8 Sticks, No Triangle

Date: 05/12/2003 at 23:13:31
From: Derek
Subject: Math test

Hi, 

A triangle can be formed having side lengths 4, 5, and 8. It is 
impossible, however, to construct a triangle with side lengths 4, 5, 
and 9. 

Ron has 8 sticks, each having an integer length. He observes that he 
cannot form a triangle using any three of of these sticks as side 
lengths. The shortest possible length of the longest of the eight 
sticks is

   a)20    b)21   c)22   d)23   e)24

We were told the answer is 21, but I don't know how they got that.
I could really use your help.

Thanks, 
Derek

Date: 05/13/2003 at 09:04:36
From: Doctor Peterson
Subject: Re: Math test

Hi, Derek.

In order to be unable to make a triangle with three sticks, one stick 
has to be at least as long as the sum of the others. (This is called 
the triangle inequality.) For this to be true of ANY set of three 
sticks in the group, the longest must be at least the sum of the next 
two highest, and so on down the line. This may suggest a familiar 
sequence to you.

Alternatively, just think about building up the set of sticks from the 
bottom. Make the first two sticks as short as possible, 1 unit each. 
The third stick must be at least 2 units long. Now add a fourth stick; 
how long must it be? Keep going for eight sticks.

This problem is a good test of insight, but is very easy if you happen 
to see it in one of these two ways. The second way is how I first saw 
the answer, because when I don't know how to solve a problem, I 
generally just start by playing with it, seeing what happens if I just 
try something. This is also an example of trying a smaller problem 
first.

If you have any further questions, feel free to write back.

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