Voldemort_007's blog

By Voldemort_007, history, 4 months ago, In English

Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle.

How to Prove it Using Pigeon Hole Principle.

 
 
 
 
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4 months ago, # |
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Do they all have different lengths?

If yes then we can prove using pigeon hole principle ,otherwise i don't know.

If all lengths are different then we can always pick two sides whose sum exceeds 3rd side proving that we can form triangle with these sides.

P.S: Correct me if i am wrong!

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4 months ago, # |
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Let a1,...,a7 be the lengths of the 7 segments. Without losing our generality, take a1 <= a2 <= ... <= a7. Let's say you can actually take a1,..,a7 so that you can't take 3 of them and from a triangle. What does this mean? It means that if you take ai, aj, ak, where i < j < k, ai + aj <= ak (*). Using (*) we get: a3 >= a2 + a1, a4 >= a3 + a2 ... or a3 >= a2 + a1, a4 >= a3 + a2 >= 2 * a2 + a1, a5 >= 3 * a2 + 2 * a1, a6 >= 5 * a2 + 3 * a1, a7 >= 8 * a2 + 5 * a1. But a1 and a2 are >= 1 => a7 >= 8 * 1 + 5 * 1 = 13. But a7 <= 10 => 13 < 10 contradiction. So, using Pigeon Hole Principle, the problem is solved.