But no elaboration? Sounds to me like the game is not solved, instead the author looked pretty hard for a winning line, and didn't find one.
The next sentence:
> Although there are numerous ways to select subsets that would prove the initial position results in a draw, we used algorithm 1 to obtain a small subset.
Algorithm 1:
> We developed an algorithm that requires predictive scores for all positions with 50 empty squares and returns a subset such that if the all positions belonging to that subset are solved, and the all solutions match the predictions, the initial position is consequently solved. It was described as Algorithm 1.
(The algorithm is also printed out)
From start of game to 50 squares left is enumerable.
From 36 moves left to end-of-game is apparently solvable.
The middle is the combinatorial explosion, and he avoids explaining how he navigates it.
A nitpick, but I don't think you meant to say "enumerable" here
"enumerable" means "able to be enumerated, or counted"
https://webstersdictionary1828.com/Dictionary/Enumerate
"innumerable" means "unable to be numbered, or counted"
The beginning and end of the game have less possibilities to consider. The midgame has the most possible moves = hardest to calculate.
If you read carefully, the midgame is the bit the author avoids explaining how to solve ;)
It's possible the author is correct, I'd need to really sit down and try to work out their reasoning but at first pass I'm skeptical.
There are other proofs like this - the 4 color theorem one, which was also reduced to a finite number of configurations which were manually colored.
Also curious: in the end of the paper they talk about having "weakly solve" Othello... the paper overall reads really strangely.
But given the overall state of that paper I think this is a side concern at best anyway.
It is the default and all that matters.
This is one of the dangers of reading papers as a non-expert. You can dismiss or be wowed by something that is totally irrelevant.
They wrote the paper very much like the Checkers paper from Science 2007.
It's the default for all reasonable games - statespace is huge (i.e. tic-tac-toe is childsplay) and simple strategies don't exist (that'd make bad human game). You can't even iterate all positions - even less prove them all for one outcome.
- the game is obviously known to be a draw
- but we don't have computational power to enumerate that
- so test a bunch of game states
- and confirm that none of them are wins
- ???
- it's solved! Trust me!
The process used generates a statistical approximation and tells you how close it is to correct, in theory a perfect solution would beat this by that amount, in practice of course Poker is a game of chance, and so over any realistic game it wouldn't matter because the deviation from correctness they've computed is tiny. Could they make an even smaller deviation with more compute used? Sure, but why bother.
https://en.wikipedia.org/wiki/Cepheus_(poker_bot)
Cepheus is instructive also because some humans have played against this and believed they were outplaying it, which indicates there are real human poker players who misunderstand their own variance so much (and/or discount real variance from others so much) that they're completely unable to successfully rate their abilities.
If you lose 12Bb over 100 hands you are not, in fact, "winning except that it sometimes gets lucky". You're just losing, of course it sometimes gets lucky, that's how luck works, it's a game of chance.
You have a ~10% chance over 100k hands to be <0 dollars earned. Likewise, 10% of time time you'll make twice that. Poker is fascinating in that there are a ton of people who never actually hit the true law of big numbers hands and walk around thinking "I'll never be good enough to play at X level" or "I'm a poker god with big winnings" not knowing how good they really are.
Professional players do actually get in statistically significant sample sizes, but for amateur players, most don't get enough hands to really understand their skill level.
The script at https://github.com/eukaryo/reversi-scripts/blob/main/reversi... plays perfectly (assuming the whole thing is correct) a bunch of data computed by other scripts in the repository using the 36-empty-squares solutions, for which a regular machine is presumably suitable.
It seems that what it does is essentially look up a <=300GB table with all positions with 37-64 empty squares reachable from the weak solution, and runs edax with "-solve" for positions with <=36 empty squares.
I don’t see a lot of people saying this (not parent either), but I know that many hope for closed-form solutions to things, even secretly.
It just dang doesn’t look that way anymore. Weird when a finite set is sufficient to prove subproblems that cover a perplexingly different or larger domain.
Whether a proof is accepted or not, we’ll still kinda wonder on the structure of special cases in computational complexity. They seem to work unreasonably well, and we rarely find them by hand/brainmeat.