You want to put several balls on $8 \times 8$ tiles, such that all $16$ ball arrangements on its rows and columns are different. What is the minimum number of balls to be put?
Two arrangements of balls are different if there exists a ball on some $i$-th tile of the first one but no ball on $i$-th tile of the second one. (For row, the $i$-th tile means the $i$-th column; for column, the $i$-th tile means the $i$-th row.)
For example, we need at least $4$ balls for $3 \times 3$ tiles and the configuration as the following:
Bonus: How about $N \times N$?