In my “intro-to-proofs” class, I like to have my students work on some induction problems that are not the usual proof of sums and products of integers.

One lovely example is the following problem about tiling a grid with “trominoes” (three squares joined in an el-shape).

Prove that for any n ≥ 1, a 2nx2n grid with one square removed can be tiled by trominoes.

Below we see 16×16 grid with one square removed. It is not too difficult to tile it with trominoes. The question is, how can you *prove *that it is always possible? We’ll leave that as an exercise.

This semester, I decided to laser cut some pieces so a student could play around with the tiles and to test conjectures. It consists of two layers of wood. The lower layer has the four grids (2×2, 4×4, 8×8, and 16×16). The top layer has the text and square holes for each of the puzzles. I also created four dark squares and enough tromino pieces to fill the board.

To play, choose one of the boards and place a dark square in any square in the grid. Then fill the board with trominoes (ideally, using an algorithm that would generalize to a proof that works for any size board).

Here are some completed puzzles.

If you want to laser-cut your own, here are downloadable pdfs (page 1 and page 2), also shown below. Note that when laser-cutting, the blue lines should be cut lines, the red lines are filled areas, and the green are etched lines.

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