Dodgson's Determinant-Evaluation Rule Proved by TWO-TIMING MEN and WOMEN

where sign(π) is the sign of the corresponding permutation, and for i = 1, . . . , n, Mr. i is married to Ms. π(i)′. Except for Mr. 1, Mr. n, Ms. 1′ and Ms. n′ all the persons have affairs. Assume that each of the men in {2, . . . , n− 1} has exactly one mistress amongst {2′, . . . , (n− 1)′} and each of the women in {2′, . . . , (n− 1)′} has exactly one lover amongst {2, . . . , n− 1}2. For each of the (n − 2)! possible (perfect) matchings σ, let