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The sequence $\lfloor n\alpha \rfloor$ for $n=1,2,\dots$, with a fixed positive irrational number $\alpha$. Complementary Beatty sequences for $\alpha ,\beta >1$ partition the positive integers when $1/\alpha +1/\beta =1$.

Let $\alpha >1$ be irrational and $\beta =\alpha /(\alpha -1)$, so $1/\alpha +1/\beta =1$. The Beatty sequences $\lfloor n\alpha \rfloor$ and $\lfloor n\beta \rfloor$, $n\ge 1$, partition the positive integers.

Indeed, the numbers of terms at most $m$ are $\lfloor (m+1)/\alpha \rfloor$ and $\lfloor (m+1)/\beta \rfloor$. Their arguments sum to the integer $m+1$, and neither is integral, so the counts sum to $m$. Subtracting the count at $m-1$ shows that every integer occurs exactly once in the two lists together. This proof interprets reciprocity as an exact complementarity of two densities.

If $\alpha ,\beta >1$ are irrational numbers with $1/\alpha +1/\beta =1$, the Beatty sequences $\lfloor n\alpha \rfloor$ and $\lfloor n\beta \rfloor$ partition the positive integers.

Willem Abraham Wythoff, also spelled Willem Abraham Wijthoff, worked on combinatorial game theory and the geometry of polytopes. His1907 modification of Nim became Wythoff's game. His work also included a published proof of the triangle-free graph bound associated with Willem Mantel.^[ref] The game illustrates how a small change to legal moves can replace a binary-arithmetic strategy by one organized through Beatty sequences and the golden ratio. His geometric work concerns the symmetry and construction of higher-dimensional polytopes, rather than only recreational games.

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