Summary: TILING RECTANGLES, CYLINDERS, AND MOBIUS
PETER G. ANDERSON
Abstract. We present a method for determining the number of ways
various two-dimensional grids can be tiled using several small tile reper-
1. Problem Background and Definitions
Figure 1. Tile sets for problems 1, 2, 3, and 4.
How many ways may we tile an m×n rectangular grid (i.e., graph paper)
using a pair of tile shapes from the repertoires show in Figure 1:
(1) Vertical dominoes and squares.
(2) L-trimonoes and squares.
(3) Large and small squares.
(4) Vertical and horizontal dominoes.
In all cases here, the short dimensions are 1 and the long dimensions are
2. We can modify the problem by identifying the top and bottom edges to
form a cylinder. We can identify the top and bottom edges with orientation
reversal to form a Mobius strip. (Exception: our problem-2, L-trimonoes,
cannot participate in the tilings of non-orientable surfaces.)
2. Our Approach