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## Tiling the integers with one prototile

Ethan Coven

Department of Mathematics

Wesleyan University

Middletown, CT

A finite set A of integers ** tiles the integers **if the
integers can
be written as a union of disjoint translates of A. In this case,
A is
called a ** prototile**. Can one decide whether a given set of
integers tiles the integers?

Very few results exist. Necessary and sufficient conditions are known
for
the case that #A is a prime power (D. Newman, J. Number Theory,
1972),
but none are known even for the case that #A=6. We will discuss
the
cases #A=p^n, giving an alternate formulation and an alternate
proof
of Newman's Theorem, and #A=pq. In the latter case, we will
discuss
an easily verified sufficient condition which holds for all known (to
me)
prototiles.

The problem is related to the problem of ``factoring'' finite cylic
groups
(writing the group as a direct sum of two subsets, not necessarily
subgroups), which goes back to Hajos and ultimately to Minkowski. The
corresponding problem of tiling the nonnegative integers is easy ---
the
solution is in your pocket/purse.

* *

Thu January 11, 1996