I once knew a fellow who was a bit long of leg and short of foot.

The soles of his shoes were, in fact, exactly 9 inches long and his stride was exactly 35 inches. He had a habit of counting his steps when they were all on the same slab of the sidewalk and saying "Boing!" every time he stepped on a crack. If he stepped over a crack his counting started again at one, and of course his counting started at one after each "Boing!"

In his neighborhood there was a sidewalk with perfectly regular slabs all the same size. He noticed that when he walked along this sidewalk he always got the following repeating pattern (where "*" stands for "Boing!"):

121231231212312312*121231231212312312*121231....etc.

How far apart, in inches, were the cracks in the sidewalk?

He takes
2.5 steps in one slab
3 in the next, 3, 2, 3, 3, 2.5
How do I know it's 2.5 and not 2.9 or something? Well, even if it were 2.9 the first time, it would be 2.1 the last time. If the two decimals don't add to a whole number, then he is stepping on the crack with at a different point on his shoe and the pattern cannot continue forever.
Add the numbers up (19) divide by the 7 slabs, and you come up with 2.714285714285714285 and so on.. average steps per slab.
Now, for his pattern to continue forever, his average steps * stride (35") = "average" slab size.. and since all slabs are of the same size, all slabs are 95"

Posted by Carleton Dunn on 2003-09-21 14:55:17