In a rectangular array of people, who will be taller: the tallest of the
shortest people in each column, or the shortest of the tallest people in each
row?

Give up?

*Yeah, I would too. Wait until you see the solution below*

Drag your cursor between the asterisks for the solution:

*****

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.

Makes perfect sense.

*****

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