In the following, "tile" means p-dimensional n-omino. Yes, I did mean that any tile isometric to another tile is to be considered the same. (This includes mirror images.) There is no requirement about physical assemblability. As usual, it is assumed that in any tiling the interiors of any two tiles are disjoint. Simpler than the phrasing of the definition below, but equivalent: We assume each tile (p-dimensional n-omino) in any position is the union T of n closed unit lattice cubes such that T is topologically equivalent to a p-dimensional closed disk. (I.e., topologically equivalent to the closed unit p-cube [0,1]^p.) This definition excludes conventional multiply-connected polyominoes, which are no doubt interesting to ask similar questions about, but it keeps the question at hand simpler. (For instance, consider the 3D octomino Q consisting of a 3x3x1 arrangement of unit cubes, except for the middle one. The only way to fill the hole in Q is with another, linked, copy of Q. (Call such a union of two linked copies of Q by the name "Q2". (Then: Can such 16-ominoes Q2 tile 3-space? Each of these is topologically equivalent to a closed unit p-cube, so fits the question at hand.) --Dan ------------------------------------------------------------------ Rich writes: << You should add "allowing tiles to be turned over, reflected, rotated, etc." to the rules. If not, the 2D answer changes. Another likely 3D issue is whether you require that the tiling be physically assembleable, without requiring some temporary interpenetration. . . . I wrote: << Polyomino non-tiling problem: Definition: A p-dimensional n-omino X is a union of integer lattice cubes (of size 1^p) such that i) The interior of X is connected, AND ii) X is topologically equivalent to a closed p-dimensional disk. . . . In p-dimensional Euclidean space, find the least N for which there exists a p-dimensional N-omino, copies of which cannot tile p-space. (E.g., it's known that F(2) = 7.) Call this N by the notation F(p) QUESTION I): Can F(p) be determined explicitly for all p ??? ------------- QUESTION II): In any case, can an asymptotic formulas be found ------------- for F(p), as p -> oo ???