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A 6x5 fault-free tiling I made of post-it notes. What other rectangles can you make fault-free tilings of?
The Problem
Say you want to tile a mxn room with 2x1 tiles. Of course you want to cover the whole room, and no tiles may overlap. But you are even more particular - you don't want any "fault lines", or a line (parallel to one of the walls) that goes all the way through the room but doesn't cut through any tiles.
Formally, a fault-free tiling of a rectangle with rectangles is a decomposition of a rectangle into smaller rectangles such that there is no line parallel to one of the sides of the rectangle that does not cut through one of the rectangles.
There are lots of questions to ask here - what other shapes can we tile? What if we use polyominoes to tile? What if we have "multi-level" tilings where a fault can be broken on any level? What questions do you have?
Sources and Background Information
We will use the following papers as our main sources to get going:
Fault-free Tilings of Rectangles by Ron Graham, http://math.ucsd.edu/~ronspubs/81_01_fault_free_tilings.pdf, 1981.
Tiling Rectangles with Rectangles Chung, Gilbert, Graham, Shearer, and van Lint, https://www.tandfonline.com/doi/abs/10.1080/0025570X.1982.11976999, 1982.
Fault-Free Tileability of Rectangles, Cylinders, Tori, and Mobius Strips with ¨ Dominoes by Emily Montelius: https://arxiv.org/pdf/1912.04445.pdf , 2019.
Desirable Experiences for Applicants
Applicants should have an introduction to proofs course and preferably at least one more proofs-based course. It may be advantageous to have some experience in coding. The main qualifications are intellectual curiosity, perseverance. There will be 2 students working on this project so it's important that you have the capability and desire to work with others.
How to Apply
Go to gvsu.edu/mathreu for more information. If you have questions about how to apply, email mathreu@gvsu.edu. If you have questions about this project, email me at keoulaur@gvsu.edu.
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