Is it solvable in a human mind? Not by knowing all the positions, but via abstract strategies at least? Can a human player play perfect?
Again prove what you have just said without ad homs. Don’t insult me.
I don’t even know what “what you have just said” is even referring to.
Anyway, pointing out that you’ve gone off course isn’t an ad hominem, it’s just true. Zermelo’s Theorem isn’t about human beings, it’s about abstract statements you can make about a game.
There are games which Zermelo’s Theorem doesn’t apply to.
I’ve just quoted it to you.
In case you did not notice the flipping obvious, l already treated the pure arithmetic of the theorem here, showing it to be solvable:
I then went on to show that this is shown to be absurd because it implies unfulfillable conditions, and therefore it is unsolvable.
The unfulfillable conditions always implicit, e.g. the cosmos itself, would make it unsolvable.
You want me to prove that you’ve gone off course? From what Zermelo’s theorem is about?
No, that l need a “full reset” and that l am irrelevant here. In fact as l’ve shown, l already solved the problem.
But proceeded to cast doubt on that finding. I have encompassed the problem fully, the opposite of the shade you are casting. You should not ad hom. It is your style, l see. It is weakness.
Asking me to name a player is a complete confusion on your part. Name a player? WHAT ARE YOU TALKING ABOUT?!?!? It’s a nonsense thing to say. Name what player? What are you actually asking? You’re just saying random words.
Name a singer. Name a singing raisin. Why are you saying to name random things? It’s not contextually relevant. The fact that you do think it’s contextually relevant to say “name a player” means you need to take a step back and reassess what the conversation is about. “Name a player” is as contextually relevant as demanding that i “play a tune”. It’s just not a thing that fits in the convo.
I have already shown chess to have a finite number of sensible games, hence it is solvable (unsolvable = infinite sensible games):
I then ad hoc-ed to show this is absurd regardless (hence "name a player where it mighht be fulfilled e.g. a computer etc. - they would not last the run of permutations). Do not ad hominem me when l have encompassed the problem more fully than yourself.
You mean Nine mens morris? No, it can actually be a surprisingly tricky game and imo no human will ever be perfect at it. But good players usually draw each other.
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The only way the run of 10^50 permutations would be possible would be to invoke the supernatural.
Here is a perfect Nine mens morris computer http://ninemensmorris.ist.tugraz.at:8080/
It even tells you if the position is draw or if either side is winning, how many more moves the game will last with perfect play.
Tell me something: How is it even a question, when the winning permutations are obviously finite?
oh wow, interesting.
I think it’s fascinating that, despite it being a solved game, there’s no abstract rules available that would allow a human to have a perfect strategy. Connect 4 is also solved, I wonder if humans can fit some version of the solution in their head for that game.
I played Nine mens morris a lot, no chance for a set of abstract rules that work every time. Games can go wildly different, some are boring drawish, some are super violent, sometimes it’s about capturing, sometimes it’s about blocking etc etc. There are some funny exceptions to strategies that usually work. I would say that strategies don’t exist at all.
I tried to look for Zermelo’s theorem. The original version was published in 1913 in German (4 pages article). I do not speak the language. It appears that the author uses other known theorems or lemmas from abstract mathematics to derive certain conclusions. Other authors have tried to draw similar conclusions. 13 years later, König argued that the original proof wan incomplete. Kalmar also worked on the topic and generalized the theorems.
In their conclusion, Schwalbe and Walker stated for the three authors:
“The common starting point for their analysis was the concept of a winning position, defined in a precise mathematical way: If a player is in a winning position, then he can always force a win no matter what strategy the other player may employ. They then sought an answer to the question: Given that a player is in a winning position, is there an upper bound on the number of moves in which he can force a win? Or, if he is in a losing position, how long can a loss be postponed?”
Source: Ulrich Schwalbe, Paul Walker, 2001. “Zermelo and the Early History of Game Theory”.
Games and Economic Behavior, 34 (1), 123-137.
At this point my remark is that this analysis treats the game of chess without considering: a) that there is time limitation in every chess game format, which needs to be taken into account while the players (potentially computers, since for humans it will be an impossible task) try to compute the perfect moves, b) the 50 move rule: if there is no pawn move and no capture for 50 moves, the game is automatically a draw.
I think for the sake of zermelo’s theorem it’s typical to set the time limit aside (since that’s not in principle something you can even take into account when analyzing the game at a theoretical level), but consider rules like 3-repetition and 50 move rule as valid draws.
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If what l cited, 10^40 sensible games is correct, then there’s a specified cap on the number of no-lose chess games.
Also, it seems logical that the number is finite anyway, given the rules.
This makes Chess solvable.
However, the won’t be enough energy to do all those games so it is by definition impossible. I focus on the word “players”, which becomes an absurd term.
I believe l have covered the problem fully. The 10^40 figure comes from Numberphile’s treatment of the Shannon number.
I have an idea, every time a mathematician talks about a number greater than 10^80, let’s smack them in the head. Because that’s a bigger number than there are atoms in the universe or whatever. That’s not allowed.
Please no one tell the cat about tree(3)
being forced into these are not a draw