Prove that no matter how each cell of a 5 x 41 table is filled with a 0 or 1, one can choose 3 rows and 3 columns which intersect in 9 cells filled with identical numbers.

Prove that 41 is the lowest possible n for 5 x n table; i.e., the statement is not true for a 5 x 40 table.

Source: Colorado math contest.

(In reply to re(2): Example needed by broll)

I'm sure it was not intended to be any more esoteric than that the three rows intersect the three columns, just as each row intersects each column.  In this case, the set of intersections numbers 9, and it is those 9 that are intended to match.

Posted by Charlie on 2015-11-30 19:09:30