# Counting Candy Crush Configurations

- Published in 2019
- Added on

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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.

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- key
- CountingCandyCrushConfigurations
- type
- article
- date_added
- 2021-01-06
- date_published
- 2019-09-30

### BibTeX entry

@article{CountingCandyCrushConfigurations, key = {CountingCandyCrushConfigurations}, type = {article}, title = {Counting Candy Crush Configurations}, author = {Adam Hamilton and Giang T. Nguyen and Matthew Roughan}, 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.}, comment = {}, date_added = {2021-01-06}, date_published = {2019-09-30}, urls = {http://arxiv.org/abs/1908.09996v1,http://arxiv.org/pdf/1908.09996v1}, collections = {combinatorics,games-to-play-with-friends}, url = {http://arxiv.org/abs/1908.09996v1 http://arxiv.org/pdf/1908.09996v1}, year = 2019, urldate = {2021-01-06}, archivePrefix = {arXiv}, eprint = {1908.09996}, primaryClass = {math.CO} }