Lights Out Puzzle in p Colors: Evolution of Quiet Patterns
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911235183738880 |
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| author | Boinde, Wisdom Minevich, Igor Poudel, Dipesh |
| author_facet | Boinde, Wisdom Minevich, Igor Poudel, Dipesh |
| contents | The Lights Out Puzzle represents a cellular automaton based on a grid of squares where clicking a square changes its state and the states of surrounding squares. A "quiet pattern" is a way to click such that in the end, no change is effected. We introduce a way to "evolve" quiet patterns in smaller grids into ones in $p$ times larger grids when the number of possible states of a square is a prime $p$. Using elliptic curves, we also find that an inverse "de-evolution" exists for most $p$. We also describe the only ways to click a grid of squares such that only 5 (the minimum) number of squares have a nonzero state. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lights Out Puzzle in p Colors: Evolution of Quiet Patterns Boinde, Wisdom Minevich, Igor Poudel, Dipesh Cellular Automata and Lattice Gases G.2 The Lights Out Puzzle represents a cellular automaton based on a grid of squares where clicking a square changes its state and the states of surrounding squares. A "quiet pattern" is a way to click such that in the end, no change is effected. We introduce a way to "evolve" quiet patterns in smaller grids into ones in $p$ times larger grids when the number of possible states of a square is a prime $p$. Using elliptic curves, we also find that an inverse "de-evolution" exists for most $p$. We also describe the only ways to click a grid of squares such that only 5 (the minimum) number of squares have a nonzero state. |
| title | Lights Out Puzzle in p Colors: Evolution of Quiet Patterns |
| topic | Cellular Automata and Lattice Gases G.2 |
| url | https://arxiv.org/abs/2510.23677 |