Home > Combinatorics, Olympiad, Problem Solving > Snails in a triangular array

## Snails in a triangular array

Suppose we have a triangular array in the shape of an equilateral triangle of side $10$, and draw all parallel lines to the edges of the triangle such that $100$ small equilateral triangles are formed.
Place a snail in each of the 100 small triangles. It is known that in $10$ seconds, each snail moves from his cell to an adjacent one. Prove that after $10$ seconds, at least $10$ triangle cells are free.

Solution: Color the array in a chessboard style, with black and white, such that any two adjacent cells have different colors. Then, notice that the three corner cells have the same color, without loss of generality, suppose that color is black. Then there are $1+2+...+10=55$ black squares and $1+2+...+9=45$ white squares. Since the “white” snails will turn “black” in 10 seconds and all the “black” snails will move from their cells after 10 seconds, at least 10 black cells will be free after 10 seconds, and this is because there are not enough white snails initially on the board.

This can be easily generalized to a triangular array with $n^2$ triangles, and then at least $n$ cells will be free after 10 seconds. This is a nice coloring argument, similar to the domino tilling of the chessboard with two opposite corners removed.