Abstract
Given a region and a collection of basic shapes (tiles), a natural question is to look at how many ways there are to cover the region using the tiles where no pair of tiles overlaps in their interiors. We show how to transform some problems of this type into counting walks on graphs. In the latter setting, there are well-known and efficient methods to count these for small cases, and in many cases recurrences and closed-form expressions can be found. We explore variations of these problems, and get to the point where the reader can set off to explore problems of this type.
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Notes
- 1.
The Fibonacci Quarterly.
- 2.
In general the OEIS is one of the best resources for people working on counting problems as it gives a way to see if other people have produced similar counts and opens up new avenues for exploration. We will give an example of this later.
- 3.
One might object that we have already done it once, and that doing it a second time will only give the same answer. But the important thing is not the answer, it is the process. A different approach might yield better insight and allow for a better generalization.
- 4.
Another way to think about this is akin to the character from The Hobbit, “Column, column,…my sweet, my love, my precious transitions-esss.”
- 5.
For our purposes it suffices to know that a directed graph consists of a collection of objects (vertices) and connections between the objects (edges). In our case our vertices will be possible column crossings and edges will indicate which pair of columns can occur consecutively (order matters).
- 6.
A walk in the graph is a sequence of moves along edges, repetition is allowed and the direction of edges must be respected.
- 7.
Woohoo!
- 8.
https://projecteuler.net/problem=161. Project Euler is a collection of mathematically based problems that require computational tools to solve them. Working through the collection of these problems becomes a good way to develop mathematical programming skills.
- 9.
The analogous problem for dominoes where we allow gluing on edges or corners is equivalent to counting the number of perfect matchings in chess king graphs. This has been tackled by Shalosh B. Ekhad, a frequent collaborator of Doron Zeilberger. http://sites.math.rutgers.edu/~zeilberg/tokhniot/oKamaShidukhim3.
- 10.
As an example, as of this writing the number of ways to tile the 4×4×4 cube with 1×1×2 tiles is not in the OEIS or found via an internet search engine.
- 11.
For more information on squaring the square visit the website http://www.squaring.net/.
- 12.
For example, when the authors were using this approach to count the number of ways to fill up a 10×20 board using Tetris pieces (the board sized used in the game) they had 447426747 different crossings to keep track of which would create an exceedingly huge matrix that even the largest computers would have a hard time working with. On a side note, the number of such tilings is: 291053238120184913211835376456587574.
- 13.
For publication you will be expected to go beyond the enumeration, look for interesting characteristics and properties that you can prove about these numbers.
- 14.
With many colorful examples.
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Butler, S., Ekstrand, J., Osborne, S. (2020). Counting Tilings by Taking Walks in a Graph. In: Harris, P., Insko, E., Wootton, A. (eds) A Project-Based Guide to Undergraduate Research in Mathematics. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-37853-0_5
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