Sources page biographical material

5.Z.3. BISHOPS
Dudeney. AM. 1917. Prob. 299: Bishops in convocation, pp. 89 & 215. There are 2ⁿ ways to place 2n 2 bishops non attackingly on an n x n board. At loc. cit. in 5.Z, he says that for n = 2, ..., 8, there are 1, 2, 3, 6, 10, 20, 36 inequivalent placings.

Pál Révész. Op. cit. in 5.I.1. 1969. On pp. 25 26, he shows the maximum number of non attacking bishops on one colour is 7 and there are 16 ways to place them.

Obermair. Op. cit. in 5.Z.1. 1984. Prob. 17, pp. 23 & 50. 8 bishops strongly, and 10 bishops weakly, dominate the 8 x 8 board.
5.Z.4. KNIGHTS
Ball. MRE, 4th ed., 1905. Loc. cit. in 5.Z.1. Says questions as to the maximum number of non-attacking knights and minimum number to strongly dominate have been considered, citing L'Inter. des math. 3 (1896) 58, 4 (1897) 15-17 & 254, 5 (1898) 87 [5th ed. adds 230 231], ??NYS.

Dudeney. AM. 1917. Loc. cit. in 5.Z. Notes that if n is odd, one can have (n²+1)/2 non attacking knights in one way, while if n is even, one can have n²/2 in two equivalent ways.

Irving Newman, proposer; Robert Patenaude, Ralph Greenberg and Irving Newman, solvers. Problem E1585 -- Nonattacking knights on a chessboard. AMM 70 (1963) 438 & 71 (1964) 210-211. Three easy proofs that the maximum number of non-attacking knights is 32. Editorial note cites Dudeney, AM, and Ball, MRE, 1926, p. 171 -- but the material is on p. 171 only in the 11th ed., 1939.

Gardner. SA (Oct 1967, Nov 1967 & Jan 1968) c= Magic Show, chap. 14. Gives Dudeney's results for the 8 x 8. Golomb has noted that Greenberg's solution of E1585 via a knight's tour proves that there are only two solutions. For the k x k board, k = 3, 4, ..., 10, the minimal number of knights to strongly dominate is: 4, 4, 5, 8, 10, 12, 14, 16. He says the table may continue: 21, 24, 28, 32, 37. Gives numerous examples.

Obermair. Op. cit. in 5.Z.1. 1984. Prob. 16, pp. 21 & 47. 14 knights are necessary for weak domination of the 8x8 board.

E. O. Hare & S. T. Hedetniemi. A linear algorithm for computing the knight's domination number of a k x n chessboard. Technical report 87 May 1, Dept. of Computer Science, Clemson University. 1987?? Pp. 1 2 gives the history from 1896 and Table 2 on p. 13 gives their optimal results for strong domination on k x n boards, 4  k  9, k  n  12 and also for k = n = 10. For the k x k board, k = 3, ..., 10, they confirm the results in Gardner.

Anderson H. Jackson & Roy P. Pargas. Solutions to the N x N knight's cover problem. JRM 23:4 (1991) 255-267. Finds number of knights to strongly dominate by a heuristic method, which finds all solutions up through N = 10. Improves the value given by Gardner for N = 15 to 36 and finds solutions for N = 16, ..., 20 with 42, 48, 54, 60, 65 knights.
5.Z.5. ROOKS
É. Lucas. Théorie des Nombres. Gauthier Villars, Paris, 1891; reprinted by Blanchard, Paris, 1958. Section 128, pp. 220 223. Determines the number of inequivalent placings of n nonattacking rooks on an n x n board in general and gives values for n  12. For n = 1, ..., 8, there are 1, 1, 2, 7, 23, 115, 694, 5282 inequivalent ways.

Dudeney. AM. 1917. Loc. cit. at 5.Z. Notes there are n! ways to place n non attacking rooks and asks how many of these are inequivalent. Gives values for n = 1, ..., 5. AM prob. 296, pp. 88 & 214, is the case n = 4.

D. F. Holt. Rooks inviolate. MG 58 (No. 404) (Jun 1974) 131 134. Uses Burnside's lemma to determine the number of inequivalent solutions in general, getting Lucas' result in a more modern form.