Saturday, September 25, 2010

Completable Paintings

An $n \times n$ checker board is painted all white except $m$ cells that are painted black. At each step, one is allowed to choose a rectangle that has three black corners, and paint the fourth corner black. A configuration is called completable if one can paint the entire board black using a sequence of steps described above.

First Question:
Find the smallest $m$ such that any configuration with $m$ black cells are always completable.

Second Question:
Find the smallest $m$ such that there is a completable configuration with $m$ black cells.