Counting Candy Crush Configurations

TL;DR

This paper introduces a randomized algorithm to approximate the number of stable colorings in Candy Crush grids, providing insights into the combinatorial complexity of the game.

Contribution

The paper presents a fully polynomial randomized approximation scheme (FPRAS) for counting k-stable c-colored Candy Crush configurations, a novel approach in combinatorial game analysis.

Findings

Approximately 4.3*10^61 3-stable colorings in a 7-color grid.

FPRAS can be applied to count weak c-colorings of general hypergraphs.

Algorithm implementation in Matlab demonstrates practical feasibility.

Abstract

A k-stable c-coloured Candy Crush grid is a weak proper c-colouring of a particular type of k-uniform hypergraph. In this paper we introduce a fully polynomial randomised approximation scheme (FPRAS) which counts the number of k-stable c-coloured Candy Crush grids of a given size (m, n) for certain values of c and k. We implemented this algorithm on Matlab, and found that in a Candy Crush grid with7 available colours there are approximately 4.3*10^61 3-stable colourings. (Note that, typical Candy Crush games are played with 6 colours and our FPRAS is not guaranteed to work in expected polynomial time with k= 3 and c= 6.) We also discuss the applicability of this FPRAS to the problem of counting the number of weak c-colourings of other, more general hypergraphs.