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board 100x100 with mallest possible number of different residues modulo 33

Source: Rioplatense Olympiad 2012 level 3 P6

September 4, 2018

number theoryboardcombinatorics

Problem Statement

In each square of a

100 \times 100

board there is written an integer. The allowed operation is to choose four squares that form the figure or any of its reflections or rotations, and add

1

to each of the four numbers. The aim is, through operations allowed, achieving a board with the smallest possible number of different residues modulo

33

. What is the minimum number that can be achieved with certainty?

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