If a game continues indefinitely, repetition is guaranteed to occur.
If repetition occurs in a game where both sides are using best play, the continuation will follow the exact same line as the first time the position was reached, eventually resulting in a third repetition....
We can split hairs about whether the rules dictate that the game be stopped at a certain point, or the repetition be left to occur indefinitely, but it's irrelevant as the game is drawn in either case.
Because some won't believe it the whole exercise is moot?
It's a good thing that Copernicus, Newton and Galileo didn't have the same additude or we would never have made it to the moon either.
Just because it's an esoteric exercise doesn't mean it's not one worth pursuing.
Understand now? People will play anyway. That is why "solving" chess is moot.
There is nothing really to solve, actually.
I do agree that when it finally is solved it will have little to no consequence on the way people play, but it's still an academically interesting exercise and one that I'm sure will yeild advancements that can be leveraged for other more practical applications.
Understand now? People will play anyway. That is why "solving" chess is moot.
There is nothing really to solve, actually.
I do agree that when it finally is solved it will have little to no consequence on the way people play, but it's still an academically interesting exercise and one that I'm sure will yeild advancements that can be leveraged for other more practical applications.
Well said.
I also agree with richie that Chess will continue to be played regardless. We still play games that involve simple math, spelling, or facts. All of these can be flawlessly done with computers yet we still have fun with them.
Understand now? People will play anyway. That is why "solving" chess is moot.
There is nothing really to solve, actually.
I do agree that when it finally is solved it will have little to no consequence on the way people play, but it's still an academically interesting exercise and one that I'm sure will yeild advancements that can be leveraged for other more practical applications.
Cyber-handshake extended.
You know I can't stay mad at you....
All of that evidence is great, but the question is not one that will be solved by evidence. For instance, it sure looks like there are an infinite amount of prime numbers, and I can keep counting until the end of time but that is in no way proof that they continue forever. However, it is relatively simple to prove deductively that they do.
Um...
Assume that there are a finite number of prime numbers. Now multiply them all together and add one. That new number is then also a prime number (because the prime factorization of that number cannot contain any of the already known prime numbers). It's not the next one, but it's a prime number that wasn't in the original finite number of prime numbers. Therefore the original assumption, that there are finite number of prime numbers cannot be true. So there are an infinite number of prime numbers.
(So yes, you can prove that there are infinite primes.)
Except for two, which is prime and which will guarantee that any product including it is even. The proof is valid.
Rainbow: Euclid proved an infinite number of primes exist. No one else need do the work again.
As a vague corollary to the discussion at hand, now that Euclid's proven that the number of primes is indeed infinite, the real work in the field is to find a proof of how to locate and determine inarguably that any given number is or is not a prime. Much like the difference between knowing that chess can be 'solved', and knowing what the best move in any given position is.
Rainbow: Euclid proved an infinite number of primes exist. No one else need do the work again.
As a vague corollary to the discussion at hand, now that Euclid's proven that the number of primes is indeed infinite, the real work in the field is to find a proof of how to locate and determine inarguably that any given number is or is not a prime. Much like the difference between knowing that chess can be 'solved', and knowing what the best move in any given position is.
An algorithm to do so is known.
An algorithm to do so in non-deterministic polynomial time is not.
If you come across one let me know -- I'll pay extremely well for it.
Rainbow: Euclid proved an infinite number of primes exist. No one else need do the work again.
As a vague corollary to the discussion at hand, now that Euclid's proven that the number of primes is indeed infinite, the real work in the field is to find a proof of how to locate and determine inarguably that any given number is or is not a prime. Much like the difference between knowing that chess can be 'solved', and knowing what the best move in any given position is.
An algorithm to do so is known.
An algorithm to do so in non-deterministic polynomial time is not.
If you come across one let me know -- I'll pay extremely well for it.
What do you mean by 'non-deterministic polynomial time'? I know what all three terms mean individually, but I'm not sure how you mean them in this context.
Yeah, a formula for the prime numbers is going to be a pain. Shall we all pool together here and try and find a formula?
Attempt to factor your candidate number by all prime numbers up to and including the square root of your candidate.
If you are successful, it is not a prime. If you are not, it is.
I was going to crack chess as soon as I get this last digit on Fermat's in place.
So, check back with me.
Hey, no need - Andrew Wiles cracked Fermat's theorem a while back.
Yeah, a formula for the prime numbers is going to be a pain. Shall we all pool together here and try and find a formula?
Attempt to factor your candidate number by all prime numbers up to and including the square root of your candidate.
If you are successful, it is not a prime. If you are not, it is.
Yes, of course it's possible to check whether any number is a prime. The method you describe is a short cut to checking if the given number can be divided by 2, 3, 4, and so on. I didn't write clearly before - I meant, to find an algorithmic 'map' of primes.
