How many chess games are possible? (opens in new tab)

Is that a new rule? I was under the impression that it had been the case for a very long time that if you went out on time but there was no possible sequence of moves leading to checkmating you, it was a draw instead. (Meaning, of course, that having more pieces could be a disadvantage in such situations, which feels a bit unfair. E.g., KvKB is a draw, but KPvKB can lead to a mate if both sides cooperate, and thus would be a time loss for white even if black would never win in practical play.)

> the game is a draw if there exists no sequence of moves that could lead to checkmate. Interestingly, although this has almost no strategic implications, it means that... it's almost impossible to write a program to detect draws that's technically correct.

I don’t see what makes that technically difficult. The number of possible positions is finite, so just enumerate the game tree and check whether it contains a checkmate situation.

I also don’t see why it would be almost impossible in practice. Aren’t the only weird situations ones where there are pawns that could be promoted to queens if they weren’t blocked by other pawns, and those pawns prevent all other pieces on the board from taking pawns and from checkmating the king?

For 7 remaining pieces or less, there are actually tablebases of all possible positions showing whether there's a possible win or loss for either side: https://en.wikipedia.org/wiki/Endgame_tablebase

> That’s over ten times as many games as estimated.

That's still a pretty good estimate of an exponentially large quantity; the exponent being off by only 1. With these estimates you cannot hope to do better than estimating the exponent.

It depends totally on the opening. You can be out of book and database far quicker than that for offbeat stuff, or in book far longer for popular openings.

Another distinction needs to be made between positions seen and positions played. Almost every viable position will have been seen in preparation well beyond 10 moves. But seeing them on the board is rarer.

I don't think the math is correct here. The 25% of positions that have been seen more than once represent more than 25% of the occurrences. Even if all of them would be seen only twice, you should already see them in 40% occurences.

A very strong player would show the novice the scholar's mate once and then move on to hanging tactics and pieces on purpose so that the novice starts seeing things, probably leading to positions that are a lot more rare.

Yes, the stronger the players, the more often they will both go deeper into established theoretical lines that have been played before.

To be fair, that's atoms in the observable universe.

The total size of the universe is unknown, and could (and likely does) have way more atoms than that.

Actually, that's a fun thought: assuming homogenuity of matter between the observable and unobservable universe, how much bigger would the unobservable universe need to be to render some of these claims no longer true?

Because you're right to point out that factorials grow absurdly quickly. It's entirely possible my caveat straight up doesn't matter.

Edit: Ok, I'm seeing Wikipedia has a (disputed) estimate for the diameter of the total universe as 10^10^10^128 megaparsecs. Then, radius cubed should be 1/2(10^10^10^128^3)=1/210^10^10^131, as opposed to the radius of the observable universe being a nice, clean 14 billion parsecs = 1410^3 megaparsecs, making the radius cubed 1410^4 megaparsecs. I don't think I have a big enough calculator for this, but for fun, let's say 128^3 is roughly 2,000,000. Then we can rewrite T, the relative volume of the total universe, as 1/210^10^10^2*(10^ 6). I guess if we call 14 close enough to 10, then our density is 10^80/10^6=10^74 atoms for every pi megaparsecs cubed.

Going off the heuristic that n!<n^n, and the total universe can trivially produce (10^10)^(10^10), we would need to rearrange >10^10 objects just to even start to think about the number of (megaparsecs cubed)/pi it might have, let alone the 10^74 those each have.

We might not have enough decks of cards for this one.

(Feel free to criticize/tear down my math or logic anywhere in this one, it's very much off the cuff and I'm sure I made at least as many egregious errors in computing exponents as I did computations. No math class I've taken yet really prepares you to handle exponents raised four deep.)

It depends what rules you're using, but there are the three-fold repetition and 50-move rules which allow a player to force the game to end in a draw. The catch is they both require one of the players to claim a draw under the rule, otherwise they can keep playing.

There is additionally the 75-move rule where the the game is forced to be over without either player claiming the rule (the arbiter just ends the game), that rule would give an upper bound regardless of the players knowledge of the rules.

Well there is. The three/five fold rule. And 50 moves rule.