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?

For the purposes of this argument, I am going to assume a crack has zero width, but is noticeable anyway.

Since the solution repeats, we realize that the initial '1' and the '1' following each boing should be equidistant from the crack which caused the 'boing'.

So, we see, that there are 7 slabs before we get back to the beginning of the cycle.

We can also count the steps between cycles, and we see there are 19 steps between each cycle.

19 steps (x 35 inches/step) = 665 inches for a complete cycle

And again, there are 7 slabs for each cycle, so 665 inches / 7 slabs = 95 inches/slab.

So, the cracks are 95 inches apart.

Edited on September 17, 2003, 12:21 pm

Posted by SilverKnight on 2003-09-17 12:21:05