# Soldiers in formation

In the parade of the independence day, an army of 10,000 soldiers maintains a formation of 100 rows with 100 soldiers each and 100 columns.

Two friends watching the parade have the following conversation:

-Look, there is my boyfriend Juan. It's funny but if we take the highest soldier in each column, he would be the lowest of them all.

-Well !, says her friend. With my boyfriend Pedro something similar happens. If we take the lowest of each row, he would be the highest of those 100.

If we know that no two soldiers are exactly the same height, Who is taller, Juan or Pedro?

#### Solution

Suppose Juan is in the column j and therefore is the highest in the column j.

Pedro will be in line i and is the lowest in that row.

If both are in the same column, Juan is taller than Pedro (since Juan is the highest in his column) and if both are in the same row, Juan is also taller than Pedro (since Pedro is the lowest of your row)

Consider that they are not in the same row or column. Let Luis be the man in line icolumn j.
Juan is taller than Luis by being in the column j and Pedro is shorter than Luis by being in line i.

By transitive property, Juan is taller than Pedro.