Puzzling Pathways

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Puzzling Pathways

When you, Anita, and Bill reach the edge of Checkerboard City (see "A Familiar Place"), you find three bicycles and three parallel bike paths (see "The Bumpy Bike Path"). A sign points to Pentagon Plaza.

All of the wheels have about the same diameter, but one bike has square wheels, another has hexagonal wheels, and the third has octagonal wheels.

A bike with a square wheel on a bumpy pathway.

All of the bike paths are bumpy, but one path has long, tall bumps, one has medium-size bumps, and one has short, relatively low bumps.

Anita picks up the square-wheeled bike, Bill takes the hexagonal-wheeled bike, and you grab the octagonal-wheeled bike.

Match each wheel with its bike path.

Which path should Anita take? What about Bill? Which path will give you the smoothest ride on your octagonal wheels (see "Different Roads for Different Wheels")?

Answers:

As the number of sides on a wheel in the shape of a regular polygon increases, the wheel fits shorter and flatter "bumps," or inverted catenaries. (The size of the wheel's radius also affects what size bumps the wheel fits, but in this case, all three bikes have wheels of roughly the same size.)

Anita's square-wheeled bike fits the path with the longest, tallest bumps. Bill's hexagon-wheeled bike fits the path with medium-size bumps, and your octagonal-wheeled bike will get the smoothest ride on the path with the shortest, smallest bumps.

NEXT: Parting Ways