Distance between Points on a Line

Date: 10/02/2002 at 20:15:37
From: CJ
Subject: Patterning/Problem solving

Five points are located on a line. When the ten distances between 
pairs of points are listed from smallest to largest, the list reads: 
2,4,5,7,8,k,13,15,17,19. What is the value of k?

Any help would be greatly appriciated!

Date: 10/02/2002 at 23:19:45
From: Doctor Peterson
Subject: Re: Patterning/Problem solving

Hi, CJ.

Interesting problem!

The first thing I see is that the smallest distance is 2 (so none of 
the points are closer than 2 units apart), and the largest is 19 (so 
the farthest two points are that far apart). Also, the distances are 
all whole numbers, so the points can all be taken as integers on a 
number line. We can starting by calling those points 0 and 19:

    o--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--o
    0                                                       19

Now it occurs to me that if we put any point between these, it will 
give us two distances whose sum has to be 19. So I look at the given 
distances to find pairs that add up to 19. These are

    2+17
    4+15

That gives us a couple of possibilities for where two of the remaining 
three points are. Either we can play with those possibilities and see 
which work, or we can look for more facts of this sort that might 
lead us to the fifth point, such as considering what values of k will 
pair up with given differences to make 19.

I'll leave the rest to you; write back if you need more help, and 
tell me what you found.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/