This is really interesting, the full story is told here: http://www.wfnmc.org/mc20101.pdf http://www.wfnmc.org/mc20101.pdf During the years 2002–2004 I was vi

This is really interesting, the full story is told here: http://www.wfnmc.org/mc20101.pdf http://www.wfnmc.org/mc20101.pdf During the years 2002–2004 I was visiting Princeton University with its fabulous mathematics department, a great fixture of which was a daily 3 to 4 PM coffee hour in the commons room, attended by everyone, from students to the Beautiful Mind (John F. Nash Jr.). For one such coffee hour, in February 2004, I came thinking—for the hundredth time in my life—about the network of evenly spaced parallel lines cutting a triangle into small congruent triangles. This time I dealt with equilateral triangles, and the crux of the matter was a demonstration that n2 unit triangles can cover a triangle of side n. I asked myself a question where the continuous clashes with the discrete: what if we were to enlarge the side length of the large triangle from n to n+ε, how many unit triangles will we need to cover it? This comprised a new open problem: Cover-Up Problem 1. Find the minimum number of unit equilateral triangles required to cover an equilateral triangle of side n + ε. During the next coffee hour, I posed the problem to a few Princeton colleagues. The problem immediately excited John H. Conway, the John von Neumann Professor of Mathematics. From the commons room he went to the airport, to fly to a conference. On board the airplane, John found a way (Figure 1) to do the job with just n2+2 unit triangles! (Area considerations alone show the need for at least n2 + 1 of them.) Conway shared his cover-up with me upon his return—at a coffee hour, of course. Now it was my turn to travel to a conference, and have quality time on 28 Mathematics Competitions Vol 23 No 1 2010 an airplane. What I found (Figure 2) was a totally different cover-up with the same number, n2 + 2 unit triangles! Upon my return, at a coffee hour, I shared my cover-up with John Conway. We decided to publish our results together. John suggested setting a new world record in the number of words in a paper, and submitting it to the American Mathematical Monthly. On April 28, 2004, at 11:50 AM (computers record the exact time!), I submitted our paper that included just two words, “n2 + 2 can” and our two drawings. I am compelled to reproduce our submission here in its entirety.