A Near Miss

Recently, I was working with my son on some of his Beast Academy homework on tangrams. Tangrams are a set of seven tiles (5 triangles of 3 different sizes along with a square and a parallelogram) that can be arranged to make a surprising variety of figures, including cats, fish, and zombies.

As part of this assignment, my little beast was supposed to make one large rectangle with the seven tiles. He called me over to check his work, and I saw this:

I was ready to check off on it, but then I paused. I knew that the entire tangram set had a total area of 8: the two large triangles have an area of 2 each, the medium triangle has area 1, the small triangles each have area 1/2, and the square and parallelogram (which can both be made with two small triangles) have area 1.

$$2+2+1+\frac{1}{2}+\frac{1}{2}+1+1=8.$$

The most natural way to make a rectangle with area $8$ would be $4\times 2$, but that’s not what he did. What was going on?

It was only after all of this area business that I found the error: he had not used the square (which he had misplaced). We found the square and he built a $4\times 2$ rectangle, but I was still stuck on his first solution.

The first shape my son made was, apparently, a rectangle with area $8-\text{(square area)}$, or $7$. Since $7$ is a prime number, it is really hard to make a rectangle with area $7$ from pieces like these. How did he manage to do it?

It turns out that what my son made was no rectangle after all. It was a near miss. These sort of things happen when we build physical models of mathematical structures. Sometimes, the physical limitations of the material allow us to “make” a figure whose existence is mathematically impossible.

My son’s rectangle of area $7$ looks convincing at first glance. But, when you consider the dimensions of the pieces, they don’t actually fit together just right. There are little gaps between the pieces, the kind of little imperfections you’d expect when working with hand-cut mildly crumpled paper tiles. But these gaps proved more than that. They were legitimate errors. Using the Pythagorean Theorem, I found that my son’s “rectangle” had a height of $2$ and a width of $2+\sqrt{2}$, for an area of

$$2\times(2+\sqrt{2})=6.828\ldots.$$

The pieces he used to make this “rectangle” had a total area of $7$, so the actual area of the tiles was larger than the figure he was making. Had he tried to eliminate the gaps by scrunching the tiles together, there would have been some overlap (notice the yellow triangle covering the upper right corner of the parallelogram, and then the orange triangle sitting atop it).

As a mathematician, I am delighted by these near misses. They are wonderful reminders of the power that math has to explain the ways things work and the ways things don’t work. But as a parent, they mean I’ll have to stay on my toes a bit more when looking over my son’s homework.