Sources page biographical material

5.Z.1. KINGS
Ball. MRE, 4th ed., 1905. Other problems with queens; Extension to other chess pieces, pp. 119-120. Says problems have been proposed for k kings on an n x n, citing L'Inter. des math. 8 (1901) 140, ??NYS.

Gilbert Obermair. Denkspiele auf dem Schachbrett. Hugendubel, Munich, 1984. Prob. 27, pp. 29 & 58. 9 kings strongly, and 12 kings weakly, dominate an 8 x 8 board.

5.Z.2. QUEENS
Here the graph is denoted Q_n, but I will denote γ(Q_n) by γ(n) and i(Q_n) by i(n).
Murray. Pp. 674 & 691. CB249 (c1475) shows 16 queens weakly dominating an 8 x 8 board, but the context is unclear to me.

de Jaenisch. Op. cit. in 5.F.1. Vol. 3, 1863. Appendice, pp. 244-271. Most of this is due to "un de nos anciens amis, M^r de R***". Finds and describes the 91 ways of placing 5 queens so as to non-attackingly dominate the 8 x 8 board. Then considers the n x n board for n = 2, ..., 7 with strong and non-attacking domination. Up through 5, he gives the number of pieces being attacked in each solution which allows one to determine the weak solutions. For n < 6, he gets the answers in the table below, but for n = 6, he gets 21 non-attacking solutions instead of 17?.

Ball. MRE, 3rd ed., 1896. Other problems with queens, pp. 109-110. "Captain [W. H.] Turton has called my attention to two other problems of a somewhat analogous character, neither of which, as far as I know, has been hitherto published, or solved otherwise than empirically." The first is to place 8 queens so as to strongly dominate the fewest squares. The minimum he can find is 53. (Cf Gardner, 1999.) The second is to place m queens, m  5, so as to strongly dominate as many cells as possible. With 4 queens, the most he can find is 62.

Dudeney. Problem 54: The hat peg puzzle. Tit Bits 33 (9 & 30 Oct 1897) 21 & 82. Problem involves several examples of strong domination by 5 queens on an 8 x 8 board leading to a non attacking domination. He says there are just 728 such. This = 8 x 91. = Anon. & Dudeney; A chat with the Puzzle King; The Captain 2 (Dec? 1899) 314-320; 2:6 (Mar 1900) 598-599 & 3:1 (Apr 1900) 89. = AM; 1917; pp. 93-94 & 221.

Ball. MRE, 4th ed., 1905, loc. cit. in 5.Z.1. Extends 3rd ed. by asking for the minimum number of queens to strongly dominate a whole n x n board. Says there seem to be 91 ways of having 5 non-attacking queens on the 8 x 8, citing L'Inter. des Math. 8 (1901) 88, ??NYS.

Ball. MRE, 5th ed., 1911, loc. cit. in 5.Z. On pp. 120-122, he considers queens and states the minimum numbers of queens required to strongly dominate the board and the numbers of inequivalent solutions for 2 x 2, 3 x 3, ..., 7 x 7, citing the article cited in the 4th ed. and Jaenisch, 1862, without a volume number. For n = 7, he gives the same unique solution for strongly dominating as for non-attacking dominating. [In the 6th ed., this is corrected and he says it is a solution.] He says Jaenisch also posed the question of the minimum number of non-attacking queens to dominate the board and gives the numbers and the number of inequivalent ways for the 4 x 4, .., 8 x 8, except that he follows Jaenisch in stating that there are 21 solutions on the 6 x 6. [This is changed to 17 in the 6th ed.]

Dudeney. AM. 1917. Loc. cit. in 5.Z. He uses 'protected' for 'weakly', but he seems to copy the values for 'strongly' from Jaenisch or Ball. His 'not protected' seems to mean 'non-attacking'. However, some values are different and I consequently am very uncertain as to the correct values??

Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 24 25, he shows 5 queens are sufficient to strongly dominate the board and says this is minimal.

Below, min. denotes the minimum number of queens to dominate and no. is the number of inequivalent ways to do so.
STRONG WEAK NON-ATTACKING

n min. no. min. no. min. no.

1 1 1 0 0 1 1

2 1 1 2 2 1 1

3 1 1 2 5 1 1

4 2 3 2 3 3 2

5 3 37 3 15 3 2

6 3 1 4 2 4 17?

7 4 4 5? 4 1

8 5 150 5 41 5 91

Rodolfo Marcelo Kurchan, proposer; Henry Ibstedt & proposer, solver. Prob. 1738 -- Queens in space. JRM 21:3 (1989) 220 & 22:3 (1989) 237. How many queens are needed to strongly dominate an n x n x n cubical board? For n = 3, 4, ..., 9, the best known numbers are: 1, 4, 6, 8, 14, 20, 24. The solution is not clear if these are minimal, but it seems to imply this.

Martin Gardner. Chess queens and maximum unattacked cells. Math Horizons (Nov 1999). Reprinted in Workout, chap. 34. Considers the problem of Turton described in Ball, 3rd ed, above: place 8 queens on an 8 x 8 board so as to strongly dominate the fewest squares. That is, leave the maximum number of unattacked squares. More generally, place k queens on an n x n board to leave the maximum number of unattacked squares. He describes a simple problem by Dudeney (AM, prob. 316) and recent work on the general problem. He cites Velucchi, cf below, who provides the following table of maximum numbers of unattacked cells and number of solutions for the maximum. I'm not sure if some of these are still only conjectured.

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Max 0 0 0 1 3 5 7 11 18 22 30 36 47 56 72 82 97

Sols 0 0 0 25 1 3 38 7 1 1 2 7 1 4 3 1
Mario Velucchi has a web site devoted to the non-dominating queens problem and related sites for similar problems. See: http://www.bigfoot.com/~velucchi/papers.html and http://www.bigfoot.com/~velucchi/biblio.html.

A. P. Burger & C. M. Mynhardt. Symmetry and domination in queens graphs. Bull. Inst. Combinatorics Appl. 29 (May 2000) 11-24. Extends results to n = 30, 45, 69, 77. Summarizes the field, with 14 references, several being earlier surveys. The table below gives all known values. It will be seen that the case n = 4k + 1 seems easiest to deal with. The values separated by strokes, /, indicate cases where the value is one of the two given values, but it is not known which.

n 4 5 6 7 8 9 10 11 12 13 14 15 16 17

γ(n) 2 3 3 4 5 5 5 5 6 7 7/8 8/9 8/9 9

i(n) 3 3 4 4 5 5 5 5 7 7 8 9 9 9
n 18 19 21 25 29 30 31 33 37 41 45 49 53 57

γ(n) 9 10 11 13 15 15 16 17 19 21 23 25 27 29

i(n) 11 13 17 23
n 61 69 77

γ(n) 31 35 39