Consider the 4(N−1) squares on the boundary of an N by N array of squares. We wish to insert in these squares 4(N−1) consecutive integers (not necessarily positive) so that the sum of the numbers at the four vertices of any rectangle with sides parallel to the diagonals of the array (in the case of a “degenerate” rectangle, i.e. a diagonal, we refer to the sum of the two numbers in its corner squares) are one and the same number.
Is this possible? Consider the cases
(a) N=3
(b) N=4
(c) N=5(VG Boltyanskiy, Moscow) combinatoricssquare tableconsecutive