Answers.Body: <p>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.</p>
  <p>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.</p>
  <p>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 &quot;depth 30&quot;, 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 &quot;mate in 50&quot;.</p>
  <p>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.</p>
  <p>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.</p>
  <p>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.</p>
  
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Answers.Text: 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?
Answers.UserDisplayName: Rewan Demontay
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Answers.Text: @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.
Answers.UserDisplayName: Inertial Ignorance
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Answers.Text: 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.
Answers.UserDisplayName: Rewan Demontay
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Answers.Text: @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).
Answers.UserDisplayName: Inertial Ignorance
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Answers.Text: @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.
Answers.UserDisplayName: Inertial Ignorance
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Answers.Text: I wish I could accept both of these answers! As a math nerd I really appreciate the hard numbers, and the analysis you give is really fascinating. Thank you so much!
Answers.UserDisplayName: Milo Moses
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Answers.Text: @InertialIgnorance something seems to be slightly off with your numbers in your first sentence.
  One extra presumably means White gets two moves as otherwise there's no "extra" there. But then 3 extra means that White has 4 moves which already is scholars mate.
  Ironically I still agree with the statement. 3 extra means scholars mate so is winning. One extra might be quite good for White but not enough. And then I also see two extra as a toss up. I suspect it may be enough to win but that's of course just guesswork.
Answers.UserDisplayName: koedem
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Answers.Text: @koedem When I said one extra I just meant assuming we start from a state where no one gets to go first (even though here the game would never begin). So one extra becomes what we know as normal chess. Maybe that was a bad way to word it in my answer.
Answers.UserDisplayName: Inertial Ignorance
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Answers.Text: @Milo Moses No problem, it was interesting analyzing this.
Answers.UserDisplayName: Inertial Ignorance
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Answers.UserDisplayName: Inertial Ignorance
Answers.Body: <p>There is a similar <a href="https://chess.stackexchange.com/questions/874/giving-moves-as-odds-whats-the-least-number-of-moves-by-white-which-forces-che">ancient question,</a> but this isn't a duplicate since White may move anywhere.</p>
  <p>For an <strong>upper bound,</strong> I can prove a guaranteed win for White in <strong>5 moves.</strong> Indeed, it is reminiscent of Scholar's Mate.</p>
  <pre><code>[FEN &quot;&quot;]
  
  1. e3 null 2. Bc4 null 3. Qf3 null 4. Nh3 null 5. Ng5
  </code></pre>
  <p>With Black to move, as White has now used up their five given moves, <a href="https://lichess.org/analysis/rnbqkbnr/pppppppp/8/6N1/2B5/4PQ2/PPPP1PPP/RNB1K2R_b_KQkq_-_0_1#1" rel="nofollow noreferrer">Stockfish</a> announces a <strong>mate in 6.</strong></p>
  <pre><code>[FEN &quot;rnbqkbnr/pppppppp/8/6N1/2B5/4PQ2/PPPP1PPP/RNB1K2R b KQkq - 0 1&quot;]
  
  1... d5 2. Qxf7+ Kd7 3. Qxd5+ Ke8 4. Qf7+ Kd7 5. Qf5+ Kc6 6. Qb5+ Kd6 7. Ne4#
  </code></pre>
  <p>(If Black plays 1... d6?, then it is instead a mate in four with 2. Bxf7+ Kd7 3. Be6+ Ke8 4. Qf7#.)</p>
  <p>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.</p>
  
