In the game of Tic Tac Toe (knots and crosses for our English brothers out there), it doesn’t take long to learn the strategies to employ so that you can guarantee that you can AT LEAST get a tie in every game, and possibly win depending on mistakes from your opponent. This is what we call a “solved game” - you can gurantee a minimum result, we know how to gurantee it, there’s no way around that strategy and it’s fully understood.
Zermelo’s Theorem is a theorem that there is a much broader category of game that you can apply that same sort of thinking about. This broader category includes games like Chess and Go, games that aren’t solved, games that are so complex and massive in the space of possibilities that they seem impossible to solve. Zermelo’s Theorem states, effectively, that Chess and Go are just like Tic Tac Toe in that way - that there is a flow-chart of perfect strategy, and if the perfect flow chart were available to both players, one of the following statements would necessarily be true:
White can gurantee a win
Black can gurantee a win
Both players can gurantee a draw-or-better (a draw, when they’re both perfect)
Zermelo’s Theorem may seem intuitive to some of you. It seems intuitive to me (I thought of the idea indepdently, though certainly not the proof). Yet intuitive as it is, it still feels… shaky, somehow. Like if I were to be challenged on it, I certainly wouldn’t be able to prove it. I’m not sure I’d even be able to make a great argument for it. Not in the context of something like Chess.
How can you, in plain language, convince another rational person that Chess is as in-principle-solvable as Tic Tac Toe is? I’ve not read the proof of Zermelo’s Theorem myself, but I doubt it’s as plain-language as I’m looking for.
The number of possibilities in chess is finite, therefore chess is in-principle-solvable. Finite number of pieces, finite board, finite game length (can’t be more than 10000 moves or so).
Circa 10^40 sensible chess games, hence it’s still beyond the human mind. It is estimated that there are between 10^{78} and 10^{82} atoms in the observable universe.
I really dont understand this question.
Chess has a limited number of moves. Its estimated to be around 10⁴⁰–10⁵⁰ legal positions.
What do you mean convincing?
If a person does not understand what it means that chess has a maximum number of possible moves, then its pointless to try and convince them of anything because they do not have the intellectual capacity to understand what you are talking about.
This is not a question about convincing, this is a question about people who are incapable of processing information.
That’s not what we’re convincing them of. We’re convincing them that there’s a possible flow chart for white to follow to gurantee a win, OR that flow chart exists for black, OR that flow chart (like tic tac toe) exists for both sides to guratantee at least a draw.
I lean toward a proof by negation, because intuitively I can’t imagine what it would look like for the theorem to be false.
One possibility would be if a chess game could be infinitely long, in which case a player could force an infinite game. But there are rules that prevent that (triple repetition and fifty-move rule draws).
The only other possibility seems to be some kind of indeterminacy. But chess is a perfect information game with no element of chance, so it can’t be indeterminate with perfect play either.
And if it can’t be otherwise, then the theorem must be true.
The theorem is false with chess (i.e. it’s unsolvable) because of the sheer number of sensible ways to win a 40-round game - 10^40. The limits of statistical probability when l were young was somewhere stated to be 1:1x10^50. It’s just not possible to cover all bases and that’s why there’s always going to be a market for new chess computers and always the chance of beating one.
What this means to me is, one can never approach a chess board and know it will end in win or draw.
Never heard of 50-move rule, and l hope l never need it lol.
From the fact that there is a fix number of moves there is a direct conclusion that you can make a flowchart.
I stand by the previous statement. You cant convince people about anything if they cant understand the concept that is being talked about.
The OP said in-principle-solvable, not in-practice-solvable. Chess is solvable in principle, but it’s unfathomably beyond humanity’s (current) ability to actually solve it.
I wouldn’t say “with perfect play”, rather that there can be a no-lose strategy. Unsure how perfect would even be defined here.
There are a finite number of moves available in a chess game - the rules see to it. Also bear in mind that the opponent is trying to win too. Sensible moves are presumed too *, bear that in mind.
So it’s a foregone conclusion that it’s solvable in principle but then, the solvability of a game would never be a question.
The question is always about players playing.
If players are playing then it’s unsolvable - especially with humans but computers too. They’d disintegrate and l’m unsure if there’d even be enough energy in the cosmos to power them up to even half the sensible games count (half of 10^40 = 5x10^39 right?)
Asterisk: Sensible moves otherwise we can move one step forward, then back , repeat eternally
It really is not my friend. It’s like saying: Can the perfect music concert ever occur?
One which caters to all tastes?
Your answer: Yes, because there are finite 5-minute songs that can be performed.
The real answer: No, because people are playing and even if it were a jukebox on stage, the jukebox would crumble to dust and the cosmos would reach heat death powering it all, before the concert is even 1% through,
Show me how this analogy is wrong.
Bear in mind that the solvability of tic tac toe was in question and dealt with in the OP. For the solvability of tic tac toe to be a matter for consideration, it is implicit that humans be playing. Because any hardened criminal could tell you there are only so many games you can get out of that.
After a few more seconds’ thought, the permutations of chess become clearly finite.
So again, that is never the question. The question is players playing.