A Chess Question Of The Late Great W.T.Tutte [closed]

In "Graph Theory As I Have Known It", p.12, Knights Errant, Tutte mentions as an aside the chess question " does either Black or White have a certain win from the initial position, given perfect play by both sides".

Is there any literature on that possibility of a Black win, that is the possibility of the initial chess position being mutual zugzwang? What is the earliest reference to the question?

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closed as off-topic by Noah Snyder, Lucia, Steven Landsburg, Yemon Choi, JoëlDec 24 '15 at 17:26

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I do not know the earliest reference to the question - I've wondered about it myself - but I believe this question is far, far outside the realm of current computational techniques and power. I just don't think there is anything even on the horizon that would make answering this question tractable. – TLss Sep 17 '12 at 14:54
This looks like a chess question, not a math question. Also, it's quite well-known. Whenever someone asks whether chess is close to solved, one of the responses is that we don't even know that black doesn't have a forced win. – Douglas Zare Sep 17 '12 at 16:39
I'm voting to close this question as off-topic because this is a chess question and not really a math question. – Noah Snyder Dec 24 '15 at 14:49

The question "does either Black or White have a certain win from the initial position, given perfect play by both sides" was first addressed by Wilhelm Steinitz in his 1896 "Theory of Perfect Play" (Chapter 6 of Modern Chess Instructor). He concluded that "by proper play on both sides the legitimate issue of a game ought to be a draw".

You can find a quite detailed overview of the literature since Steinitz in Wikipedia. The advantages of Black over White seem to be largely psychological ("underdog").

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Actually, if you look at opening databases, and condition on games being decisive (that is, not a draw), black seems to be running a massive disadvantage, so the underdog thing is of limited benefit, it seems... – Igor Rivin Sep 17 '12 at 16:15
The reference cited in this answer does not address the question posed. – TLss Sep 17 '12 at 16:25
Just to be clear, the results of practical play are not relevant to the question of the theoretical status of a Black win. – TLss Sep 17 '12 at 16:27
For that matter, I highly doubt one could prove using current techniques that from the initial position with Black's queen removed, that Black has no win. Doing so would, in fact, be a major advance in the state of the art. – TLss Sep 17 '12 at 16:30
Almost everyone who plays competitive chess would view White as having some practical advantage over Black, other thins being equal. Well, maybe Adorjan would disagree... – Yemon Choi Sep 17 '12 at 17:04

In the German chess problem magazine feenschach, Heft 214 Band XXIV, July-August 2015 pp 178-180, I pose & solve the following puzzle: "Consider the 25 chess positions deriving from the 25 sequences of moves of the following form: 1. e4 e5 2. L~ L~. Show that at most six of these are wins for Black."

This is solved with a simple strategy-stealing argument, but I don't think this fragile approach can be generalized to the Holy Grail of showing that chess is not a win for Black. The population of 5x5 (or more generally nxn) positions examined here is kind of a best case, where there is maximal stealing possible. If one considers e.g. the population of 400 positions after Black's first move, then then we can show that at most 304/400 can be wins for Black. This is a much higher proportion, and shows the limitations of the approach.

27/12/2015: After the closing of this question for not being mathematical, I added the following to emphasize that it can very well be interpreted mathematically.

Thanks very much, Mr Calvert, for your support. I was a bit bemused that the question was suddenly closed for not being mathematical immediately after I posted an entirely mathematical response. The underlying article has simple lemmas and everything :-) The question of whether chess is a win for Black may be interpreted in terms of chess strategy, but my interest is only mathematical.

Claude Shannon wrote in his well-known 1950 paper (Philosophical Magazine, Ser.7, Vol. 41, No. 314 - March 1950. XXII. Programming a Computer for Playing Chess)

"It is interesting that a slight change in the rules of chess gives a game for which it is provable that White has at least a draw in the initial position. Suppose the rules the same as those of chess except that a player is not forced to move a piece at his turn to play, but may, if he chooses, 'pass'."

I was really looking for partial strategy-stealing solutions, in the spirit of Shannon, but in ordinary chess. So for White to lose a tempo takes at least two moves. Interestingly, parity concerns are particularly significant in the chess starting array.

In "More Games of No Chance" by MSRI Publications, Volume 42, 2002, Unsolved Problems in Combinatorial Games #29, p 465, Richard Guy also raised the question whether chess could be a Black win. However he misstated the result that I had emailed him, which has only now been published (feenschach, above).

As this discussion is closed, I'm happy to discuss this with anyone offline: there is an email contact given at the bottom of http://anselan.com/chess.html.

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It is sad that this comment, from a new poster, will not be more widely seen in view of the later "put on hold" decision. – Ian Calvert Dec 25 '15 at 12:03

In my decade of chess-playing, I have never come across anything remotely resembling an answer to this question. If the perfect play question had already been answered by example, chess would be an exercise in memorization -- the absolute perfect path(s) of the game could all be played out down to the endgame, and whoever deviates first loses material. In the event that one side has a forced win, the other side would always be the one forced to lose material. Brute-forcing a solution to chess is nowhere near possible at the moment, given the amount of possible game positions (~ $10^{43}$, according to Claude Shannon). I've never heard of any sound way to argue this question other than brute-force.

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That $10^{43}$ excludes promotions. – Douglas Zare Sep 17 '12 at 16:43
Why the definite article before "perfect play"? It's imaginable that either e4 or d4 (for example) can be the opening move of a "perfectly played game (and similarly, for many positions, it's imaginable that there is no one best move). Actually, when I was more active in chess, I used to like opening with Nf3 as White, and it's not clear to me that that is inferior to e4 or d4. – Todd Trimble Sep 17 '12 at 18:24
I believe the meaning is the "perfect play question" not "the perfect play" question – Lee Mosher Sep 17 '12 at 23:42
Ah, thanks Lee. Happily, I see "path(s)" in the next line, so this would support that reading as well. – Todd Trimble Sep 18 '12 at 0:05

No one in the Chess world believes Black wins, hence no literature on what you highlight.

Of course, that isn't mathematical proof, but if we were to ever get an answer and I had to gamble on it, I'd say it's a draw with optimal play.

Now if we look at stats, certainly Black loses more often than white loses, with draws becoming more common at higher levels of play.

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