Tilings with T and Skew Tetrominoes
" Quercus: Linfield Journal of Undergraduate
Quercus: Linfield Journal of Undergraduate Research
Cynthia Lester 0
Recommended Citation
0 [3] M. Korn, Geometric and algebraic properties of polyomino tilings, Ph. D Thesis, Massachusetts Institute of Technology , Cambridge MA , USA
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Acknowledgements
This research was supported by the Linfield Faculty-Student Collaborative Research Grant.
1
Introduction
In this paper we answer two tiling questions involving the tiles appearing in
Figure 2. In particular, we prove the following result: An a b rectangle can be
tiled by the set T (Figure 2) if and only if a; b 4 and either one side is divisible
by 4, or a; b 2 (mod 4) and a + b > 16. We prove this result in section 2 after
providing background and context for this work. In section 3 we tackle related
tiling questions involving modi ed rectangles.
In the last several decades, tiling problems have been attracting the attention
of mathematicians. Their allure is easy to understand: the questions are often
simple and tangible, but the answers may require abstract mathematics. Most
tiling questions take place in the integer lattice, i.e. the tiles and regions are
both made of squares, like those on graph paper. One pop culture example of a
tiling problem is the game Tetris, where instead one tries to completely ll the
region. We say a region R is tileable by a tile set T and that T tiles R if R can
be covered without gaps or overlaps by at least some of the tiles in T and all
tiles used to cover R are contained in the region.
Tiling questions usually appear in the form: Can a region of some nite
dimension be tiled with a given tile set? If a region can be tiled, the proof of
this could be as simple as providing a tiling of the region; however, this does
not imply that nding a tiling of the region is easy. If the region cannot be
tiled, then the question becomes: How does one prove a region is untileable?
Obviously, going through every possible tiling of the region would be tedious to
both read and write. A few useful ways to prove the nonexistence of a tiling for
a given region are local considerations, coloring arguments, and tile invariants.
Local considerations are physical constraints speci c to a region that make
it untileable by a given set. For example, an a b rectangle missing one corner
can obviously not be tiled with copies of a 2 2 square because of the missing
corner (consider trying to tile the squares near the missing corner). Although
local considerations can be obvious, they can also be complicated or tedious to
prove.
Coloring arguments have a rich history and typically involve modular
arithmetic. For example, in 1958 George Gamow and Marvin Stern posed in [2] the
following well-known question: Can dominoes tile a chessboard whose
upperleft and lower-right corners have been removed? If one colors a chessboard in
the normal way, see Figure 1(a), then no matter where a domino is placed it
covers one black square and one white square; however, this chessboard has two
more black squares than white squares and therefore dominoes can never tile
this region.
In the previous example, Figure 1(a), consider replacing the white squares
with 0 and the black squares with 1, as in Figure 1(b). Now each domino will
sum to 1 (mod 2) regardless of where it is placed on the modi ed chessboard;
however, the region sums to 0 (mod 2). Suppose the region is tileable. Since the
region has 62 squares, it must be covered by 31 dominoes. Since each domino
sums to 1 (mod 2) and the region must use 31 dominoes, then the sum of the
region is 31 1 (mod 2); this contradicts that the region sums to 0 (mod 2)
argumodular
and therefore the region cannot be tiled by dominoes.
There is only one requirement for a coloring argument: a tile in the tile set
must sum to the same number modulo n, for some xed natural number n, no
matter where it is placed on the coloring. In the domino example, every domino
uses exactly one black and one white square, or sums to 1 modulo 2, no matter
where it is placed on the modi ed chessboard. Obviously, coloring arguments
depend on the tile set, but they can also depend on the region.
Tile invariants also depend on both the tile set and the region. If all the tiles
are made of n squares, then any region tileable by that set must have an area
(the number of squares in the region) divisible by n. Since every region has a
constant area, the number of tiles used in a tiling of the region is an invariant;
speci cally called the area invariant. The following is an example of how the
area invariant can easily prove a region is untileable by a set: Let the 1 7
rectangle be the region and the 1 2 rectangle be the tile. There are seven
squares in the region and two in the tile. Since two does not divide seven then
this tile cannot tile the region.
All our regions and tiles live in the integer lattice and therefore are called
polyominoes. A polyomino is a nite set of squares in the integer lattice such
(...truncated)