete theory for impartial loopy games, but in this chapter we’ll see that the theory of partizan loopy games is completely different.

The ideas in this chapter are hardly used elsewhere in the book.

Infinite Hackenbush

You may have wondered why numbers like $\frac{2}{3}$ don’t appear as values. The answer is, they do! To see how, let’s look at some Hackenbush positions that get near $\frac{2}{3}$.

Maybe the value of an infinite alternating beanstalk will be exactly $\frac{2}{3}?$ If so, Fig. 1 should be a second-player win, since its value is

$\frac{2}{3}+\frac{2}{3}+\frac{2}{3}-2=0.$

Figure 1. Two-Thirds of a Move Can Take a Long Time.

After any Left opening move the value will be

$\left(\frac{2}{3}-\epsilon \right)+\frac{2}{3}+\frac{2}{3}-2$

and by moving one higher up in another beanstalk, Right c...