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1,2,3 in 100x100 square table , balanced 2x2 subtables

Source: Czech-Polish-Slovak Junior Match 2017, Team p5 CPSJ

March 6, 2020
combinatoricstablenumbers in a table

Problem Statement

In each square of the 100×100100\times 100 square table, type 1,21, 2, or 33. Consider all subtables m×nm \times n, where m=2m = 2 and n=2n = 2. A subtable will be called balanced if it has in its corner boxes of four identical numbers boxes . For as large a number kk prove, that we can always find kk balanced subtables, of which no two overlap, i.e. do not have a common box.
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