this is a problem I wanted to share with you that I just saw today. There is a board square board (think of chess) with $10$ columns and $n$ rows. Each square contains a digit (an integer between $0$ and $9$.The board satisfies the property that for any row $c$ and any columns $a$ and $b$ there is another row $d$ which is different from row $c$ exactly in columns $a$ and $b$ (so they have the same digits in all columns, except for columns $a$ and $b$ in which they are different. Prove the board has at least $512$ rows.
This problem is from an olympiad from my university, called the Galois-Noether competition.