5

(Inspired by this cool problem)

Let’s say it’s White to move. Let A = the number of squares currently attacked by White, and B = the number of squares attacked by White after White’s next move.

What is the maximum possible value of B - A?

A “attacked” square’s definition is just as you’d expect. More rigorously, a square is attacked by White if, after Black skips the next move, a White piece can legally capture anything on that square. Specifically, if it were empty, then after teleporting a dummy black piece onto that square, a White piece must be able to legally capture that dummy piece. (A dummy piece does not attack). If it were occupied by a white piece, then it must be legally defended by another white piece (the sequence “black captures the white piece with a dummy black piece, and white recaptures the dummy piece with some white piece” must be legal). If it were occupied by a black piece, it must be able to be legally captured by a white piece.

For example, in the linked problem above, we have A = 26 and B = 58. This means B - A = 32, which is really high. But is this the best we can do?

Another example: after opening moves 1. e4 c5 2. Bc4 a6 3. Nf3 we note that Black’s c-pawn attacks b4 and d4, but he does not attack f7, because the King cannot move into check. Furthermore, after a random move (say 3. … h6), f5 is attacked by White’s e-pawn.

In the solution, preferably, the “current position” should be legal, but if there’s a really nice illegal solution we should see it too.

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2
  • Any interest in looking at the number of moves rather than controlled squares? Apart from greater ease of definition, it means there is no ceiling at 63
    – Laska
    Oct 5, 2022 at 10:06
  • @Laska that would be interesting too. Feel free to ask your own question and mention me in a comment. It might be related to this though chess.stackexchange.com/questions/4490/…
    – Benjamin Wang
    Oct 5, 2022 at 22:11

2 Answers 2

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4

The original problem may be reduced to

[Title "White to move"]
[FEN "5nRB/6PR/8/8/8/8/8/k2K4 w - - 0 1"]

White controls f8, h8, g7, h6 to h1 (6 squares) and 5 squares around the king, for a total of 14 squares.

[Title "After gxf8=Q"]
[FEN "5QRB/7R/8/8/8/8/8/k2K4 w - - 0 1"]

White now controls row 8 (8 squares), row 7 but h7 (7 squares), 4 squares on row 6, 5 on each of the three next rows, 7 squares on second row, 6 squares on first row, for a total of 47 squares.

Thus B - A = 47 - 14 = 33.

In the example given in the original post, I count A = 26 and B = 58, thus B - A = 32. Or, if you do not count squares c2, d2, f2 in your problem as attacked by white, then you would have A = 23 and B = 55, thus B - A = 32 still.

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2
  • Nice reduction of the problem. I’ve changed the counting error in my question. Well done for improving the result by 1. But is it possible to prove that B - A = 33 is the maximum? (By the way I fixed a minor typo in your answer — please approve the edit)
    – Benjamin Wang
    Oct 3, 2022 at 18:37
  • 2
    FYI, the matrix itself is well known: Cf. e.g. the chess problem yacpdb.org/#524394
    – Hauke Reddmann
    Oct 4, 2022 at 8:33
3 
[Title "White to move"]
[StartFlipped "0"]
[fen "1q4Q1/P7/8/8/4B3/8/5k1K/R7 w - - 0 1"]

1.axb8=Q

In the diagram position, WTM controls 4 squares: b8, g3, h3, h1. After 1 axb8=Q, White controls 48: a8-h8 (8), a7, b7, c7, f7, g7, h7, a6-e6 (5), g6, a5, b5, d5, e5, f5, g5, a4, b4, c4, f4, g4, a3, b3, d3, f3, g3, h3, a2, b2, c2, g2, b1-h1 (7).

A=4. B=48. B-A=48-4=44.

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4
  • 1
    This approach can be dramatically improved by placing the white king on A8. This makes B8 the only legal move. Then additional pieces can be scattered around the board in a way that they will attack as much as possible after the forced B8 move due to check has been eliminated. For this we can use 8 rooks (since other pawns may have promoted). And we need to hide the black king somewhere from those rooks such that it is only being attacked after the promotion with the pawn.
    – Sextus Empiricus
    Oct 4, 2022 at 8:45
  • 1
    [Title "White to move"] [FEN "1Kq5/RP6/R7/R7/Q7/R5pk/R7/R7 w - - 0 1"] This will be a 63 improvement.
    – Sextus Empiricus
    Oct 4, 2022 at 8:57
  • Nice use of check to reduce A and thus increase A - B. I didn't anticipate this. I guess we have now generalised the problem to be completely different than the source of the inspiration. Also thank you @SextusEmpiricus for that one :)
    – Benjamin Wang
    Oct 4, 2022 at 18:53
  • 1
    This answer uses the @BenjaminWang's definition of attacked squares (based on performing an actual capture on the attacked square) while my answer views a square as controlled by white if a black king on the given square would be in check.
    – Étienne
    Oct 5, 2022 at 7:12

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