Comment by arutar | Hacker News Reader

Comment by arutar | Hacker News Reader

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More fun facts:

After identifying solutions up to rotation and reflection there are only 49 solutions. No solutions have rotational symmetry, and there is exactly one solution with reflection symmetry (already mentioned by an earlier commenter).

Out of the 49 solution classes, there are 18 distinct queen layouts. The layouts have between 1 and 5 ways to place the bishop to complete the solution. Interestingly, there is exactly one queen layout (up to rotation / reflection) for which there are exactly 2 ways to place the bishop to complete the puzzle.

Even more fun facts: if you change the problem instead to 4 queens and one knight, there is exactly one solution up to symmetries (rotations and flipping). Here it is:

    ..N.....
    ........
    ........
    ....Q...
    .....Q..
    ...Q....
    ......Q.
    ........

Edit: even more fun facts: if we take the standard piece values of Q=9, R=5, B/N=3, then we can ask for the smallest piece budget that attacks every square. The cheapest possible configuration is 24 points, you can see one with 8 bishops:

    ........
    ....B...
    ....B...
    .B......
    ...B.B..
    .....B..
    ..B.....
    ....B...

Which has pleasing symmetry when you view it as a composition of light-square bishops and dark-square bishops.

wow. oddly that seems like an easier problem, despite the solution space being smaller. not saying i woulda solved that, because i doubt it, but it’s much closer to what my raw intuition woulda spit out as a solutuon to the OG problem.

You can't have rotational symmetry with 5 pieces since that would require a piece in the center but the chess board has an even number sized.

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