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arXiv · 1709.00789

The combinatorics of the colliding bullets problem

Nicolas Broutin·Jean-François Marckert
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Abstract

The finite colliding bullets problem is the following simple problem: consider a gun, whose barrel remains in a fixed direction; let $(V_i)_{1\le i\le n}$ be an i.i.d.\ family of random variables with uniform distribution on $[0,1]$; shoot $n$ bullets one after another at times $1,2,\dots, n$, where the $i$th bullet has speed $V_i$. When two bullets collide, they both annihilate. We give the distribution of the number of surviving bullets, and in some generalisation of this model. While the distribution is relatively simple (and we found a number of bold claims online), our proof is surprisingly intricate and mixes combinatorial and geometric arguments; we argue that any rigorous argument must very likely be rather elaborate.

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Nicolas Broutin, Jean-François Marckert. 2020-02-05. The combinatorics of the colliding bullets problem. https://arxiv.org/abs/1709.00789

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