Subtraction games with FES sets of size 3

TL;DR

This paper proves that subtraction games with a finite set of three elements as the FES set have a G-sequence that is purely arithmetic periodic, extending previous results for smaller sets.

Contribution

It establishes that for subtraction games with FES sets of size three, the G-sequence is purely arithmetic periodic, filling a gap in the existing theoretical understanding.

Findings

G-sequence is purely arithmetic periodic for |X|=3

Extension of Siegel's results to sets of size three

Sequences are not always purely arithmetic periodic for |X|≥4

Abstract

This paper extends the work done by Angela Siegel on subtraction games in which the subtraction set is N \ X for some finite set X. Siegel proves that for any finite set X, the G-sequence is ultimately arithmetic periodic, and that if |X| = 1 or 2, then it is purely arithmetic periodic. This note proves that if |X| = 3 then the G-sequence is purely arithmetic periodic. It is known that for |X| \geq 4 the sequence is not always purely arithmetic periodic.