Every game of Red-Blue Hackenbush represents a surreal number. Is the converse true? Assuming that it is false, what can be said about the class of surreal numbers that are representable by such games?


For instance, it is obvious that this class is a group under addition, but I can't visualize why it should even be closed under multiplication.


**Edit (by request):** A game of Red-Blue Hackenbush is a rooted graph G whose edges are either red or blue. It's helpful to think of the surreal value of the game G as the number of moves by which Blue is ahead. The addition operation is simply the disjoint union with roots identified. The additive inverse of a game is found by changing the color of every edge. [Of course, this is only an inverse after passing to equivalence classes, which are surreal numbers.]


A finite path graph beginning with a blue edge at the root represents a positive dyadic rational number, which you find as follows:


* Each blue edge that appears before the first red edge is worth 1.
* Beginning with the first red edge, every edge is worth half as much as the edge before.
* The value of the game is the sum of the values of the blue edges minus the sum of the values of the red edges.


This algorithm is presented by Tom Davis in pp. 11-13 of [this paper](http://www.geometer.org/mathcircles/hackenbush.pdf), where you can find a full explanation of what's going on.


By using graphs with infinitely many edges, we can represent many surreal numbers. If I understand correctly, the game {1|9} (for example) is not literally a Red-Blue Hackenbush game, but it is equivalent to 2={-1,0,1|3}, which is represented by a 3-edge blue path joined at the root to a single red edge.


Is there a surreal number that is not equivalent to any Red-Blue Hackenbush game?



What surreal numbers are representable by Red-Blue Hackenbush games?
All surreal numbers are representable by a Red–Blue Hackenbush game. This is discussed in *On Numbers and Games*, although it is left to the reader to fill in the details of the proof for the transfinite case. In Chapter 3 it is explained that every (surreal) number has a sign expansion. Then in Chapter 8 it is explained that the value of a Red–Blue Hackenbush chain is the number whose sign expansion is given by reading off the colors of the edges of the chain from the ground up, interpreting black edges as + and white edges as –. Of course one has to allow chains (or "beanstalks") whose edges are arranged in accordance with the structure of an arbitrary ordinal.


