Mathematical Cover-Up

Great mathematician John H. Conway passed away Saturday April 11th, as a result of complications from COVID-19.

He had a slyly bent sense of humour. Together, he and Princeton University colleague Alexander Soifer managed to get the world record for the shortest paper ever published in a serious Math Journal.

The paper was published in the January 2005 issue of the American Mathematical Monthly. Numberphile recalls the event.

The paper asks the question: Can $n^2+ 1$ unit equilateral triangles cover an equilateral triangle of side $> n$, say $n + \epsilon$? then proceeds to give the answer: $n^2+ 2$ can, together with figures showing two different ways of achieving the result¹².

The folks at numberphile worry that Conway and Soifer have written nothing about the possibility of an $n^2 + 1$ unit triangles cover-up: Tony Padilla (at 4 min 40): « Yes, so that’s what I think about it. I don’t think they even answered the question. »

Actually, in his book How Does One Cut a Triangle?, Soifer recollects the $n^2 + 1$ case was the easy part (« Area considerations alone show the need for at least $n^2 + 1$ of them »). So they needed not bother to encumber their proof with it.

It could have gone like this:

Consider the « canonic » cover-up of the triangle of side $n + \epsilon$ with $n^2$ small triangles of side $1 + {\epsilon \over n}$.
Consider the combined $n^2 + 2$ vertices of these small triangles.
The distance between any two vertices is greater than or equal to $1 + {\epsilon \over n}$, so no two vertices belong to the same unit triangle.
Consequently a cover-up of the $n + \epsilon$ triangle must be comprised of at least $n^2 + 2$ unit triangles. $\Box$

1. Figure 1 is easy enough to understand: take the bottom horizontal « strip » of the size-$n$ triangle (comprised of $2n – 1$ triangles); by squeezing it horizontally and adding $2$ triangles, we increase its height until it covers up the $n + \epsilon$ triangle.
2. Figure 2 is slightly more difficult: take the bottom horizontal strip of the size-$n$ triangle; squeeze it vertically so as to make it a little wider (to fit an $n + \epsilon$ side); meanwhile its height will decrease a little; do the same with all remaining horizontal strips (but the triangle on top), you end up covering up the $n + \epsilon$ triangle except for a triangular area on top that is slightly larger than a unit triangle; this remaining area can be covered-up using three unit triangles, as figure 2 shows. I believe there is a typo in the figure though: the length on the bottom left should read $1 – {\epsilon \over n}$ instead of $1 – \epsilon$.