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# Line Up

## Question

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?
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Firstly, switching entire columns to the left and right does not make any difference to the result and neither does switching entire rows up or down.

Assume x is the guy who is the TOSC (tallest person of the shortest in each column)

Imagine the table looks like below (after I sorted the rows and columns to get x in the bottom right):
A B
x C

A has a bunch of people in the same column as x
C has a bunch of people in the same row as x
B has a bunch of people who are not in the same row or column as x

The SOTR (shortest of tallest person from all the rows) can be someone from A, B, C or x
If SOTR is from A, then that person is taller than x (because x is the shortest from his column)
If SOTR is from B, then that person is taller than the person in the same row from A and everyone in A is taller than x
If SOTR is from C, then that person is taller than x
If SOTR is x, then that person is same height as TOSC.

Therefore the TOSC is always shorter or same height than SOTR.
Edited by vigmeister
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Would they be the same person?

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