Celebratio Mathematica — Berlekamp

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E. R. Berlekamp: “Sums of $N \times 2$ Amazons,” pp. 1–34 in Game theory, optimal stopping, probability and statistics: Papers in honor of Thomas S. Ferguson. Edited by F. T. Bruss and L. M. Le Cam. IMS Lecture Notes Monograph Series 35. Institute of Mathematical Statistics (Beachwood, OH), 2000. MR 1833848 Zbl 0988.91012 incollection

Amazons is a board game which has enjoyed some popularity since its appearance on the Internet a few years ago. It can be played on boards of arbitrary size. Like subtraction games studied by Ferguson [1974], Amazons is two-person, perfect-information, and zero-sum. Nevertheless, sums of games such as Amazons offer very interesting examples of “central limit theorems” which turn out to be considerably stronger than those encountered in probability theory.

As an illustration of the power of combinatorial game theory, we offer an analysis of all starting positions of Amazons played on sums of $ N\times 2 $ rectangles, in which each player has one Amazon in each rectangle. The analysis reveals two common yet important phenomena which many players overlook.

Although we assume no special background, seasoned combinatorial game theorists will find that Amazons provides our motivation for a novel exposition of the subject, starting with “results-oriented” thermography. This approach gets to the main results about hot games much more directly than prior expositions, which begin with Conway’s theory of canonical forms and a special emphasis on numbers. From the new perspective, numbers appear much later as a special case: they are the games whose temperatures are negative.

Thomas S. Ferguson	Related
Franz Thomas Bruss	Related
Lucien Marie Le Cam	Related