From Earlham Cluster Department
This problem focuses on tilings of tapered rectangles by squares and rhombi. Here, a ‘(a × b) tapered rectangle’ is defined as a + sqrt(2) by b + sqrt(2) rectangle with the corners cut off to form an octagon with horizontal edges of length a, vertical edges of length b, and the remaining 4 tilted edges of unit length.
Tilings of (a × b) tapered rectangles can be represented by two intersecting paths on an a by b grid. The paths travel from corner to corner, the first going NW to SE and the second going from SW to NE. In this representation, each path represents a region of connected rhombi, the intesction represents a tilted square (diamond), and the grid represents squares. This is best understood by an example...
An example for a 4 × 2 grid can be seen below. Note that the example below is somewhat distorted, with two different edge lengths present. These errors don’t change the tiling configurations, so you can ignore them in both the example below and in any hand drawings you make.
The entropy of a tapered rectangle of a given size is proportional to the logarithm of the number of valid tilings. An equivalent but potentially more convenient quantity is the entropy density, the entropy per tile.