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Product of n numbers in scattered cells gives same residue.

Source: Serbia NMO 2010 problem 5

March 11, 2011
linear algebramatrixcombinatorics unsolvedcombinatorics

Problem Statement

An n\times n table whose cells are numerated with numbers 1,2,⋯ ,n21, 2,\cdots, n^2 in some order is called Naissus if all products of nn numbers written in nn scattered cells give the same residue when divided by n2+1n^2+1. Does there exist a Naissus table for (a)n=8;(a) n = 8; (b)n=10?(b) n = 10? (nn cells are scattered if no two are in the same row or column.)
Proposed by Marko Djikic
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