This is a list of possible counterexamples to Conjecture 4.5 of this paper by Ezra Miller and Alan Guo.

Notation: Let 0 denote the misere game with no moves, let * denote the misere game which can only move to 0, and for any n, let *n denote a misere Nim heap of size n. In other words, 0 = {}, * = {0}, *n = {0, *, *2, ..., *(n-1)}. If G is a misere game, let nG denote the game G + ... + G, with n copies of G, and let G* denote G+*.

Let g denote the game {0, *2+*3, *2+*2+*2}. Exercise: for n greater than 6, whether ng is a win or a loss for the first player depends on the congruence class of n modulo 3.

Games which look aperiodic:

{*3, g*, 2g} - subject of this question, seems to be eventually periodic with period 2.

{0, g, g*} - looks promising! Converted to a cellular automaton (rules here, colors here, and some screenshots here) in a similar way as the previous one, but this time the hashlife algorithm fails to provide an exponential speedup.

{g*, 2g}

{*, *2, 2g}

{*, *2, g*, 2g}

{*, g, g*, 2g}

{0, *, *2, g, g*}

{0, *, *3, g, g*}

{0, *, g, g*, 2g}

**Update:** Found several games which are provably aperiodic!

More notation: Let [147] be the game {*, *2+*2, *2+*2+*3}, let [057] be the game {0, *2+*3, *2+*2+*3}, and let [01234] be the game {0, *, *2, *3, *2+*2}. (In this notation, the game g above would be written as [056].)

{[147]*, 3[147], [147]+*2} - very simple exponential pattern, easily provable.

{0, *2, 2[147]*} - exponential pattern, provable.

16[057] + n[01234] is aperiodic in n with a simple exponential pattern, provable.

{[057]+*2, [057]*, [057], *2, *1} - simple exponential pattern, looks provable.

{[057]+*3, *2+*2, *3, *2} - simple exponential pattern, essentially 1d, disturbance moves down.

{[057]+*2+*2+*2, *2+*2+*2, [057]+*3} - diagonals form a 1d pattern, disturbance moves down.

In a few cases, we get strange 1-dimensional automata that appear aperiodic:

{0, *, [147]*, 3[147]} - deceptively simple-looking.

{0, *, [057]*, 3[057]}, {0, *, [057]*, 4[057]+*2}, {0, *, [057]*, 5[057]+*2+*2} - all seem to use the same mechanism, looks like part of a larger pattern.

{0, *, *2+*2+*2, [147]*} - appears to eventually become a 1d automaton.