The game of Flipping Coins

TL;DR

This paper analyzes a variant of a coin-flipping game, revealing that its positions are valued as numbers and can be decomposed using ordinal sums, contrasting with similar combinatorial game structures.

Contribution

It introduces a novel analysis of the Flipping Coins game, showing its positions are numbers and detailing the decomposition method involving ordinal sums.

Findings

Game values are numbers after reduction

Positions decompose into iterated ordinal sums

Contrasts with Hackenbush Strings decomposition

Abstract

We consider Flipping Coins, a partizan version of the impartial game Turning Turtles, played on lines of coins. We show the values of this game are numbers, and these are found by first applying a reduction, then decomposing the position into an iterated ordinal sum. This is unusual since moves in the middle of the line do not eliminate the rest of the line. Moreover, when $G$ is decomposed into lines $H$ and $K$, then $G=(H:K^R)$. This is in contrast to Hackenbush Strings where $G= (H:K)$.