IMO 1984 SL 7

Origin: FRG

Problem

(a) Decide whether the fields of the 8 \times 8 chessboard can be numbered by the numbers 1, 2, . . ., 64 in such a way that the sum of the four numbers in each of its parts of one of the forms is divisible by four. (b) Solve the analogous problem for

Solution

It clearly suffices to solve the problem for the remainders modulo 4 (16 of each kind). (a) The remainders can be placed as shown in Figure 1, so that they satisfy the conditions. 1 0 3 2 1 0 3 2 2 3 0 1 2 3 0 1 3 2 1 0 3 2 1 0 0 1 2 3 0 1 2 3 1 0 3 2 1 0 3 2 2 3 0 1 2 3 0 1 3 2 1 0 3 2 1 0 0 1 2 3 0 1 2 3 p q r s t Fig. 1 Fig. 2 (b) Suppose that the required numbering exists. Consider a part of the chessboard as in Figure 2. By the stated condition, all the numbers

p + q + r + s, q + r + s + t, p + q + r + t, p + r + s + t give the same remainder modulo 4, and so do p, q, r, s. We deduce that all numbers on black cells of the board, except possibly the two corner cells, give the same remainder, which is impossible.