= Hackenbush stalk
{c}
A <Hackenbush stalk> is a <Hackenbush> position whose <graph> is a single path attached to the ground at one endpoint. Read its blue and red <graph edges> upward from the ground; Left may cut blue edges and Right red edges, with every edge above the cut falling away. Finite red-blue stalks are <number games> with <dyadic rational> values.
Here is the explicit value rule. Give blue sign $+1$ and red sign $-1$. If the initial monochromatic run has length $m$ and sign $\varepsilon$, it contributes $\varepsilon m$. If more edges remain, edge $j>m$, of sign $\varepsilon_j$, contributes $\varepsilon_j/2^{j-m}$. Thus
$$
G=\varepsilon m+\sum_{j=m+1}^{n}\frac{\varepsilon_j}{2^{j-m}}.
$$
For example, a blue-red stalk has value $1-1/2=1/2$, blue-red-red has value $1/4$, and blue-blue-red has value $3/2$. The recursive <number game> rule verifies the first example directly: its Left option is $0$, its Right option is $1$, and the simplest <surreal number> between them is $1/2$. Repeating this comparison after successive cuts gives the dyadic rule.
Back to article page
