Building Shapes out of Polyominoes - HW2

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Now that we've learned what polyominoes are in HW1, the questions in HW2 ask us to try building some different shapes out of them. We also start talking about how you can actually prove whether or not you can make a particular shape out of specific polyominoes.

The goal for these questions is two-fold: In the early questions, I wanted to encourage experimentation. The value of false starts, dead ends, and general mucking about is a good thing to learn about mathematical thinking. In the later questions, however, I wanted to try and introduce the idea that sometimes you can work something out without having to try lots of experiments. I do this here using the classic "Mutilated Chessboard" problem, which I hope is a clear enough example of a simple proof for kids to get the idea.

If you're still loving the polyominoes, you really need to get a copy of Solomon Golomb's book "Polyominoes: Puzzles, Patterns, Problems and Packings."

Intro to Polyominoes - HW1

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Link to Answer Key

Here is the first proper Bespoke Mathematics Problem Set! In these questions, you'll meet up with some very fun shapes called "polyominoes" which are a mainstay of combinatorial geometry. There's a TON of material about polyominoes out there and if you've played Tetris, you already know what these things are.

This problem set is mostly about teaching kids how we pick a definition for a new class of mathematical objects (establishing the "rules" for polyominoes) and getting them to play within those constraints to make new things. One thing I wanted to emphasize here is that it's okay to try out different rules, but being consistent once you've decided how you want things to work matters a lot.

For a great introduction to polyominoes, I can't think of anything better than the original Martin Gardner article about them. You can find this in the collected volumes of his work (which are unbeatable) published by the AMA. Alternatively, many places online have nice articles about these - try Wolfram or www.mathpuzzle.com for some pointers.