Billiards with Bombs

TL;DR

This paper introduces a new billiards variant where a ball erases walls on a grid, proving the existence of both periodic and non-periodic tunnels from any starting point, and explores complex behaviors including large-period tunnels.

Contribution

It establishes the existence of various tunnel types in the new billiards variant and discusses conjectures about non-tunneling starting conditions, supported by simulations.

Findings

Existence of periodic tunnels with arbitrarily large periods

Existence of non-periodic tunnels from any starting point

Conjecture and simulation evidence for starting conditions without tunnels

Abstract

In this paper, we define a variant of billiards in which the ball bounces around a square grid erasing walls as it goes. We prove that there exist periodic tunnels with arbitrarily large period from any possible starting point, that there exist non-periodic tunnels from any possible starting point, and that there are versions of the problem for which the same starting point and initial direction result in periodic tunnels of arbitrarily large period. We conjecture that there exist starting conditions which do not lead to tunnels, justify the conjecture with simulation evidence, and discuss the difficulty of proving it.