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.

Three white markers are placed on the first three squares of a row of seven squares. There is a space of one square and then three black markers.

White markers can only move right.

Black markers can only move left.

Markers can move forward one square, or can jump over a marker of either colour if there is an empty square to land on.

Markers are not removed from the board if jumped.

You DO NOT have to alternate moving black and white markers.

How can you get the markers to exchange places?

+---+---+---+---+---+---+---+
| W | W | W | | B | B | B |
+---+---+---+---+---+---+---+