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
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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.
I would even say that paper is shorter, but I don't see an objective way to weight and measure the length of figures presented in a paper so it's kind of subjective.
I'm personally a fan of John Nash's paper, "Equilibrium Points in n-Person Games" [1]. It's the bottom of page 48 and the top of page 49. It's quite high on the impact to word count ratio.
My understanding (which I haven't verified) is that this is the main paper resulting in his Nobel prize.
If you like GOL, check out 'golly', the game-of-life simulator. It has tons of amazing examples included, including a cpu emulator. On sourceforge and probably in your favourite OS repos, too.
what is this? I really don't understand how figure 2 has to do with figure 1, what the question is or what the construction is of the narrative I'm supposed to follow...
The comments on the left explain it (I didn't understand it either). Basically, it's known that an equilateral triangle of side length n needs n^2 unit triangles to perfectly tile it (e.g. you need 16 unit triangles to tile an equilateral triangle of side length 4).
Now, if you make the side length of the large triangle just a teeny bit longer (that's what epsilon is), what is the minimum number of unit length triangles it takes to cover the larger triangle? The two different figures show two ways this can be done with n^2 + 2 triangles: the first figure essentially adds 2 triangles to the base row (the comments show how they overlap a little bit), while the second figure uses 3 overlapping triangles to make up the "tip", instead of just 1 (again, the comments show how this works because the base row doesn't need any additional triangles).
Thus, these two examples show how it can be done with n^2 + 2 triangles. It's still an open question if it can be done with n^2 + 1.
The two figures illustrate two different ways to cover an equilateral triangle with sides slightly larger than n with n^2+2 unit size equilateral triangles. Other than that, there is no link between the two.
Unrelated to the published paper. Fermatslibrary is fascinating. Is there something similar for CS papers? I would love this for some distributed systems papers.
Fermat's Library is not only for math, it has a few CS papers already, for instance http://fermatslibrary.com/s/bitcoin. If there is a particular paper you would like to be annotated, you can suggest it and maybe you'll get lucky :)
It would have benefited from being a little longer. It takes about as long to figure out that they leave open the question whether n^2+1 can, as it would to come up with the same or similar constructions on your own!
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.