MatheMUSEments

Articles for kids about math in everyday life, written by Ivars Peterson for Muse magazine.

June 2, 2007

Knight's Tour

"I'm going to be a knight," said Ron. With that remark, Ron Weasley took charge of a deadly, larger-than-life game of wizard chess, played near the climax of Harry Potter and the Sorcerer's Stone.

Deciding to be a knight was an interesting choice. Chess has six different kinds of pieces: king, queen, bishop, knight, rook, and pawn. Each piece moves in its own special way. The knight stands out because, unlike the other pieces, it doesn't move in a straight line. It makes L-shaped moves, jumping over anything in its way to reach an empty square on a chessboard.

For example, a knight can move two squares forward, then one square sideways, or it can move one square forward, then two squares sideways.

The special way in which a knight moves not only adds drama to chess games but also suggests various puzzles mathematicians and others have spent untold hours trying to solve.

One of the oldest of these puzzles is called the knight's tour: Given the ways in which a knight is allowed to move, is it possible for a knight to start at one square on an empty chessboard, then land on each of the other squares, visiting each square exactly once?

Try it. Get out a chessboard and a knight. Pick a starting point. Make legal knight moves, keeping track of each square the knight visits.

In fact, it turns out there's a huge number of ways a knight can make such a tour. In some cases, the knight ends up at its starting point, after visiting all the other squares once.

Here's another, tougher puzzle. What is the smallest number of knights that you would need to place on a chessboard so that every vacant square could be reached by a knight in one move? For a standard chessboard, the answer is 12. Any idea where you would put them? The answer can be found at www.contestcen.com/kn3.htm.

As you can see, board games such as chess can keep you busy in all sorts of ways.

Muse, April 2004, p. 35.