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### 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/
```
Associated Topics:
High School Triangles and Other Polygons
Middle School Triangles and Other Polygons

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