Papers › The Generalized Bergman Game
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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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(n²) time, and we also prove that the shortest possible Generalized Bergman game on an initial state terminates between Ω(n) and O(n²) time. We also show a linear bound on the maximum length of the tuple used throughout the game.
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