Perfect Polyominoes
While at Saint Ann's School, my 5th graders and I studied polyomino divisibility and played with problems of division and tiling. Polyominoes offer a beautiful blend of number theory and geometry, because they have size and shape with interesting restrictions. About a year after my move away from Brooklyn, I found myself pondering them again.
Imagining a geometric analog for perfect numbers, I defined the following: A perfect polyomino is one that may be reassembled as a sum of its proper factors. I enlisted the help of Justin Lanier, and together we came up with some interesting results, especially as they compare to the ancient search for perfect numbers. We are currently preparing a first paper to post to the ArXiv, but for now you can read an overview of our work, which we are presenting at the 2015 MOVES conference. |
Overview of our work:
Perfect Polyominoes (MOVES proposal) Background: Polyomino Number Theory (I) Examples: The picture above shows the smallest known perfect polyomino of odd size. 
This "Q" is one of ten perfect hexomino configurations.
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Imbalance Puzzles
Puzzle: Order the three types of shape by weight,
given that shapes of the same type weight the same.
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See Sam Loyd's Puzzling Scales series for a classic example of the "balanced scale" genre. Inspired by them, I invented a new type of puzzle that uses unbalanced scales to give information about order and inequality.
I presented my puzzles at the 2013 MOVES Conference and published a handful of them. I hosted an online problem-writing contest, the results of which can be seen here, along with my original 15 puzzles. |
Problem set:
MOVES Conference handout Published: New York Times Numberplay The College Mathematics Journal MAA FOCUS Magazine Inspiration: Puzzling Scales by Sam Loyd |
My mathematical work can also be found on my old blog, Lost in Recursion.