Some Things You May Not Know About Chess
Chess and mathematics are intimately related
By Dan Talpalariu, Science Editor
1st of December 2008
When it comes to chess, most people see themselves as falling into the category labeled “I have some skill, I may even beat my neighbors or my little niece, but I’m not really a grandmaster.” The last part is perhaps even more true, when it comes to maths. Well, to prove this to you beyond doubt, here’s a couple of facts that will show you how chess and maths are related and, most likely, that there are still many things you don’t know about either of them.
We’re not going to provide a list of the top Russian Olympic winners, but speak about the beginnings of the sport instead. According to the legend, the basics of the game were developed by an Indian mathematician, and soon the game was very appreciated and famous. Everybody – including the King of India – was playing it. In fact, the King liked it so much that he summoned the mathematician in order to offer him a prize – any prize – for his feat.
So as not to seem greedy, he only asked that he was payed in rice grains placed according to the chessboard squares: one for the first, 2 for the second, 4 for the third, and so on, doubling with each new square. There were only 64 squares, so how hard could that be to pay him, surely the King (and maybe you) thought. However, by the 16th square there was already a kilogram of rice and 16 kilos at the 20th. The last square was never reached, but if the King could afford it, it would have summed 18,446,744,073,709,551,615 grains.
The second fact involves the total number of possible chess games expressed in chessboards. So, the mathematicians said, if we were to line up chessboards in a line that would stretch from one end of the observable universe to the other end, the number of chessboards would be somewhere in the 28 digits. Yet, that’s only about a fifth of the total calculated chess game possibilities, which rise to approximately 10120 (1 followed by 120 zeros). Hopefully, this indicator of the complexity and uniqueness of each chess game will rather attract than deter you from playing it.
Source: http://news.softpedia.com
An amazing connection between chess and computer memory: Take any program P that runs using s memory registers, and which gives yes/no answers. Working generally much faster than it takes to run P, I can produce a chess position P’ on a roughly s x s board, such that White to move can win if-and-only-if P will answer “yes”.
Thus a lightning-fast chess-problem solver could jump the gun on any computer program. Want to know if there’s a way you can save another $100 on a big & complicated tax return? If it’s big enough, I can turn this into an equivalent chess problem quicker than TurboTax (TM) can tell you the answer. And if TurboTax tries to speed up their program, they will inevitably make mistakes (unless the answer to this Millennium Prize Problem is a shocking “equal”).
The caveats that keep this from being practical are: (1) the s x s board needs much bigger area than the original size-s computer chips, (2) the speedup translating your taxes into chess would only be noticeable for Star Wars Galactic Republic-sized tax returns (no wonder it fell:-), and finally (3), the resulting chess problems would be rather harder than the likes of John Nunn are good at solving.
Still, in terms of my field, this is “within the realm of practicality”, and is summed up by saying that “all Polynomial-Space problems are reducible to Chess.” Thus chess was taken as the exemplar of my field in this item in a major CS theory blog. If you play without any adaptation of the 50-move rule to larger boards, then an ostensibly-bigger class called Exponential Time reduces to Chess. Thus the computer Deep Thought in the Hitchhiker series really could have been the chess program that was later named for it! However, before we get too prideful about chess, all of what I’m saying is also true of other games, including generalized tic-tac-toe!
I think in such a complex problem the simple calculations should be carefully done to avoid simple mistakes that take the beauty out of the story.
It sais: “…the number of chessboards would be somewhere in the 28 digits. Yet, that’s only about a fifth of the total calculated chess game possibilities, which rise to approximately 10120 (1 followed by 120 zeros).”
Now, a number in the 28 digits IS NOT a fifth of 1 followed by 120 zeros..its the fifth ROOT. The first number is almost zero near the second..not a fifth.
hmmm maybe I’m missing something, but the way I see it, there should be 9,223,372,036,854,780,000 grains of rice by the time the 64th square is reached. The number mentioned in the article would be for the 65th square.
Yes, i think you have missed that you must sum the grains on each square to get the number of grains on the board. And the number of grains on the 65th square is one more than the number of grains on the chessboard.
anon: I guess I had a different understanding. I thought the article was talking about the number of grains on the 64th square, not the cumulative total.
Yes, the article is ill written, as pointed out here before.
Yes, the article is ill written, as pointed out here before.