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We define cut-and-paste, a construction which, given a quadriculated diskobtains a disjoint union of quadriculated disks of smaller total area. Weprovide two examples of the use of this procedure as a recursive step. Tilingsof a disk $\Delta$ receive a parity: we construct a perfect or near-perfectmatching of tilings of opposite parities. Let $B_\Delta$ be the black-to-whiteadjacency matrix: we factor $B_\Delta = L\tilde DU$, where $L$ and $U$ arelower and upper triangular matrices, $\tilde D$ is obtained from a largeridentity matrix by removing rows and columns and all entries of $L$, $\tilde D$and $U$ are equal to 0, 1 or -1.

Cut-and-paste of quadriculated disks and arithmetic properties of the adjacency matrix