Two hundred students are arranged in 10 rows of 20 children. The shortest student in each column is identified, and the tallest of these is marked \(A\). The tallest student in each row is identified, and the shortest of these is marked \(B\).
If \(A\) and \(B\) are different people, who is taller?
\(B\) is taller.
Proof
There are three different cases:
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If \(A\) and \(B\) stand in the same row, then \(B\) is taller, since we know that \(B\) is the tallest student in their row.
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If \(A\) and \(B\) stand in the same column, then again \(B\) is taller, since we know that \(A\) is the shortest student in their column.
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If \(A\) and \(B\) share neither a row nor a column, then let \(C\) be the student who’s in the same column as \(A\) and the same row as \(B\). Then \(C\) is shorter than \(B\) (who is the tallest in their row) and taller than \(A\) (who is the shortest in their column), so \(B > C > A\).
In every case, \(B\) is taller than \(A\).