Sticks in a Bag

Each stick in a certain bag has integral length. If any three sticks are selected from the bag, they will not form a triangle. The longest stick in the bag has length 100. What is the maximum number of sticks in the bag?
Source: NCTM Mathematics Teacher, December 2005

SOLUTION
In a triangle the sum of the lengths of any two sides is greater than the length of the third side (Triangle Inequality theorem). One way of not forming a triangle is to have the sum of the lengths of any two sides equal to the length of the third side.
Let’s start with 3 sticks of lengths 1,1,2. They do not form a triangle because 1+1=2.
Continue the process and make the fourth stick of length 1+2=3, etc. and we end up with the following 12 Fibonacci numbers
1,1,2,3,5,8,13,21,34,55,89,144,\cdots
Since the longest length is 100, we cannot have 12 sticks in the bag. But, we can use the first ten sticks plus the 100 stick
1,1,2,3,5,8,13,21,34,55,100
The maximum number of sticks in the bag is 11.

Answer: 11

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About mvtrinh

Retired high school math teacher.
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One Response to Sticks in a Bag

  1. Pingback: Re-Blog – mada360blog

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