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## Senior Prank

A university has had $50$ presidents, and their portraits line a hallway of the student center. Every year, on April Fool’s Day, $50$ students from the senior class get together to pull a prank: they paint over the portraits in an elaborate way. At the end of the prank, some portraits are yellow and some are blue — but how can we tell which color a given portrait will be? It turns out we can find the answer without walking down the hallway and checking.