Problem of the Week 801:

90 x 5 = 0

Anyone can take a piece of paper and apply four 90-degree rotations so that it returns to its starting position. Show how one can take a piece of paper and apply FIVE 90-rotations so that it ends up in the same position it started in.

Note: As observed by Dan Velleman (Amherst), this is in fact possible with 3 rotations, though the solution is not as pretty as the 5-rotation solution. I have started, but not completed, the task of working out the exact rotation matrices that work in the 3 case.

© Copyright 1996 Stan Wagon. Reproduced with permission.

The Math Forum

2 October 1998