## Wednesday, June 11, 2008

### Triangles Are Easy

Take a stick and cut it twice so that you have three sticks. If you make the cuts randomly, what are the chances you can make a triangle out of your three sticks?

1. I think it is 100%

2. No, I believe it is 50%. The basic principle at work here is that no single side of a triangle can be longer than the other two sides combined. For example, if you have one side that is 3 ft, and two 1 ft sides, you can't make a triangle out of those (try it if you don't believe me).

To make it easier to explain, let's say you put dots on the places you will cut (so we can just talk about one stick instead of two and three as we cut it). Now imagine you put your first dot somewhere. It will either be on the left side or the right side of the stick. If the second dot is on the same side of the center of the stick, then you won't be able to make a triangle, since you'll have one piece that is at least half of the total length of the stick, so the other two sides can't add up to more than it's length. Since the first cut has to be on one side or the other, you have a 50% chance of getting the same side with the second cut.

3. or is that what they want you to think. technically, if the stick was 10 cm long, and you cut 1cm 1cm 8cm you could place them such that a triangle appeared, but with a longer side, like this:

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just a thought

4. Gavin, you make a very good point about triangles. In this case, I was thinking of triangles that connect, so that Abe's explanation makes sense. If we're trying to be tricky, you are right, and the answer is 100%.

If we aren't trying to be tricky, then you need to calculate the probability the longest piece is shorter than the two other pieces combined.

5. consider a 1 meter stick
what if the first cut is in the 1st cm?????

6. As long as the longest piece is longer than the other two combined, it can work. If the first piece is 1 cm long, and the second is 50 cm long, then it won't work because the third piece is too long.

7. so the answer 50 % is wrong?
now the first cut is important too!

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