perplexus.info :: Numbers : BOING!

BOING! (Posted on 2003-09-17)

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?

Submitted by DJ

Rating: 4.1538 (13 votes)

Solution: (Hide)

95 inches.
The simplest way to approach this is to notice that because the pattern repeats itself, each iteration must be exactly the same. The total length covered in each repetition is equal to the length of his stride (call this l) multiplied by the number of steps he took (n); it is also equal to the number of slabs (s) times the length of each slab (which is, of course, the distance between the cracks, call this c).
Thus, we have that:
l×n = s×c.
From this:
c = l×n÷s
His stride is 35 inches long, and we see that he takes 19 steps each time, crossing seven slabs, so:
c = 35 × 19 ÷ 7
= 95
So, each slab must be 95 inches wide, and the cracks are 95 inches apart.
This is the only possible solution; and while the length of his shoes are not necessarily used to solve the problem, that is also the only value for which the given pattern is possible. Detailed explanations for this, as well as other methods for solving, can be found in the problem comments.

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
answer	K Sengupta	2010-06-06 19:34:06
??solution??	Bry	2003-09-23 21:06:44
Solution	Carleton Dunn	2003-09-21 14:55:17
re: solution(possibly)	abc	2003-09-19 15:22:18
solution(possibly)	brandi	2003-09-19 13:51:02
re(3): Solution	abc	2003-09-18 13:00:21
re(2): Just a thought	Sam	2003-09-18 11:43:35
re: Just a thought	SilverKnight	2003-09-17 23:25:13
Just a thought	Sam	2003-09-17 22:36:42
re(2): more thoughts (update to problem)	SilverKnight	2003-09-17 18:21:36
re: more thoughts (update to problem)	Brian Wainscott	2003-09-17 18:10:32
more thoughts (update to problem)	SilverKnight	2003-09-17 15:38:01
re(2): Solution	SilverKnight	2003-09-17 15:21:00
re: Solution	abc	2003-09-17 14:31:17
Solution	SilverKnight	2003-09-17 12:21:05

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