I cannot make any triangle with sticks

Question

There are $13$ sticks with different lengths, and you try to form a triangle by using $3$ sticks (making every stick is a edge of a triangle) but somehow whatever $3$ sticks you choose, you cannot form any triangle.

What is the minimum value of the length of the longest stick assuming the smallest stick length is $1$ unit.

Note: Lengths are only integer values.

Answer 1

The answer is

377 units

We can find this result using the following:

If we have three sticks that cannot be formed into a triangle, that means that the third stick's length is greater than or equal to the combined length of the first and second stick, or:

$ \begin{equation} a_1 + a_2 \geq a_3 \end{equation}$

We want to minimize the value of the longest stick, so let's assume

$\begin{equation} a_1 + a_2 = a_3 \end{equation}$.

We can continue these equations for the following 13 sticks:

$\begin{align} a_2 + a_3 &= a_4 \\ a_3 + a_4 &= a_5 \\ a_4+a_5 &= a_6 \\ a_5 + a_6 &= a_7 \\ a_6 + a_7 &= a_8 \\ a_7+a_8 &= a_9\\a_8+a_9 &= a_{10}\\ a_9 + a_{10} & = a_{11}\\ a_{10} + a_{11} &= a_{12} \\ a_{11} + a_{12} &= a_{13} \end{align}$

As stated in the problem, no two sticks can be the same length, and the smallest stick, $a_1$ is 1 unit. To minimize the value of the largest stick we want to pick $a_2$ as small as possible. Since $a_2 \neq 1$ and all sticks must have integer unit lengths, we can pick $a_2 = 2$ units. It follows that $a_3 = 3$, $a_4 = 5$, $a_5 = 8$, $a_6 = 13$...and so on - this pattern follows the Fibbonacci sequence. The 14th Fibbonacci number is $377$, hence the answer to this question.

Answer 2

If we assume that all of the following must be true:

1. All thirteen sticks must be different lengths
2. No three sticks may make a triangle
3. The smallest stick is 1 long

then the longest stick is at least 377 long.

The smallest stick is 1. The second smallest stick is therefore 2. The sum of these is 3, which means that the third stick must be no shorter than this, or the three sticks would form a triangle. From this point forward, add the lengths of the two longest sticks whose lengths have been determined; this gives the length of the next stick.

Note that this is the Fibonacci series, with the initial term dropped.