Hackenbush

A partizan game played on a graph attached to the ground. Left removes blue edges and Right removes red edges; after a removal, all edges disconnected from the ground disappear. The player with no legal move loses under the normal-play convention.

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 $\epsilon$, it contributes $\epsilon m$. If more edges remain, edge $j>m$, of sign ${\epsilon }_j$, contributes ${\epsilon }_j/2^{j-m}$. ThusFor 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.