5.I.2. COLOURING CHESSBOARD WITH NO REPEATS IN A LINE
New section. I know there is a general result that an n x n board can be n coloured if n satisfies some condition like n 1 or 5 (mod 6), but I don't recall any other old examples of the problem.
Dudeney. Problem 50: A problem in mosaics. Tit Bits 32 (11 Sep 1897) 439 & 33 (2 Oct 1897) 3. An 8 x 8 board with two adjacent corners omitted can be 8 coloured with no two in a row, column or diagonal. = 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.
Dudeney. AM. 1917. Prob. 302: A problem in mosaics, pp. 90 & 215-216. The solution to the previous problem is given and then it is asked to relay the tiles so that the omitted squares are the (3,3) and (3,6) cells.
Hummerston. Fun, Mirth & Mystery. 1924. Q.E.D. -- The office boy problem, Puzzle no. 30, pp. 82 & 176. Wants to mark the cells of a 4 x 4 board with no two the same in any 'straight line ..., either horizontally, vertically, or diagonally.' His answer is: ABCD, CDEA, EABC, BCDA, which has no two the same on any short diagonal. The problem uses coins of values: A, B, C, D, E = 12, 30, 120, 24, 6 and the object is to maximize the total value of the arrangement. In fact, there are only two ways to 5-colour the board and they are mirror images. Four colours are used three times and one is used four times -- setting the value 120 on the latter cells gives the maximum value of 696.
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