Variations on a Theme of Tricky Math Problems

Tyson posted a couple of cute problems on his xanga today. (I’m not sure how to link to a particular post on xanga, so if you follow the link in the distant future, you will just see whatever has been on his mind most recently.)

Here, I present them with skins that I like a little better.

1) I rode my bike to the Berkeley area today in hopes of finding someplace slightly more hospitable than a gutter to live next year. I took the same route there and back. I rode 14 mph on all the uphills, 21 mph on all the downhills, and some constant speed “x” on all the flat parts. It took me five hours to get there and back combined. If I were to tell you “x”, you would be able to figure out how long my route was. And, now that I told you that, you can figure out how long my route was without any more information.

2)There is a castle with four identical circular towers arranged with their centers in a square, so that there’s a small courtyard in the middle. The edges of each tower are tangent to their two nearest neighbors.

Each tower has a lookout who stands at the center. To avoid confusion, the king has given the lookouts very strict rules. They can only walk in one of the four cardinal directions (fortunately these are exactly the directions between the centers of the towers). They must always use the same step size. When they walk North or East, they must count their steps aloud, starting at zero and going up. But when they walk South or West, they must count down.

One day the lookout in the Northeast tower thinks he sees something suspicious on the wall of the Southwest tower. However, it’s at a spot so that his line of sight is tangent to the wall of the Southwest tower there. He calls over to the lookout of the Southwest tower to go check it out.

If it is 100 strides from the center of one tower to the center of the next, what number will the lookout of the Southwest tower call when he gets to the spot indicated by the lookout in the Northeast tower?

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