In this chapter, we explore three generalizations of Nim: Bamboo Stalks, Lumberjack, and Green Hackenbush. Each game is played by hacking away edges from a graph.
In the previous chapter, we considered variations of Nim that still dealt with heaps of beans (or tokens or coins or pins). We considered more restricted "taking rules" and added "breaking rules" that split heaps apart. But still, the components of our game were heaps of things. In this chapter, we explore ways to generalize heaps into more exotic structures.
The game of Bamboo Stalks starts with a set of bamboo stalks planted in the ground. We represent the ground by a horizontal dashed line. A \bf bamboo stalk of height \(n\) is a path with \(n\) edges (and \(n+1\) vertices) where one of the endpoints is planted on the ground. Figure 4.1.1 shows the Bamboo Stalks position consisting of stalks of heights 4, 1 and 2.
Figure4.1.1.The Bamboo Stalks position \(\pile{4} \oplus \pile{1} \oplus \pile{2}\)
To move in Bamboo Stalks, you hack away one of the edges of the graph. You remove the hacked edge, along with the part of the stalk that is no longer connected to the ground. As always, the last player to move wins. For example, there are seven edges in the position of Figure 4.1.1, so there are seven followers for this position. These followers are shown in Figure 4.1.2.