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Answers.Text: 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.
Answers.UserDisplayName: Milo Moses
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Answers.Text: @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.
Answers.UserDisplayName: koedem
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Answers.Text: @koedem 1.Nf3 2.Ne5 3. Nxf7 4.Nxd8 keeping the bishop pair might be slightly better.
Answers.UserDisplayName: bof
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Answers.Text: @bof maybe, though the d4 line claims the center. But it does not matter of course, either is easily winning for White.
Answers.UserDisplayName: koedem
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Answers.Text: But isn't 1. e3 null 2. Bc4 null 3. Qf3 null 4. Qxf7# already a mate in 0?
Answers.UserDisplayName: Hagen von Eitzen
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Answers.Text: The question seems to allow the last move to be a mating move, which would make 4 head-start moves a forced win. But, if we do not allow this, are you sure that 4 is then not possible? With 4 moves, White can capture Black's queen with a knight, which gives +8.6 in Stockfish.
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Answers.UserDisplayName: Rewan Demontay
Body: <p>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 &quot;headstart moves&quot;?</p>
  <p>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.</p>
  <p>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.</p>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2020-11-21T00:17:09.163Z
Comments.Text: 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?
Comments.UserDisplayName: kattmanndoo
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Comments.CreationDate: 2020-11-21T00:45:32.093Z
Comments.Text: 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.
Comments.UserDisplayName: Milo Moses
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Comments.CreationDate: 2020-11-21T03:09:54.657Z
Comments.Text: If you make a rule that White can't cross the midline, 14 moves are enough, as in this old question: https://chess.stackexchange.com/questions/874/giving-moves-as-odds-whats-the-least-number-of-moves-by-white-which-forces-che
Comments.UserDisplayName: bof
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Comments.Text: @bof That's already been linked in my answer :-)
Comments.UserDisplayName: Rewan Demontay
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Comments.CreationDate: 2020-11-22T00:33:01.457Z
Comments.Text: @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.
Comments.UserDisplayName: bof
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Comments.CreationDate: 2021-01-21T11:30:30.260Z
Comments.Text: Let's assume you play 1.e4 2.d4 3.Lc4 or whatnot. I play the Hippo then, I stood worse in the opening (in about ANY of my games :-) Does anyone have actual Stockfish or Leela values and the best continuation for Black?
Comments.UserDisplayName: Hauke Reddmann
Tags: <opening><checkmate><chess-variants><puzzles>
Title: How many moves of head start does white need for a guaranteed win?
ThreadText: # How many moves of head start does white need for a guaranteed win?
  
  (Asked by: username_0 on 2020-11-20)
  
  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.
  
  **Tags:** opening, checkmate, chess-variants, puzzles
  
  > **Comments:**
  > **username_1**: 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?  
  > **username_0**: 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.  
  > **username_2**: If you make a rule that White can't cross the midline, 14 moves are enough, as in this old question: https://chess.stackexchange.com/questions/874/giving-moves-as-odds-whats-the-least-number-of-moves-by-white-which-forces-che  
  > **username_3**: @username_2 That's already been linked in my answer :-)  
  > **username_2**: @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.  
  > **username_4**: Let's assume you play 1.e4 2.d4 3.Lc4 or whatnot. I play the Hippo then, I stood worse in the opening (in about ANY of my games :-) Does anyone have actual Stockfish or Leela values and the best continuation for Black?  
  
  ---
  
  ## 2 answers
  
  ### Answer by username_3 on 2020-11-21. Score: 13 | Accepted answer
  
  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.
  
  > **Comments:**
  > **username_0**: 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.  
  > **username_6**: @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.  
  > **username_2**: @username_6 1.Nf3 2.Ne5 3. Nxf7 4.Nxd8 keeping the bishop pair might be slightly better.  
  > **username_6**: @username_2 maybe, though the d4 line claims the center. But it does not matter of course, either is easily winning for White.  
  > **username_7**: But isn't 1. e3 null 2. Bc4 null 3. Qf3 null 4. Qxf7# already a mate in 0?  
  > **username_8**: The question seems to allow the last move to be a mating move, which would make 4 head-start moves a forced win. But, if we do not allow this, are you sure that 4 is then not possible? With 4 moves, White can capture Black's queen with a knight, which gives +8.6 in Stockfish.  
  
  ---
  
  ### Answer by username_5 on 2020-11-21. Score: 3
  
  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.
  
  > **Comments:**
  > **username_3**: 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?  
  > **username_5**: @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.  
  > **username_3**: 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.  
  > **username_5**: @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).  
  > **username_5**: @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.  
  > **username_0**: I wish I could accept both of these answers! As a math nerd I really appreciate the hard numbers, and the analysis you give is really fascinating. Thank you so much!  
  > **username_6**: @InertialIgnorance something seems to be slightly off with your numbers in your first sentence. One extra presumably means White gets two moves as otherwise there's no "extra" there. But then 3 extra means that White has 4 moves which already is scholars mate. Ironically I still agree with the statement. 3 extra means scholars mate so is winning. One extra might be quite good for White but not enough. And then I also see two extra as a toss up. I suspect it may be enough to win but that's of course just guesswork.  
  > **username_5**: @username_6 When I said one extra I just meant assuming we start from a state where no one gets to go first (even though here the game would never begin). So one extra becomes what we know as normal chess. Maybe that was a bad way to word it in my answer.  
  > **username_5**: @username_0 No problem, it was interesting analyzing this.  
  
  ---
  
  