In [BNRR], it was shown that tiling of general regions with two rectangles is NPcomplete, except for a few trivial special cases. In a different direction, Rémila [Rém2] showed that for simply connected regions by two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10 6 rectangles for which the tileability problem of simply connected regions is NP-complete, closing the gap between positive and negative results in the field. We also prove that counting such rectangular tilings is #P-complete, a first result of this kind.Theorem 1.2 There exists a finite set R of at most 10 6 rectangular tiles, such that counting the number of tilings of simply connected regions with R is #P-complete.Although #P-completeness is known for tilings of general regions with right tromino and square tetromino [MR], nothing was known for tilings with rectangles. We refer to Section 7 for the history of the problem, references, and further remarks.

Consider all geodesics between two given points on a polyhedron. On the regular tetrahedron, we describe all the geodesics from a vertex to a point, which could be another vertex. Using the Stern-Brocot tree to explore the recursive structure of geodesics between vertices on a cube, we prove, in some precise sense, that there are twice as many geodesics between certain pairs of vertices than other pairs. We also obtain the fact that there are no geodesics that start and end at the same vertex on the regular tetrahedron or the cube.

Tiling planar regions with dominoes is a classical problem in which the decision and counting problems are polynomial. We prove a variety of hardness results (both NP-and #Pcompleteness) for different generalizations of dominoes in three and higher dimensions.

DefinitionsLet V be a n-set (set of size n). Let E be the collection of all possible k-subsets (subsets of size k) of V , 2 ≤ k ≤ n, each taken in one of its k! possible permutations. A pair T = (V, E) is called a hypertournament, or a k-tournament. Each element of V is a vertex, and each ordered k-tuple of E is a hyperedge or, simply, an edge. For vertices u, v ∈ V and an edge e = (x 1 , . . . , x k ) ∈ E, we say u dominates v via edge e if u precedes v in e, that is if u = x i , v = x j , 1 ≤ i < j ≤ k. We denote this by uev. A path consists of an alternating sequencex 0 e 1 x 1 e 2 x 2 . . . x −1 e x of distinct vertices x i and distinct edges e i so that x i−1 dominates x i via e i , i = 1, . . . , . Such a path has length . A cycle is a path when all vertices are distinct except x 0 = x . A path (cycle) of T is Hamiltonian if it contains all vertices of T . A k-tournament T is pancyclic if it contains cycles of all possible lengths. It is vertex-pancyclic if each vertex of T is contained in cycles of all possible lengths. It is d-disjointvertex-pancyclic if each vertex of T is contained in d edge-disjointcycles for each possible length . A k-tournament T is strong if there is a path from u to v for each pair u, v ∈ V . The vertex set V and the edge collection E is also denoted V (T ) and E(T ), respectively. A hypertournament T is d-edge-connected if, for any two vertices u, v ∈ V , there are d pairwise edge-disjoint paths from u to v. For an ordinary graph (V, E), Γ(v) is the neighbors of v. For A ⊆ V , we denote by Γ(A) the set a∈A Γ(a). Known ResultsTheorem 1 (Redei). Every tournament has a Hamiltonian path.Theorem 2 (Moon). Every strong tournament is vertex-pancyclic.Gutin and Yeo [2] proved the following.

Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this problem. However, we present an algorithm for testing whether the complement of a finite region is tileable by a set of rectangles.We discuss this connection and some curious consequences of Theorem 1.2 in Subsection 7.2. It is worth noting that if we are given a finite region to tile as opposed to the entire plane, the