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/
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