INMO - 1992
Find the number of ways in which one can place the numbers on the squares of chess board, one on each, such that the numbers in each row and each column are in arithmetic progression. (Assume n ≥ 3).
The answer is 8.
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1 solution
case 1: n is odd
subcase: when n=3
clearly, 1 and 9 must be diagonally opposite, since if it is not the case then their A.Ps will be both strictly increasing and decreasing
this fixes the position for 5, and now it is easy to check that we have 2*4=8 orientations
now, for any n=2k+1, take the middlemost [3 by 3] matrix and you would recognize that this is exactly the same as that for n=3(apart from starting term), thus there 8 orientations and working backwards from middle we get the desired configurations.
case2: n is even
subcase: n=2
easy to check that there are 8 orientations
do the same trick with the middlemost[2 by 2] matrix and you are done.