Problem 1:  Pick any integer greater than or equal to 1.  Multiply it by 9.  Permute its digits in any way.  Show that the number you get is a multiple of 9.

 

Problem 2:  "The Infected Checkerboard"  An infection spreads among the squares of an n by n checkerboard in the following manner:  If a square has 2 or more infected neighbors, then it becomes infected itself.  Neighbors are orthogonal only, so each square has at most 4 neighbors.  Show that you cannot infect the whole board if you begin with fewer than n infected squares.