Solution for #24

This is a tongue twister of an explanation, but bear with me.

The shortest of the tallest people in each row will be taller than, or the same height as, the tallest of the shortest people in each column. There are four cases. The first is that the shortest of the tallest and the tallest of the shortest are the same person, so obviously in this case the shortest of the tallest and the tallest of the shortest would be the same height.

The second case is that the shortest of the tallest and the tallest of the shortest are in the same row. The shortest of the tallest people in each row is obviously the tallest person in his row, so he's taller than the tallest of the shortest, who is also in his row.

The third case is that the shortest of the tallest and the tallest of the shortest are in the same column. The tallest of the shortest people in each column is obviously the shortest person in his row, so he's shorter than the shortest of the tallest, who is also in his column.

The fourth case is that the shortest of the tallest is neither in the same column nor the same row as the tallest of the shortest. For this case, consider the person X who is standing in the intersection of the row containing the shortest of the tallest and the column containing the tallest of the shortest. X must be taller than the tallest of the shortest, since the tallest of the shortest is the shortest in his column, and X must also be shorter than the shortest of the tallest, since the shortest of the tallest is the tallest in his row. So TofS < X < SofT.

So the shortest of the tallest in each row is always taller than, or the same height as, the tallest of the shortest in each column.