Finding Golden Nuggets by Reduction

TL;DR

This paper studies a class of combinatorial games called Complementary Subtraction, focusing on the Golden Nugget game, and develops methods to determine game values using Fibonacci properties and polynomial-time characterizations.

Contribution

It introduces the Golden Nugget Subtraction Game, analyzes its values using Fibonacci words, and provides a polynomial-time method to identify certain game states.

Findings

Values are either numbers or switches in reduced form.

Switches are invariant under Fibonacci shifts.

Numbers have a polynomial-time characterization via ternary Fibonacci representation.

Abstract

We introduce a class of normal play partizan games, called Complementary Subtraction. Let $A$ denote your favorite set of positive integers. This is Left's subtraction set, whereas Right subtracts numbers not in $A$. The Golden Nugget Subtraction Game has the $A$ and $B$ sequences, from Wythoff's game, as the two complementary subtraction sets. As a function of the heap size, the maximum size of the canonical forms grows quickly. However, the value of the heap is either a number or, in reduced canonical form, a switch. We find the switches by using properties of the Fibonacci word and standard Fibonacci representations of integers. Moreover, these switches are invariant under shifts by certain Fibonacci numbers. The values that are numbers, however, are distinct, and we find a polynomial time bit characterization for them, via the ternary Fibonacci representation.