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?
(In reply to
more thoughts (update to problem) by SilverKnight)
I'll take a run at it....
Clearly. SilverKnight's original analysis is correct. If in fact the pattern given by DJ repeats forever (long sidewalk there...can he ever reach the end? Maybe if he runs faster and faster....always covering twice the distance in the same time... Oh, sorry).
Anyway, if it does repeat forever, then the distance between cracks must be EXACTLY 95 inches. Otherwise the discrepancy will accumulate, ruining the pattern.
Ignore the size of the foot, and consider only where the heel lands. To simplify things, I'm going to do everything modulo the slab distance (as if he were walking on a treadmill with a belt 95 inches long).
Starting at 0 and going on for 19 steps, we have the heel landing at these distances:
0 35 70 10 45 80 20 55 90 30 65 5 40 75 15 50 85 25 60
Sorting these, we arrive at these locations within a slab, where his heel will land:
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Now, if his foot is > 5 inches long, he will obviously cover every point on the slab, and hence hit a crack, somewhere in his 19 steps. Since he completely covers the slab, in fact it does not matter WHERE he starts -- the above covering can be shifted by any amount, and it will still cover the slab.
In the original sequence, 60 is the last landing point for his heel. If we want him to miss the crack on the first 18 steps, we can offset his starting point by 30 inches, and the series becomes:
30 65 5 40 75 15 50 85 25 60 0 35 70 10 45 80 20 55 90
If we consider 0 as a "miss", then in this case, with a foot >5 inches long, he will hit the crack on the 19th step. (Or offset by 30.00001 inches if you'd rather).
Now, to answer SilverKnight's question. Suppose his foot is >= 10 inches long? Going back to the original sequence, we find that he will hit every point on the slab within 18 steps. His last step, covering 60-70, was fully covered by steps 8 (55-65) and 11 (65-75).
So no matter where he starts, he covers the whole slab in less than 19 steps, which violates DJ's pattern.
re: more thoughts (update to problem)
