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by dllu26 days ago |
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upvoteby sobellian26 days ago|
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I used CP-SAT to enumerate the solutions. A heuristic for "interesting" solutions is those which only admit one valid bishop placement. For example:

    ........
    Q..B..Q.
    ........
    ........
    .......Q
    ........
    ........
    ...Q....
Where the bishop lies at the intersection of three queens' horizontal attacks. With these queens, no other bishop placement works.
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upvoteby aaron69526 days ago|
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[dead]
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upvoteby arutar26 days ago|
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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.

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upvoteby sobellian26 days ago|
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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.
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upvoteby keeganpoppen26 days ago|
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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.
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upvoteby dllu26 days ago|
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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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upvoteby NooneAtAll326 days ago|
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is there a solution where all the pieces are covered as well?
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upvoteby arutar26 days ago|
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Unfortunately there are none
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upvoteby JKCalhoun26 days ago|
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I also tried the "place the queens on unique rows and columns".

That got me down to 6 free spaces.

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upvoteby keeganpoppen26 days ago|
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yeah i had that same intuition and then realized that you _have_ to have pieces that overlap responsibility to fully “cover”. amazing puzzle. i can’t claim i solved it, because i didn’t, but i did give it 15 minutes of focused time, which is pretty good.
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upvoteby LeifCarrotson26 days ago|
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It's a bit disappointing, then, that the page's action on clicking the "solution" button is just to set the pieces to:

    const solution = {
      a8: "b1",
      b8: "q1",
      f7: "q2",
      a4: "q3",
      e3: "q4"
    };
It would be cool if it randomly selected one of those 388, so you could click repeatedly and develop an intuition for what kinds of distributions were a valid solution.
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upvoteby Eswo26 days ago|
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ciao, Alex(https://rutar.org/) sent me a super cool visualization of the solutions and I added it.
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