Interestingly enough, I saw a program on the BBC a while back with Marcus du Sautoy (Oxford Maths prof.) which looked briefly at some research into the connection between the Riemann hypothesis and the resonance distribution of a struck piece of a real quartz crystal. The very close correlation suggested that the distribution of primes might have some direct relationship with the properties and actions of the physical world.
Rainbow: Euclid proved an infinite number of primes exist. No one else need do the work again.
As a vague corollary to the discussion at hand, now that Euclid's proven that the number of primes is indeed infinite, the real work in the field is to find a proof of how to locate and determine inarguably that any given number is or is not a prime. Much like the difference between knowing that chess can be 'solved', and knowing what the best move in any given position is.
An algorithm to do so is known.
An algorithm to do so in non-deterministic polynomial time is not.
If you come across one let me know -- I'll pay extremely well for it.
What do you mean by 'non-deterministic polynomial time'? I know what all three terms mean individually, but I'm not sure how you mean them in this context.
Basically, if the time required to solve the problem increases exponentially in relation to the size of the number you have not solved the problem in "Non Deterministic Polynomial time" (NP).
Almost all current encryption systems rely on the fact that prime factorization (the core test in the prime detection algorithm) cannot be solved in polynomial time, so if you manage to crack this problem let me know....
Yeah, a formula for the prime numbers is going to be a pain. Shall we all pool together here and try and find a formula?
Attempt to factor your candidate number by all prime numbers up to and including the square root of your candidate.
If you are successful, it is not a prime. If you are not, it is.
Yes, of course it's possible to check whether any number is a prime. The method you describe is a short cut to checking if the given number can be divided by 2, 3, 4, and so on. I didn't write clearly before - I meant, to find an algorithmic 'map' of primes.
Interestingly enough, I saw a program on the BBC a while back with Marcus du Sautoy (Oxford Maths prof.) which looked briefly at some research into the connection between the Riemann hypothesis and the resonance distribution of a struck piece of a real quartz crystal. The very close correlation suggested that the distribution of primes might have some direct relationship with the properties and actions of the physical world.
Very cool, although I don't think we should be terribly surprised....
For a proof to be complete, you need to address every possible condition. For chess to be completely solved and declared draw, then every single possible line of play will have to be analyzed as ending in draw.
Incorrect.
Only best play needs be shown to draw, as in Tic-Tac-Toe and draughts.
Mistakes lead to decisive results.
Even a cursory look at large databases (40 million games+) gives more evidence than that evidence which is used to justify and accept biological evolution. I am puzzled by people that accept the one, with much less evidence, but continue to argue the other.
No it is you who is wrong.
The only way to determine best play is by proving that all other lines of play are inferior, thereby requiring you to address every possible condition. You agree with me yet say I'm wrong. I'm not arguing that chess is inheritantly won, but the only way to prove it beyond any doubt that that is true is to show that every possible line of play is indeed drawn.
If you call this wrong, you have no background in mathematics or proofs.
A solution of chess would indeed be required to be exhaustive.
The idea that with best play chess cannot be infinitely long rests on a logical fallacy. How can a player use judgment to determine that a position is drawn, when "best play" is only defined in computer terms of being able to compute every possible variation to its end? The argument begs the question. For "best play" to be determinable, it depends on a finite game.
Once again, that is incorrect. If a position is reached that could be duplicated indefinitely, then that position is a draw. By the current rules, there are protocols that must be followed in order to claim the draw, but on a mathematical level, a repetition never creates winning conditions for either side.
It can be easily shown that Chess moves are infinite, but the game itself is not transfinite. If a game exists with N moves, then a game must exist with N+1 moves, i.e. all the way up to infinity. However, at some point, one of 3 conditions will be met, 1) white wins, 2) black wins, 3) draw. There is an upper bound at which this condition will be satisfied for every game. So yes, it is possible to have a game where a position is repeated indefinitely, but one of the conditions (draw) will be satisfied. The fact that an infinite number of positions exists does not change Chess's solvability.
As far as judging indeterminate positions as a draw, again it can be shown that with best play on both sides, eventually a position will be reached where neither side computes an advantage, no matter how far the depth. This function is further simplified by the fact that as the number of pieces goes down, there are less legal positions to consider. In fact, all games of 6 pieces or less have been solved by endgame tablebases.
http://en.wikipedia.org/wiki/Endgame_tablebase
I know that you have a strong conviction that Chess is unsolvable, but my (and others) conviction is that it is. While I don't expect you to change your mind about it, there is an overwhelming amount of evidence pointing to the eventuality, Moore's Law notwithstanding. As I have pointed out before, it is still possible that other systems of computation have yet to be discovered, as human thought, for example, is clearly different than binary computing.