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 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 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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