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We prove an asymptotic for the number of additive triples of bijections$\{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$, that is, the number of pairs ofbijections $\pi_1,\pi_2\colon \{1,\dots,n\}\to\mathbb{Z}/n\mathbb{Z}$ such thatthe pointwise sum $\pi_1+\pi_2$ is also a bijection. This problem is equivalentto counting the number of orthomorphisms or complete mappings of$\mathbb{Z}/n\mathbb{Z}$, to counting the number of arrangements of $n$mutually nonattacking semiqueens on an $n\times n$ toroidal chessboard, and tocounting the number of transversals in a cyclic Latin square. The method ofproof is a version of the Hardy--Littlewood circle method from analytic numbertheory, adapted to the group $(\mathbb{Z}/n\mathbb{Z})^n$.

Additive triples of bijections, or the toroidal semiqueens problem