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We have to be careful though, because while the growth rate is hyperastronomical even by the 3rd move by black, as games progress some branches die (checkmates appear).
Plus we have to account for draw control. And stalemates.
Here's my question: How many checkmates are there? (Of any natural number of moves.)
I think I'm funny.
Chess isn't infinite. It can be counted, but the numbers are very high. And there are also other factors affecting it including checks.
There are so many possible moves yes. Due to the static and dynamic nature of the game many possibilities exist on a small board
Due to the static and dynamic nature of the game many possibilities exist on a small board
That doesn't make sense.
You should lose some of that belly fat if you wanna look ripped.
There are an infinite number of different sizes on infinity. it might help if we could define which sort of infinity we are talking about here.
It's not infinite when there clearly are only a finite number of possible moves.
Only two things are infinite: the universe and human stupidity; and I'm not sure about the universe.
I haven't read this entire thread by any means (mainly the initial question), but first define "[i]s it infinite?".
Think of it this way. If each player had two pieces and the board had four squares, would "it be infinite?" No. (Kind of cramped.)
If each player had four pieces and the board had 16 squares, would "it be infinite?" No.
If each player had six pieces and the board had 36 squares, would "it be infinite?" No, but we are getting into lots of permutations now (relative to what our threy grey cells can comprehend).
Theoretically, it seems a chess game could be desinged wherein "it could approach infinity" in the limit. Those are my first thoughts.
Even the alcohol in this world is finite. You'd think it's infinite.
But I just pulled the last beer out of my fridge so I can sing a song about it.
No no no. Let's make something clear:
As referred to, yes (given the axiom of choice), we may define at least two and quite possibly many (see continuum hypothesis) cardinalities (sizes of things, here, infinity). Certainly there are at least countable and uncountable. Countable is the size of all the natural (whole) numbers. Uncountable is the size of all the real numbers, or in fact, all the real numbers in, say, the arbitrary interval (0,1). You should not think of infinity as a process (say, in the loosehanded and corrupt reference to limits - recall that .9999... = 1). Instead, like limits (where they exist), they are (best considered) as actual quantities. Be careful of how you think about infinity. Consider, for example, that it is a well-proven theorem that there is a 1-1 relationship that will take every point in the punctured plane (say, the Euclidean plane set minus the origin) and map it to a unique point in the unit circle, and vice versa, such that we say, "The punctured plane is isomorphic to the unit circle."
In terms of chess, the possible number of games is certainly not uncountable, but seemingly not countable, either, and is finite. This makes sense because, in considering the games to count, we should only count once the games that draw, and every game should follow the current rules (of course). Note that, for draws, the repetitions do not have to be consecutive.
How many 960 games are there?
Huh? That was pretty random.
Chess.com may force a draw when 50 moves are met
LongIslandMark is right, if we did not have the three-fold repetition move nor the 50-move rule. Players can play the same moves over and over again non-stop,
Yeah, perpetual check etc. We all know.
Chess would be "infinite" if there were not the 50-move and three-fold repetition rules.
I count repetition as the same. Let's ignore 50-move rule
If you're asking about possible positions, then chess is finite. There are only 64 squares on the board, and under 96 possible pieces. (between 2 and 32 on the board at a time, but 16 of them can promote to any of 4 pieces. On the other hand, some of the pieces are indistinguishable, which reduces the distinguishable positions). Some positions are illegal, for example you'll never see a white pawn on row 1, or both kings in check, though those are a small proportion of the possible positions.
The numbers are so mind-bogglingly huge that the various complications make no practical difference: the number of positions is somewhere in the neighborhood of a forty-digit number, without even considering promotions. That's over a thousand billion billion billion billions. But that's still a very long way (infinitely far, in fact) away from infinite.
If you're talking about *games*, then it's still finite, if you impose a mandatory draw after threefold repetition. Since there is a finite number of positions and every move changes from one position to another, eventually you must repeat your position (if the game does not end by checkmate or resignation or agreed draw before then). [Or by fifty-move rule.]
Without a mandatory threefold repetition rule, you could have a game where two players agree to circle each other aimlessly (with, say, a queen each) for an arbitrary number of moves, then one player wins. That's one game. Repeat that game, but circle one more time before winning. That's another game. Clearly, there is an infinite number of games like that. Most of them would take longer than a human lifetime to play, but that wasn't part of the question.
Clearly, there are a finite number of types of "games like that": a queen each, a rook each, two rooks each, a queen vs a rook, ... So the number of possible games (without a mandatory threefold repetition rule) is countably infinite.
(Instead of a mandatory threefold repetition rule, you could have a mandatory fifty-move rule, but that seems too easy. Fifty moves + pawn moves, reset if there's a capture (but there's only 32 pieces to capture) - anybody can see that's finite.)
I love maths.
Me too, lol. :D
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