In the German chess problem magazine feenschach, Heft 214 Band XXIV, July-August 2015 pp 178-180, I pose & solve the following mathematical puzzle: "Consider the 25 chess positions deriving from the 25 sequences of moves of the following form: 1. e4 e5 2. B~ B~. (I.e. two bishops are developed.) Show that at most six of these positions are wins for Black."
E.g. one position out of the 25 is:
This is solved non-constructively with a simple strategy-stealing argument, which places an upper bound on the possible number of Black wins as 25/4. However, 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 allegedly 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 allegedly not being mathematical immediately after I posted an entirely mathematical response. The underlying article has simple lemmas and everything :-) Whether Black wins chess may be addressed by chess players in terms of chess strategy. Indeed the accepted answer here is such an opinion! However 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 (= "Mathematical Sciences Research Institute") Publications, Volume 42, 2002, Unsolved Problems in Combinatorial Games #29, p 465, the mathematician Richard Guy also raised the question whether chess could be a Black win. However he misstated the result that I had emailed him. The correct result has only now been published (feenschach, above).
Another possible angle of attack, given that chess is so hard, is to ask what properties a simpler symmetric positional combinatorial game would have to satisfy to offer non-trivial strategy-stealing. Intuitively, Zugzwang is a feature in cold positions, but the chess in the opening appears to be hot. What other games can one set up so that e.g. a two-move tempo loss can exist, without being apparently as (literally!) self-defeating as in chess.
As this mathematical question remains incorrectly 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.