Given the rule that “no set of moves may fully repeat itself three times or the game is a stalemate,” is the game of chess finite, given that one eventually will have to repeat oneself, or does that rule still allow an infinite set of games?
It can’t be infinite with a finite set of possible moves, even if that number is hard to contemplate for a human.
No, it is not infinite but it the possible moves in a chess game is enormous. The chances of repeating a game are slim but it can be done.
Whoops…forgot the copy/paste method didn’t bring over the superscript in my quote above properly. That 1 with all those zeros after it is meant to be 10[sup]120[/sup].
Dang…and…
10[sup]26[/sup] nanoseconds since the Big Bang.
10[sup]75[/sup] atoms in the entire universe.
I’m thinking the rule is that after 3 consecutive identical moves the game is a stalemate. You could imagine, sort of, a game going on forever, being passed down from generation to generation. This assumes some collusion between the players.
But if you make the more reasonable assumption that all games either eventually end in a victory or stalemate in fifty moves you end up with a whole butt-load of possible games, but somewhat short of infinite.
There are twenty possible first moves for both black and white. As pieces are moved more pieces come into play. Let us assume that, on average there are forty possible moves each turn for both sides (as pieces are taken you will start to lose possible moves).
So in a forty move game you would have (30^40)*(30^40) possible positions, which is 1.5X10^118.
With the entire population of the earth playing at 100 moves per second it would take 1.6X10^99 years to run through them.
None of which exactly answers your question. But it kept me from some chores I’d just as soon put off doing.
I would say that it is infinate to some degree. You could get down to a situation where you are down to 2 kings only - each king can move and does not have to repeate a pattern but it will never end. This could also happen with a king & bishop vs king & bishop - and I’m sure many other combinations.
Not only is the game of chess finite, it also has a solution.
Mathematician John NAsh – (“A simple mind”) wrote that, if I remember correctly, “any game with a finite number of moves has at least one equlibrium outcome” That is, there is at least one result (or strategy) from which neither player would wish to deviate. In other words, there is at least one strategy, or algorithm, for playing chess that will be favored, no matter what.
Somebody, somwhere, is going to build a big enough computer to figure it out, and take all the mystery and strategy out of the game. Whether the equilibrium result(s) will be a stalemate or a victory for one side, I don’t know.
Think of it this way. You used to play tic tac toe until you figured out how to play in order to guarantee a stalemate no matter what. Then it got boring, and you haven’t played tic tac toe since.
I am sorry if I am not making sense. I hope a mathematician, a chess player or computer person can make better sense of what I’m saying than I can,
So there is not enough space in the universe on which to actually record all the possible moves (unless of course we develop a way of storing data at subatomic bit scales) - I take it that it is possible to mathematically define the tree though, is it? (so that one could calculate and examine any given portion)
The primary rule which constrains long chess games is not the “draw by repeated position” rule but the “50 move rule”. Either player may claim a draw after 50 moves have passed without a pawn move or a piece capture.
So the longest possible chess game would be to prance your knights around for 50 moves, move a pawn, prance your your knights around for 50 more moves, move another pawn . . . and so on. When the pawns can’t move any more start capturing pieces.
According to “The Complete Chess Addict” by Mike Fox and Richard James, if you carry this process out to the limit, the longest possible chess game is 5,999 moves. And this obviously puts a ceiling on the number of possible games.
However . . . if you want to be hyper-technical, both the 50-move rule and the repeated position rule allow, but do not require, a draw to be claimed. So you can, in principle, create an infinite number of possible chess games just by moving knights around for an indefinite length of time, with neither player claiming a draw.
IIRC, this rule has been recently withdrawn from the international chessrules, since it could be proven that winning strategies are possible with more than 50 moves and not moving a pawn or piece capture.
No that is not the rule. The rule is - if the same position happens 3 times in a game a player can call for a draw.
FIDE laws of chess
Didn’t know that, but I do see that the rule is still mentioned in the FIDE laws of chess (see section 9.3 at the link I just posted.)
Yeah, pretty much. In theory, it’d be possible to write down a strategy for chess, so that no matter what your opponent does, you just have to look up the board arrangement and do the move written down.
Of course, looking it up could take a very long time.
Oh, and 10[sup]120[/sup] isn’t a large number.
I think it’s the rule regarding repetition that keeps it from being infinite, not the finite number of moves. Checkers, for instance, has an infinite number of possible games, because you’re allowed to repeat positions (AFAIK, anyway; I’ve never played professional checkers).
Also, while the fact that chess has an optimal strategy is a good point, that fact itself does not make the game finite, because even when you’re not playing by the optimal strategy, you’re still playing the game.
What would you write it on; even if you managed to write each move on a single atom (and assuming you had all the atoms in the universe at your disposal), you’d run out of space about halfway.
How big is 10[sup]120[/sup] in relation to googol?
It’s 100,000,000,000,000,000,000 googol.
A googol is 10[sup]100[/sup], so 10[sup]120[/sup] is a googol multiplied by 100,000,000,000,000,000,000
A googol is 10[sup]100[/sup]. The game of chess has about 20 orders of magnitude more moves than a googol.