Beatty Games Big and Small

A Beatty sequence is a sequence of integers formed by taking the floor of the positive integral multiples of a positive irrational number $\alpha$.  The complementary sequence is formed in a similar manner using $\beta$, where $\beta$ satisfies the equation $\frac{1}{\alpha} + \frac{1}{\beta}  = 1$.  For a given $\alpha$, we investigate the partizan subtraction game with left and right subtraction sets given by $(1, \alpha)$ and $(1, \beta)$, respectively.  We analyze this family of games using the Atomic Weight Calculus.

We will also report results for the non-atomic version, where the left and right subtraction sets are given by $(\alpha)$ and $(\beta)$, respectively.

Octal games are impartial games that involve removing tokens from heaps of tokens.  These types of games are interesting in that they can be described using an octal code.  Historically, research efforts have focused almost exclusively on octal games with finite codes.  We consider octal games based on infinite octal codes where the heap sizes corresponding to elements of a Beatty $\alpha$ sequence are played according to some fixed removal rule and the heap sizes corresponding to elements of a Beatty $\beta$ sequence are played according to some other fixed removal rule.  Interesting periodicity seems to occur in most cases.

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