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.