The Big Knight Switch--Hints

1. Make sure you understand your goal. If you want to move the knights on the top to the bottom, and the ones on the bottom to the top, how many ways can they end up?

If you try to get them to end up one way and that doesn't seem to be working try for a different ending.

2. The knights like symmetry.

3. Different ways of representing the problem are often helpful. One approach is to put a dot for each location that the knights can have. Connect two dots with a line segment, if a knight at one dot can move directly to the other dot. (The dots don't have to lie in a checkerboard.)

Any such collection of dots and line segments is called a graph. Graphs are often used to study transportation problems.

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