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Chess Inexhaustible

by Andrew Sacks

Certainly a game featuring a specific number of pieces that move — although variously in their functions — in a limited number of ways on a board of 64 squares only, must be finite in its possibilities of variation. The number of legal positions and of potential differently played games simply must be absolute and measurable.

All true enough. We need no mathematician or statistician to demonstrate the point; logic dictates the conclusion. Chess cannot be as limitless as life itself.

Or can it?

I would offer that the average intelligent non-chessplaying layman, if asked about such matters, would respond in something like the following manner: “Well, obviously there must be thousands of possible games and positions, maybe even hundreds of thousands, but I think that may be pushing it...”

In the initial array — the set-up of the 16 pieces for each side in a traditional chess game — the player with the White pieces chooses among 20 possible moves to start the contest. Regardless of the move selected, Black has 20 in reply. How many possible positions have we after only one move on each side? Not 20 plus 20, but 20 times 20 or 400. The surprises of cold, calculating mathematics are already beginning to assert themselves.

What about after two moves on each side? 800? Don’t be silly. Rather, 71,852 seems to be the number most mathematicians and statisticians currently agree upon. And after four moves on each side we jump well into the millions, about 988 of them, all possible and perfectly legal within the rules of play.

Are we dizzy yet? Just wait.

The average chess game lasts somewhere around 40 moves for each player. The number of possible 40-move games has been estimated at between 10 to the 43rd and 50th power, staggering totals that approximate the number of electrons in the known universe.

In 1950, American mathematician and information theory pioneer Claude Shannon, who was advising about chess computer programming, arrived at what is now known as the Shannon Number. It approximates the game-tree complexity of chess. It includes battles that may last far more than 40 moves — always a distinct possibility in both tournament and offhand games — at over 10 to the 120th power. Such a number is so overwhelming as to defy any comparison we can easily comprehend.

The number far exceeds the number of atoms in the entire observable universe — estimated at 10 to “only” the 80th power — meaning that for each and every atom in the known universe one could count for it, individually, some 10,000,000,000,000,000,000,000,000,000,000,000,000,000 distinct and different chess games.

So, is chess finite? Must be. Logic dictates that it is. But maybe it’s just a little broader than we thought.

Copyright © 2016 by Andrew Sacks

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