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For each positive integer n ≥ 2, there is a well-known regular orientable Hamiltonian embedding of K n,n , and this generates a regular face 2-colourable triangular embedding of K n,n,n . In the case n ≡ 0 (mod 8), and only in this case, there is a second regular orientable Hamiltonian embedding of K n,n . The current paper presents an analysis of the face 2-colourable triangular embedding of K n,n,n that results from this. The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.doi:10.1017/s0017089510000376 fatcat:lmxzeu4w6zfcvbvgp6ocov4z6y