The shortest paper ever published in a serious math journal explained
fermatslibrary.com
fermatslibrary.com
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.
I realize it's vague (to put it mildly), but it was 20-something years since it came up in the math class in the Uni.
Edit - Haha, found it! It was in Littlewood's Miscellany - Picard's Theorem [1].
[1] https://books.google.ch/books?id=MjVgeT7Laf8C&pg=PA40&lpg=PA...
My understanding (which I haven't verified) is that this is the main paper resulting in his Nobel prize.
[1] http://web.mit.edu/linguistics/events/iap07/Nash-Eqm.pdf
$(".comment-form").remove()The banner on their homepage is a [low res version of a] valid game-of-life set: http://golly.sourceforge.net/
As mathematics papers go, this is easy to comprehend (incredibly so) but I can see that people who aren't used to the culture in mathematical literature of leaving out the obvious (http://math.stackexchange.com/questions/151782/when-is-somet...) and the trivial (https://en.m.wikipedia.org/wiki/Triviality_(mathematics)#Tri...)
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.
OH. Thank you. This is what I was missing; I understood "cover" to mean "tile" and thought the bottom row was just misprinted slightly.
They had a presentatino on the Paxos paper about a month ago: http://www.meetup.com/Papers-We-Love-London/events/225736762...
But then again, that book begins with a description that visual arguments aren't truly proofs.
The Russian school is basically `something that convinces humans'. The French school is all about formality.
Since computer became a thing, the field has blossomed. Look at zero knowledge proofs and various forms of interactive proof systems as examples.
And I guess one way to think about it is that terms in an expression can be viewed as letters, so the expression then becomes a word.
(x = (-b±sqrt(b^2-4 a c))/(2 a))