May be you know about famous problem because of its answer.

> Place two billiard balls ($B_1$ and $B_2$) on the half-line $[0,\infty)$: the $B_1$ is at rest, the $B_2$ is to the right of it and is moving the left. At $x=0$ there is an elastic wall. Assume perfectly elastic collisions: the kinetic energy is preserved. How many collisions will happen depends on the masses of the ball, or rather their ratio. 

 For more details see [Here][1].
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Let me change condition of this problem a bit.

> Place two billiard balls ($B_1$ and $B_2$).
Both are at rest. Next, place one more ball $B_0$ between $B_1$ and $B_2$ which moves to the ball $B_2$. Every collision is elastic:
 kinetic energy is preserved. The same question.

We can begin with simple case when masses of $B_1$ and $B_2$ are equal to $M$ and mass of $B_0$ is equal to $m$.
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I tried to solve this and my result is a system of recurrences which 
I am not able to solve. I will post my solution below.

 


  [1]: https://calculus7.org/2012/03/14/galperins-billiard-method-of-computing-pi/