given a diagram with 5 columns and 33 rows. each box in the diagram needs to be colored black or red.

using the pigeonhole principle, prove that there must at least exist 2 rows which are colored exactly the same pattern.

each column has its own pattern, but each square has 2 choices so it is 2^5, or 32, and since 32 is less than 33, then by pigeonhole principle at least 2 rows have the same pattern.

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