Problem:
Twenty-five people of different heights stand in a grid of squares, with one person in each square. We know that each row has a shortest person; suppose Ana is the tallest of these five people. Similarly, we know that each column has a tallest person; suppose Bev is the shortest of these five people.
Assuming Ana and Bev are not the same person, who is taller: Ana or Bev? Prove that your answer is always correct.
Solution:
Bev is taller. We consider three possible cases.
Case 1: If Ana and Bev are in the same row, then Bev is taller because Ana is by definition the shortest in that row.
Case 2: If Ana and Bev are in the same column, then Bev is taller because Bev is by definition the tallest in that column.
Case 3: If Ana and Bev are in neither the same row nor the same column, then let "Casey" be the person at the intersection of Ana's row and Bev's column. Then Ana is shorter than Casey, and Casey is shorter than Bev, so Ana is shorter than Bev.