Black & White Nim games

We present a new family of Nim games where the rules depend on a given `coloring' of the tokens, each token being either black or white. The rules are as in Nim with the restriction that a white token on top of each heap is not allowed. We resolve the winning strategies of two disjoint game families played on two heaps. The heap-sizes with black tokens correspond to the numbers $\lfloor \beta n \rfloor$, where $\beta>2$ is an integer for one of the families and irrational for the other, and where $n$ ranges over the positive integers. In the process, new notions of \emph{invariant games} are introduced.