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I have intuitively thought that one of the following must be true:

  1. White has a forced mate from move one.
  2. White doesn't have forced mate, but does have a forced draw from move one (and in turn, black also has a forced draw)
  3. Black has a forced mate from move one.

So, either one of the two colours (probably white) has a forced mate, or they both have forced draws, from the starting position on the board.

I know that I'm not alone in this thought process, but here's the things I don't know:

  1. What is this thought called? Is there a name of this hypothesis or theorem or idea?
  2. Has this been formally proven? Can it be formally proven?
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  • 1
    You might be interested in Combinatorial Game Theory. Chess technically doesn't quite count as a combinatorial game under the strict definition, but it's similar enough that most of the same concepts apply. Commented Jun 13, 2023 at 22:54
  • I believe the term you're looking for is that the game has been 'solved' (e.g. that if you can sufficiently map the potential move space, that you can determine the outcome of perfect play). > A two-player game can be solved on several levels: > > Prove whether the first player will win, lose or draw from the initial > position, given perfect play on both sides > > <Sub><Sup>en.wikipedia.org/wiki/Solved_game</Sub></Sup> This, for example is how you know that given one perfect player of tic-tac-toe, that person will never lose, and two
    – Valorum
    Commented Jun 14, 2023 at 1:06
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    here's a joke channeling Bobby Fischer: 1. e4!! (now white has a forced mate)
    – Michael
    Commented Jun 14, 2023 at 22:41
  • 3
    In game theory this property is called determinacy or determinateness; one says that chess is a determined game.
    – bof
    Commented Jun 15, 2023 at 2:31

3 Answers 3

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Before 50-move rule was introduced infinite games were possible. This was proven by mathematician and chess world champion Max Euwe.

With 50-move rule infinite games aren't possible, so they must end. If they end regularly (so not due to death of one of the players or something similarly drastic) they end with one of three possible results - which can only be victory for either white or black or draw.

So you are right, one of your options indeed has to be correct. To find such forced win or draw is theoretically possible (chess is indeed one of solvable games) but practically impossible due to vast scope of possibilities.

For formal proof see Zermelo's theorem.

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    An alternate statement is that for a game meeting all of these conditions except the condition that a draw is now possible, then either the first-player can force a win, or the second-player can force a win, or both players can at least force a draw. - Zermelo's Theorem looks to be what I was looking for. Fantastic answer, thank you.
    – TKoL
    Commented Jun 13, 2023 at 14:54
  • 14
    In theory, as long as you have something like the rule of triple repetition you don't even need the 50-move rule -- though it might take astronomically long before the first repetition happened. Commented Jun 14, 2023 at 4:39
  • 6
    If you didn't have either the 50 move rule or the threefold repetition rule then infinite games would be possible. However, because there are only a finite set of possible game states, Zermelo's theorem still applies. (See the details section in the link above.) In that case there are exactly four possibilities: a win for white, a win for black, a draw, or the board position eventually repeats and the game never ends.
    – N. Virgo
    Commented Jun 15, 2023 at 1:58
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    The 50-move rule is not enough, because it doesn't end the game automatically -- one of the players has to claim a draw to end the game. Hovwever, the 75-move rule, introduced in 2014, does end the game automatically, and this is enough to guarantee that all games are finite.
    – TonyK
    Commented Jun 15, 2023 at 9:51
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Zermelo's theorem proves forced win or draw for chess, and generally, enumerating all of a game's outcomes (by proof) is done by "solving" the game. Chess currently isn't solved. Other answers (1, 2) elaborate further, but I'd like to provide another angle to 3-fold repetition enabling this.

If asking about finished games, neither 3-fold repetition nor 50 move rule are needed; "impossible to mate" and "no legal moves" rules (yielding draw) suffice. That the game can go on infinitely, with positions repeated, doesn't change the fact that a mate is still possible. This would differ in a variant of chess where a piece is removed every 60 moves; then, end by draw can be forced. If pieces are added, a checkmate can be forced. But in plain chess, this is only semantic: an infinite game is forced if perfect players produce a position where first to not repeat loses.

The 50 move rule also isn't needed if there's 3-fold repetition, it only accelerates the result (a lot): it's 3 total, not consecutive, repetitions, and the total number of possible positions is finite. And it's not necessarily 3-, but 5-fold, which is handled differently between chess.com and lichess.org.

Note, the 50 move rule (rather 75 for forced as opposed to claimed) does influence the outcome of "perfect chess", since a mate may be possible but only after (say) 100 moves, which actually motivated the rule (and a term "cursed win"):

in the 20th century it was discovered that certain endgame positions are winnable but require more than 50 moves ... [so 100 were allowed] ... However, winnable positions that required even more moves were later discovered, and in 1992, FIDE abolished all such exceptions and reinstated the strict 50-move rule over the board

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    The 50-move rule might make it impossible for White to force a win, even if it would be possible for White to force a win in the absence of the rule. If that happens to be the case, the rule could be viewed as improving the game by ensuring that perfect play by Black would be able to force a draw even given perfect play by White, but imperfect play by Black would allow a more skillful White player to win.
    – supercat
    Commented Jun 14, 2023 at 21:26
  • @supercat Good point, I thought of this of 3- vs 5-fold repetition but wasn't sure - this is much more definitive. Commented Jun 15, 2023 at 22:26
  • A key feature of the draw being three-fold rather than two-fold repetition is that the first time a position gets repeated (second time it appears) puts each players on notice that the other player might force a draw if allowed to do so, and let a player who doesn't want a draw know that the opponent must be prevented from causing another repetition. If a player given such notice allows a position to occur a third time, that would imply that either the opponent was able to force a draw, or that the player who allowed the draw wasn't seeking to prevent it.
    – supercat
    Commented Jun 16, 2023 at 15:55
  • I imagine 4+ fold making a difference since the same position can be reached from different followups. Unsure if there'd ever be need in perfect play though. Commented Jun 16, 2023 at 16:08
  • Except in cases where perfect play by both sides would yield a draw, every position must be a mate in N for one side or the other. If White has a mate in 23 (but not a mate in 22 or less) and plays perfectly, there will never again be any situation in which there isn't a mate in 22 or less, and thus the original position could not occur again.
    – supercat
    Commented Jun 16, 2023 at 17:15
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That's something that has been theorized (surely after many others) by Max Deutsch, who also invented an algorithm to prove that. His objective, after the invention of the algorithm, was to memorize the "checkmate from move one" (if I understood him correctly) to become an invincible player. His strategy was tested against WCC Magnus Carlsen.

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