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As we all know, it is not known whether white can always win in chess with optimal play. If, however, white got to make two moves before black made their first move would they be able to win with perfect play? What would those first two moves be? How about if white gets three "headstart moves"?

If white is allowed four moves of head start then it is clearly a forced mate as can be seen from the fact that a scholar's mate can be achieved in four moves if black waits patiently.

I feel like three moves would be enough of a headstart, even though mate can't be achieved immediately, though I could be wrong. Any ideas or strategies are appreciated.

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    No way to answer that. You could try putting it in various engines with 3 and 2 moves and see what they indicate. But it would still not be definitive unless they can declare mate in x moves after that head start. Only about 112 first two legal moves worth considering. About 120 total if you tried all legal 2 move head starts. A little over 13000 3 move head starts. Is it really worth the effort to see an answer for something that will never happen in real life? Nov 21 '20 at 0:17
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    The idea was more to be able to muse about what three-move combinations lead to forced mate, in the way one might muse about a chess puzzle. If the exercise doesn't seem interesting to you then you are free to not consider it.
    – Milo Moses
    Nov 21 '20 at 0:45
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    If you make a rule that White can't cross the midline, 14 moves are enough, as in this old question: chess.stackexchange.com/questions/874/…
    – bof
    Nov 21 '20 at 3:09
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    @bof That's already been linked in my answer :-) Nov 21 '20 at 15:48
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    @RewanDemontay Oh, right. Missed that. The reader who follows the link will probably read no further than the 15 move solution in the accepted answer and not notice the 14 move solution which I mentioned in my comment.
    – bof
    Nov 22 '20 at 0:33
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There is a similar ancient question, but this isn't a duplicate since White may move anywhere.

For an upper bound, I can prove a guaranteed win for White in 5 moves. Indeed, it is reminiscent of Scholar's Mate.

[FEN ""]

1. e3 null 2. Bc4 null 3. Qf3 null 4. Nh3 null 5. Ng5

With Black to move, as White has now used up their five given moves, Stockfish announces a mate in 6.

[FEN "rnbqkbnr/pppppppp/8/6N1/2B5/4PQ2/PPPP1PPP/RNB1K2R b KQkq - 0 1"]

1... d5 2. Qxf7+ Kd7 3. Qxd5+ Ke8 4. Qf7+ Kd7 5. Qf5+ Kc6 6. Qb5+ Kd6 7. Ne4#

(If Black plays 1... d6?, then it is instead a mate in four with 2. Bxf7+ Kd7 3. Be6+ Ke8 4. Qf7#.)

The problem for a setup four is that, with the Scholar's Mate setup, f7 needs to be attacked thrice, which doesn't seem possible. Additionally, there is no obvious way to do it without Scholar's Mate.

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    Thank you for your input! I was working on the assumption that captures were allowed during the beginning grace period, but this sort of "all sides attack" is definitely what I had in mind. Your input is truly appreciated.
    – Milo Moses
    Nov 21 '20 at 2:28
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    @RewanDemontay Presumably 1.d4 2.Bg5 3.Bxe7 4.Bxd8 is a forced win though it's obviously not feasible to prove that. I would guess that two starting moves aren't enough but I could see three moves being enough to be theoretically winning. Though it might also not be, I suspect it is on the border of winning. Of course both 2 and 3 are unlikely to be proved any time soon if ever.
    – koedem
    Nov 21 '20 at 6:44
  • @koedem 1.Nf3 2.Ne5 3. Nxf7 4.Nxd8 keeping the bishop pair might be slightly better.
    – bof
    Nov 21 '20 at 11:53
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    But isn't 1. e3 null 2. Bc4 null 3. Qf3 null 4. Qxf7# already a mate in 0? Nov 21 '20 at 12:52
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    Linguistic quibble: you proved that white can guarantee a win with 5 moves, you did not prove that white "needs" 5 moves.
    – bof
    Nov 22 '20 at 0:37
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If I had to guess I'd say that one extra move for White (normal chess) is objectively a draw, 3 extra moves is probably a win, and 2 extra moves is a tossup.

The problem is, for anything short of 4 extra moves, there's no way to definitively prove any of the above results. Yes, Stockfish gives some evaluation that's much better for White, but unless it can calculate a forced mate then there's no absolute certainty.

There's also the issue that engines prune the search tree a lot in order to be fast. So when an engine says it's at "depth 30", it's not actually covering every variation up to 30 ply that has a chance of affecting the evaluation. Doing that would take very long, even with optimizations such as alpha-beta pruning and a transposition table. So technically if you're searching for a guaranteed win, you couldn't even trust an engine that says "mate in 50".

We can look at it more theoretically. In chess the average number of possible moves in a given position is 30, but to be very generous let's use 20 (because maybe White can launch a mating attack early on). And let's assume that with 3 extra moves, with a ton of luck White might have a forced win in 40 ply. So to calculate everything 40 ply ahead, this would be 20^40 positions. Let's assume alpha-beta pruning turns the exponent into around 3/4 of what it was: 20^30. Now, let's say the transposition table increases the speed by a factor of around 5 (that might be slightly generous). So we're left with 20^29.46..., or about 2.1 * 10^38.

So in the best case scenario of White actually having a forced win with 3 extra moves (and it only taking 40 ply), maybe you'd need to calculate something on the order of 10^38 positions to be definitively sure. This is obviously an extremely rough estimate, as the actual number might be a lot lower (again, in the best case scenario). For example, if White is trying to mate Black, the branching factor could be lower due to Black having only moves at certain times. Also, you could try only considering the best few moves for White in each position. However, it's likely Stockfish is already doing pruning somewhat similar to this for White and Black, and so far it finds nothing after depth 40 for a few different positions I set up with 3 extra moves.

You could argue that a win might exist in something much shorter, such as 30 ply instead of 40 ply. I think this is very unlikely though, considering Stockfish gives an evaluation of +1 and change, when it's over depth 40. Sure, like I said it could absolutely be missing something, but I'm not using Stockfish for a proof here. I'm just mentioning it to say that a very quick win likely doesn't exist, and then if that's the case it will be near impossible to prove a longer one does due to the discussed time complexity.

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  • This is a very good argument for the "impossibility" of knowing of a win in 2, 3, or 4 moves. Do you suppose that any other 5 move setups exist as well? Nov 21 '20 at 14:41
  • @RewanDemontay Oh well for my answer I was assuming you could capture on a move, allowing scholar's mate for 4 moves. The impossibility of knowing a win applies to 3 moves or less. Nov 21 '20 at 15:15
  • I had no doubt that captures were allowed, but I assumed that giving checkmate with the given moves, a "mate in 0' per se, was an unintentional oversight, hence why I left it out. Nov 21 '20 at 15:59
  • @RewanDemontay Alright fair enough. Well under this framework, since you proved an upper bound of 5, the interesting question is whether there's a forced win in 4 moves if you can't immediately give mate. I think what I wrote would potentially also apply for 4 moves, as long as there's no way for White to develop an extremely fierce attack that ends things quickly (before the computational complexity gets too high). Nov 21 '20 at 16:38
  • @RewanDemontay For other 5 move setups, I can't think of any off the top of my head that don't involve the queen and knight+bishop from the kingside. Nov 21 '20 at 16:42

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