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Treebeard's Stumper Answer
4 February 2000

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 nAll solutionsUnique solutions
1 11
2 00
3 00
4 21
5 102
6 41
7 406
9 35246
10 72492
11 2680 341
12 14200 1787
13 73712 9233
14 365596 45752
15 2279184285053
16 14772512 1846955
17 95815104 11977939
18 666090624 83263591
19 4968057848621012754
20 390291888844878666808
21 314666222712 39333324973
22 2691008701644336376244042
23 242339376844403029242658210

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 queensn x n boardAll solutionsUnique solutions
2 43 
3 537 
3 61 
4 75 

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?

Bishops 148
Kings 169
Knights 3212
Queens 85
Rooks 88

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 non-dominating 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 Non-Dominating 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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