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arXiv · 2109.00117

The Generalized Bergman Game

Benjamin Baily·Justine Dell·Irfan Durmić·Henry Fleischmann·Faye Jackson·Isaac Mijares·Steven J. Miller·Ethan Pesikoff·Luke Reifenberg·Alicia Smith Reina·Yingzi Yang
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Abstract

Every positive integer may be written uniquely as a base-$β$ decomposition--that is a legal sum of powers of $β$--where $β$ is the dominating root of a non-increasing positive linear recurrence sequence. Guided by earlier work on a two-player game which produces the Zeckendorf Decomposition of an integer (see [Bai+19]), we define a broad class of two-player games played on an infinite tuple of non-negative integers which decompose a positive integer into its base-$β$ expansion. We call this game the Generalized Bergman Game. We prove that the longest possible Generalized Bergman game on an initial state $S$ with $n$ summands terminates in $Θ(n^2)$ time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between $Ω(n)$ and $O(n^2)$ time. We also show a linear bound on the maximum length of the tuple used throughout the game.

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Benjamin Baily, Justine Dell, Irfan Durmić, Henry Fleischmann, Faye Jackson, Isaac Mijares, Steven J. Miller, Ethan Pesikoff, Luke Reifenberg, Alicia Smith Reina, Yingzi Yang. 2021-09-20. The Generalized Bergman Game. https://arxiv.org/abs/2109.00117

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