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Find the greatest natural number N

Source:

November 4, 2010
symmetryceiling functiongeometryrectanglecombinatorics unsolvedcombinatorics

Problem Statement

Find the greatest natural number NN such that, for any arrangement of the numbers 1,2,…,4001, 2, \ldots, 400 in a chessboard 20×2020 \times 20, there exist two numbers in the same row or column, which differ by at least N.N.
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