Nimbers in partizan games

Summary

“The chess board is too small for two queens.”

–Victor Korchnoi, challenger for World Chess Championship

We propose some classes of games particularly useful for constructing nimbers in partizan games. We exhibit a *4 in Amazons with the help of a particular algebraic table.

Before reading this article one can consider the following Amazons position:

In Amazons there are a lot of hot positions: the players want to play to gain some territory. However, the position shown is not of this kind. If we use [Siegel 2011] to analyze it, this position proves to have value *3 + *2 = *. We will attempt to analyze the options of such a position and construct nimbers in partizan games. For instance, *3 = {0, *, *2 | 0, *, *2}, however, when we study the players options with [Siegel 2011] we see that the options are not {0,*,*2}. In a partizan game we have a bigger number of possible options to construct a nimber than in an impartial game. For instance, we know that a game like { ↑ | 0} has value * too. When we think about higher stars the number of possibilities is just gigantic. So it’s important to make some mathematical considerations to classify the games that “can act as nimbers”. In this article, we prove some useful results about the construction of nimbers and show some interesting examples in Amazons.