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 . What is the maximum number of sticks in the bag?
Source: NCTM Mathematics Teacher, December 2005
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 sticks of lengths . They do not form a triangle because .
Continue the process and make the fourth stick of length , etc. and we end up with the following Fibonacci numbers
Since the longest length is , we cannot have sticks in the bag. But, we can use the first ten sticks plus the stick
The maximum number of sticks in the bag is .