# What is the probability x triangles can be formed?

We have a stick of length $g$ which is dropped and is broken into $n$ pieces. The choice of the $n-1$ breaking points are chosen randomly and independently on the stick.

What is the probability that with $n$ breaks some number of triangles $x$ can be formed?

Note that we say that a triangle can be formed if some three lengths we choose satisfy the triangle inequality. A piece can be used in multiple triangles.

I'm not expecting a perfect solution but if you have any cases or something that'd be great.

Thanks.

For the case of two breaks, one triangle, see the earlier MO question, If you break a stick at two points chosen uniformly, the probability the three resulting sticks form a triangle is 1/4. Is there a nice proof of this?. One answer there, by Michael Lugo, answers a more general question—not yours—but rather forming an $n$-gon rather than many triangles:

One reference for a solution to this problem is Carlos d'Andrea and Emiliano Gomez, "The broken spaghetti noodle", American Mathematical Monthly 113 (2006), p. 555, JSTOR, author's website. More generally the probability that an interval broken at $n-1$ points chosen uniformly at random is broken into pieces which can be rearranged to form an $n$-gon is $1 - n/2^{n-1}$.