A Fibonacci Set of Sticks

As the first two sticks have length $1$ , in order to form a triangle is required that the third stick has a length ${ l }_{ 3 }<2$ , because of Triangle Inequality. Thus, the least possible value for ${ l }_{ 3 }$ , that does not form a triangle with the other sticks, is $2$ . Continuing with the same reasoning, we deduce that if the ${ n }^{ th }$ stick has length ${ l }_{ n }$ and the ${ (n+1) }^{ th }$ stick has length ${ l }_{ n +1}$ , then, in order to form a triangle, the ${ (n+2) }^{ th }$ stick must have a length that satisfies ${ l }_{ n+2 }<{ l }_{ n+1 }+{ l }_{ n }$ .

As we're looking for the least possible value that does not satisfy the previous inequality, we conclude that the length of the ${ l }_{ n+2 }$ stick is ${ l }_{ n+1 }+{ l }_{ n }$ . We see that this is how Fibonacci sequence is defined. Therefore, we are looking for ${ f }_{ 14 }$ , which is $377$ .

Fibonacci value for 14