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Prove that it is possible, from any starting KBN-K position, to accomplish the mate by manoeuvering to arrive at key positions from a well-known method or strategy whilst the losing king actively tries to avoid reaching the key positions that would lead to mate within 50 moves. In terms of the players' goals, the winning player's goal is to reach the key positions and still have moves to mate within 50 moves, and the losing player's goal is to avoid getting mated within 50 moves.

Given a starting KBN-K position, there exist paths through the tree of variations satisfying the winning player's goal with optimal play by the losing player. Is this satisfied for ALL legal KBNK positions using that particular stratagem only? How would one go about attempting to prove this using KBN-K tablebases?

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    Unless you state what a "key position" is for this particular endgame, I don't see how you would prove anything.
    – user1583209
    Commented Oct 20, 2018 at 15:02
  • A key position is a member of the set of all positions associated with a method or strategy to mate in this endgame.
    – candor
    Commented Oct 20, 2018 at 15:40
  • Sure. I understood what you meant. However, in the end, there is a strategy to mate from any position in this endgame. Given, no human is going to learn a strategy from any position, but a beginner might know only a strategy from a position where the king is already in the right corner. A better player might know what to do once the king is on the rim, while an even better player might be aware of even more key positions.
    – user1583209
    Commented Oct 20, 2018 at 15:54
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    It takes around 20 moves to push the king from the wrong corner to the correct corner and mate. That leaves you about 30 moves (50-20) to force the king to the side of the board, which seems feasible to me. In the most difficult position (33 moves to mate), the king is forced to the side of the board after around 15 moves.
    – user1583209
    Commented Oct 20, 2018 at 15:58

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Is this satisfied for ALL legal KBNK positions using that particular stratagem only?

Absolutely not! Please don't try to prove something that's impossible. Consider this position.

[startflipped ""]
[FEN "3kB3/4N3/8/8/8/8/8/3K4 b - - 0 1"]
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    Good point! Perhaps I should also add "that does not immediately draw for the opponent".
    – candor
    Commented Oct 21, 2018 at 4:23
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With the tablebase you can prove that all positions that can be won, can be won within 50 moves. But they're not designed to guarantee that certain "key positions" are visited in between (and if you pick them badly, then there are positions that can't be won in time anymore).

So you'd need to get your list of key positions and note in how many moves checkmate can be forced from there, using the existing tablebases.

And, for each key positions, you need to create new tablebases where the goal is not to checkmate, but to reach that key position.

Then for each KNB v K position, there needs to be at least one key position in the list for which the minimum moves to force it plus the minimum moves to checkmate from there is 50 or below.

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the diagram is not a N and B ending as one of them is lost immediately. dropping a piece on the first move should disqualify these types of positions.

similarly any moves considered at any time should not drop a piece.

i assumed this was a serious and not a trick question.

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