It's easy to show, for p>2, F(p)<= 2 3^(p-1) - 1. -Veit On Jun 13, 2013, at 2:54 AM, Dan Asimov <dasimov@earthlink.net> 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.
[Note: I'm adding the nonstandard condition ii) in order to avoid trivial cases of non-tiling due to holes, such as the heptomino with its squares' centers arranged thus:
* *
* *
* * *
.]
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 ???
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun