Treebeard's Stumper Answer

Peaceful Queens and Warriors Spoilers
It's a rainy weekend, so I have time to play! Here are some answers and hints for the additional questions posed on my Peaceful Queens and Warriors answer page. Some of these solutions are from Eric Weisstein's World of Mathematics.Most generally, how many ways can n queens be placed on a n x n chessboard? And what's the least number of queens (<= n) that can cover every square? Each of the remaining questions can also be generalized for n x n boards.
The number of ways that n queens be placed on a n x n chessboard is given by Sloane's A000170.
n x n All solutions Unique solutions 1 1 1 2 0 0 3 0 0 4 2 1 5 10 2 6 4 1 7 40 6 8 92 12 9 352 46 10 724 92 11 2680 341 12 14200 1787 13 73712 9233 14 365596 45752 15 2279184 285053 16 14772512 1846955 17 95815104 11977939 18 666090624 83263591 19 4968057848 621012754 20 39029188884 4878666808 21 314666222712 39333324973 22 2691008701644 336376244042 23 24233937684440 3029242658210 I couldn't find much info on the general problem of the least number of warrior queens that can cover every square. Eric Weisstein gives these values from Dudeney:
k queens n x n board All solutions Unique solutions 2 4 3 3 5 37 3 6 1 4 7 5 5 8 4860 638 We can ask the same questions about rooks, bishops, knights, and kings. (Pawns are excluded since they can only move in one direction.) What are the most peaceful pieces, and the fewest warrior pieces for each chess piece?
Piece Most Least Bishops 14 8 Kings 16 9 Knights 32 12 Queens 8 5 Rooks 8 8
The five warrior queens question can be made more specific. Can you place five peaceful queens on a chessboard that attack every square, but not each other? Can you place five warrior queens on a chessboard that attack every square and each other?
None All Graybear, Donna, and I all wondered if it's possible to cover the board with just four queens. It's not. But what's the least number of unattacked squares for four queens on an 8x8 board?
I believe the best that can be done leaves 2 unattacked (yellow) squares.
The eight queens stumper can be turned inside out into the nondominating queen stumper. Find an arrangement of n queens on a n x n chessboard that leaves the greatest number of unattacked squares.
Mario Velucchi has a quirky postscript paper on the NonDominating Queens Problem that gives 7 solutions with 11 free squares on an 8 x 8 chessboard with 8 queens. Here's one of his solutions. The yellow squares are unattacked.
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