Shown below is the shortest math pater ever published. It appeared in the American Mathematical Monthly, 2004, and would have appeared in a more honored journal if the authors were willing to add more words as an editor requested. You’ll see that the paper itself has only pictures and one sentence with one English word: n² + 2 can:, I thought I might as well try to explain it because as the editor commented, this is too few words for most readers.

The trick to understanding this at all is that most of the background is in the title, which is in the form of a question. The text of the article is in the form of an answer with the diagrams serving as proof. Even with this insight, you’ll likely need more background, but that’s the start.

Here’s the background: Most folks notice that you can make a big equilateral triangle of side-length n, out of n² unit subtriangles, that is of subtrangles where the side lengths =1. For example, to make an equilateral triangle with length 10, requires 100 unit equilateral triangles, n².

Now the question in the title involves what happens if all these n² component triangles are made slightly larger, the sides of each becoming 1+ε/n, where ε is some very small amount. The side of the new big triangle is now n+ε. The question in the title now is can you cover this bigger, super triangle with n²+1 unit triangles. The authors provide two, half answers to this question by their drawings, suggesting two different ways that you can cover the bigger super-triangle with n²+2 unit triangles. That would be 102 for the case where you start with 100 unit triangles and expanded each by ε/n.

The first solution is the bottom of figure 1. This shows what happens if you add two more unit triangles to the bottom row of the old super triangle, and squish a bit from the sides so that the top of the new row matches the bottom of the old row. Doing this leaves you with a row that’s n+ε long at the bottom with wings at the top that expand the sides to n+ε as well. The drawing shows that this new row has effective height, 1+ε.

Now, take every other row and push them together slightly from top-down so that the height becomes (1-ε) but the length expands to n(1+ε). Adding rows like this, you’ll be able to cover the entirety of the bottom space of the new, larger super triangle. Notice that the thickness of each line now 1-ε as shown. Use these longer lines to cover the rest of the bigger super triangle. And that’s the end of the paper. Once again you needed n²+2 unit triangles to cover the bigger super-triangle.

An extension to the above paper was discovered since the original paper. It’s shown in the figure below. Here the original requirement of equilateral triangles is relaxed. For highly elongated triangles, you still find that a normal super-triangle requires n² sub-triangles. But now, from this figure, you see that an expanded super-triangle (each side expanded by 1+ε/n say) can be covered using only n²+1 of the original size subtriangles.

The proof is clear enough that no words are needed. It’s conceivable that the authors could have published this as an even shorter paper, if it were ever published, but it was not. Instead, I saw this extension as a result from a math competition, here. These insights of geometry come from Princeton University, a top notch place where I was a grad student (in engineering). The school has gone somewhat to seed, IMHO, because of political correctness.

There are shorter published papers, BTW, though this was the shortest published math paper. The shortest technical paper ever is this one from the journal of behavioral sciences.

Robert Buxbaum, March 6, 2025. I’d like to add a joke: To make a long story short, I became an editor.