Ordinal Sums, Clockwise Hackenbush, and Domino Shave

TL;DR

This paper introduces two related combinatorial game rulesets, Domino Shave and Clockwise Hackenbush, providing explicit formulas for their values and demonstrating their connections to ordinal sums and existing Hackenbush variants.

Contribution

It establishes the equivalence of Clockwise Hackenbush and Domino Shave and derives explicit formulas for their game values, extending previous methods like van Roode's signed binary numbers.

Findings

Clockwise Hackenbush values are numbers with explicit formulas.

Clockwise Hackenbush is equivalent to Domino Shave.

The formulas generalize van Roode's signed binary number method.

Abstract

We present two rulesets, Domino Shave and Clockwise Hackenbush. The first is somehow natural and, as special cases, includes Stirling Shave and Hetyei's Bernoulli game. Clockwise Hackenbush seems artificial yet it is equivalent to Domino Shave. From the pictorial form of the game, and a knowledge of Hackenbush, the decomposition into ordinal sums is immediate. The values of Clockwise Blue-Red Hackenbush are numbers and we provide an explicit formula for the ordinal sum of numbers where the literal form of the base is $\{x\,|\,\}$ or $\{\,|\,x\}$, and $x$ is a number. That formula generalizes van Roode's signed binary number method for Blue-Red Hackenbush.