Optimal Strategies to Catch Randomly Walking Cat

TL;DR

This paper develops optimal strategies for catching a randomly moving cat in various grid environments, considering escape routes and minimizing game duration or escape probability, with mathematical proofs and formulas involving Fibonacci and Lucas numbers.

Contribution

It introduces new optimal strategies for different environments and provides mathematical proofs and formulas for expected escape times and success probabilities.

Findings

Optimal strategies minimize escape chances when exits are present.

Strategies for 1D, ring, and 2x m grid environments are derived.

A formula involving Fibonacci and Lucas numbers predicts escape times in grids.

Abstract

The optimal strategies to catch a randomly walking cat in various environments are presented. All games have a player that opens a box at step $i$. If the cat is in this box the player wins, if not, the cat moves randomly to an adjacent box, or in case the box is open to the outside, the cat may escape. If the cat is not escaping, the next step of the game is played. In case the cat has doors to escape, the optimal strategy is determined that minimizes the escape chances of the cat. If there are no doors to the outside, the strategies are calculated that minimize the game duration. The environments are 1) a one-dimensional array of up to 9 boxes in a line 2) this line is connected to a ring and 3) a $2 \times m$ grid (with $2\le m\le 4$) of boxes. In cases 1) and 3) the boxes may or may not have exits to the outside. Numerical proofs for the optimality of the presented strategies are outlined. Also a formula is presented for the average number of steps it takes a randomly walking cat to exit a $2\times m$ grid when no player is involved. This formula is only based on Fibonacci numbers if $m$ is even and Lucas numbers if $m$ is odd.