5
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Similarly to this question: How many Queens to threaten themselves.

What it the maximum number of queens you can place on an 8*8 board such that each queen is attacked precisely twice:

  1. with no other pieces on the board?

  2. other pieces are allowed?

I thought this was a legitimately interesting variation and it seems to be a more or less undocumented result as well.

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2 Answers 2

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10
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Update:
I was looking through old puzzles and saw this one and was always annoyed I couldn't get a 14th queen. So I found a way.
14 queens:
enter image description here

Original Answer:
I've managed 13 queens. I started in one corner and then put them on opposite diagonals. I then tweaked it so that all the queens were threatened twice. I was hoping for 14, but I'm starting to think 13 is the maximum.

13 Queens

I've tried a few different ways with other pieces. The most I've managed is 32. Here's an alternative example:

32 Queens

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8
  • $\begingroup$ I'd move the corner queens one square just to provide more symmetry to the second solution. Also, there's no penalty to introducing more black pawns. $\endgroup$
    – John Dvorak
    Oct 13, 2014 at 8:36
  • $\begingroup$ Only one more black pawn would be ok without ruining it. I'm a bit surprised about the lack of symmetry in the main solution. Also, what do you reckon about casting the problem as a graph problem to prove that 13 is the max? $\endgroup$
    – d'alar'cop
    Oct 13, 2014 at 9:04
  • $\begingroup$ I'd be more inclined to brute force it, as I'd have to read up on casting as a graph problem. Just been too long since I've done that. $\endgroup$
    – Joel Rondeau
    Oct 13, 2014 at 14:24
  • $\begingroup$ I'm curious. Has anyone made progress towards brute forcing this? It is not an easy problem to do that with as there are 47 trillion arrangments and no extremely easy was to shorten it. $\endgroup$
    – kaine
    Oct 13, 2014 at 18:59
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    $\begingroup$ I confirmed via integer linear programming that $14$ is optimal. $\endgroup$
    – RobPratt
    Jul 10, 2023 at 18:48
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I found a solution based on the other version of this problem: 12 queens.

enter image description here

I also found a solution where other pieces are permitted: 32 queens:

enter image description here

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6
  • $\begingroup$ Can you get more than 32 if you try it on a diagonal, instead of horizontal? $\endgroup$
    – warspyking
    Oct 12, 2014 at 22:35
  • $\begingroup$ @warspyking Try it. But I don't think so $\endgroup$
    – d'alar'cop
    Oct 12, 2014 at 22:42
  • $\begingroup$ I don't have a program to try it with. $\endgroup$
    – warspyking
    Oct 12, 2014 at 23:12
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    $\begingroup$ @warspyking WinBoard for windows, xBoard for linux $\endgroup$
    – d'alar'cop
    Oct 12, 2014 at 23:14
  • $\begingroup$ I don't really like downloading things, have any online-mobile friendly sites? $\endgroup$
    – warspyking
    Oct 12, 2014 at 23:19

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