The Wednesday Morning Math Challenge: Week 7

Welcome to Week 7 of the Wednesday Morning Math Challenge. You can find solutions to the Week 6 puzzles here, along with — more importantly — discussions of some ways to approach them. Remember that the real goal here is to think creatively.

This week we’re going to have some fun with chess. I can’t claim the three puzzles this week. These puzzles were suggested by child prodigy and chess grandmaster Robert Hess. Besides being one of my best friends, Robert achieved the grandmaster title at the age of 19 and currently helps run, the largest chess site in the world. Without further ado, here are three puzzles involving chess, specifically knights.

1.A closed knights tour is a sequence of moves of a knight on the chess board where the knight visits each square on the board only once, and returns to its original square. Can you find a knight’s tour on an eight-by-eight chess board?

2. Consider the same concept of a closed knights tour, except now on a five-by-five board. Can you find one?

3. Lets generalize problem 2. For any rectangular n x m board, (n,m>5)  with an odd number of tiles, does there exist a closed knights tour? Why or why not?