Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an object in three dimensions. Each row or column has at most one integer on it, and the integer indicates how many cubes in the corresponding 1D slice remain when the object is complete. Kusano et al. showed that Picross 3D is NP-complete and Kimura et al. showed that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and Σ2P-complete, respectively, where those results are shown for the restricted input that the rectangular parallelepiped is of height four. On the other hand, Igarashi showed that Picross 3D is NP-complete even if the height of the input rectangular parallelepiped is one. Extending the result by Igarashi, we in this paper show that the counting version, the another solution problem, and the fewest clues problem of Picross 3D are #P-complete, NP-complete, and Σ2P-complete, respectively, even if the height of the input rectangular parallelepiped is one. Since the height of the rectangular parallelepiped of any instance of Picross 3D is at least one, our hardness results are best in terms of height.

In this paper, we propose a new counting scheme for m n contingency tables. Our scheme is a modification of Dyer and Greenhill's scheme for two rowed contingency tables. We can estimate not only the sizes of error, but also the sizes of the bias of the number of tables obtained by our scheme, on the assumption that we have an approximate sampler.