10th Mexico 1996 Problem B1 For which n can we arrange the numbers 1, 2, 3, ... , 16 in a 4 x 4 array so that the eight row and column sums are all distinct and all multiples of n?
1,2,4 Solution 1+2+...+16 = 2 3 17, so if the row sums are all multiples of n, then n must divide 2 3 17. But if all the row and column sums are multiples of 17, then the largest is at least 8·17 and hence the numbers add up to more than 8·17. Contradiction. So n must be 1, 2, 4 or 8. If n = 8, then the largest row/column sum must be at least 64. But the largest possible row/column sum is 16+15+14+13 = 58 < 64. So n must be 1, 2 or 4. The example below shows that these are all possible. 1 8 3 4 16 5 6 7 2 20 9 10 11 14 44 13 16 15 12 56 28 40 36 32 10th Mexican 1996 © John Scholes jscholes@kalva.demon.co.uk 9 March 2004 Last corrected/updated 9 Mar 04