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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.10981 |
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| _version_ | 1866908504852267008 |
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| author | Cashman, Caroline Miller, Steven J. Shuffleton, Jenna Son, Daeyoung |
| author_facet | Cashman, Caroline Miller, Steven J. Shuffleton, Jenna Son, Daeyoung |
| contents | Zeckendorf proved that every positive integer can be written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith, Epstein, Flint, and Miller converted the process of decomposing an integer $n$ into a 2-player game, using the moves of $F_i + F_{i-1} = F_{i+1}$ and $2F_i = F_{i+1} + F_{i-2}$, where $F_i$ is the $ith$ Fibonacci number. They showed non-constructively that for $n \neq 2$, Player 2 has a winning strategy: a constructive solution remains unknown.
We expand on this by investigating ``black hole'' variants of this game. The $F_m$ Black Hole Zeckendorf game is played with any $n$ but solely in columns $F_i$ for $i < m$. Gameplay is similar to the original Zeckendorf game, except any piece that would be placed on $F_i$ for $i \geq m$ is locked out in a ``black hole'' and removed from play. With these constraints, we analyze the games with black holes on $F_3$ and $F_4$ and construct a solution for specific configurations, using a non-constructive proof to lead to a constructive one. We also examine a pre-game in which players take turns placing down $n$ pieces in the outermost columns before the decomposition phase, and find constructive solutions for any $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10981 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Black Hole Zeckendorf Games Cashman, Caroline Miller, Steven J. Shuffleton, Jenna Son, Daeyoung Number Theory Zeckendorf proved that every positive integer can be written as a decomposition of non-adjacent Fibonacci numbers. Baird-Smith, Epstein, Flint, and Miller converted the process of decomposing an integer $n$ into a 2-player game, using the moves of $F_i + F_{i-1} = F_{i+1}$ and $2F_i = F_{i+1} + F_{i-2}$, where $F_i$ is the $ith$ Fibonacci number. They showed non-constructively that for $n \neq 2$, Player 2 has a winning strategy: a constructive solution remains unknown. We expand on this by investigating ``black hole'' variants of this game. The $F_m$ Black Hole Zeckendorf game is played with any $n$ but solely in columns $F_i$ for $i < m$. Gameplay is similar to the original Zeckendorf game, except any piece that would be placed on $F_i$ for $i \geq m$ is locked out in a ``black hole'' and removed from play. With these constraints, we analyze the games with black holes on $F_3$ and $F_4$ and construct a solution for specific configurations, using a non-constructive proof to lead to a constructive one. We also examine a pre-game in which players take turns placing down $n$ pieces in the outermost columns before the decomposition phase, and find constructive solutions for any $n$. |
| title | Black Hole Zeckendorf Games |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.10981 |