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The shortest paper ever published in a serious math journal explained
- elbigbad 11y agoThis 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.
- jacquesm 11y agoI read the story. So much for publishers not doing at least 80% of the work when it comes to publishing papers.
- fermigier 11y agoI count 14 words (11 in the title, 1 in the body + 2 times "figure"), not 2. And 14 more if you count the authors names and affiliation as words.
- nmc 11y agoThe statement about the two words only concerns the body, and the formula n² + 2 counts for one word.
- gohrt 11y agoWhy would you count a compound expression, with internal spaces, as one word?
- Marat_Dukhan 11y agoAnother example of a short peer-reviewed paper (and related to Computer Science, yay!) is "Counterexample to Euler's conjecture on sums of like powers": http://fermatslibrary.com/s/counterexample-to-eulers-conjecture-on-sums-of-like-powers http://fermatslibrary.com/s/counterexample-to-eulers-conject...
- fermatslibrary 11y agoHi Marat_Dukhan! By the way that paper has also been annotated on Fermat's Library: http://fermatslibrary.com/s/counterexample-to-eulers-conjecture-on-sums-of-like-powers http://fermatslibrary.com/s/counterexample-to-eulers-conject... :)
- Marat_Dukhan 11y agoThanks! I updated the link
- c3534l 11y agoI 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.
- pc86 11y agoIt varies by publication, I've recently seen them counted as 250 and 450 characters (each), which would make the LA longer.
- idlewords 11y agoIs there a way to get rid of the 'Sign up with Facebook/Google' overlay that blocks the bottom part of the comments?
- aczerepinski 11y agoRight click on it, inspect element, press delete.
- gosub 11y ago$(".comment-form").remove()
- CatsoCatsoCatso 11y agoThere's a very clear hide link to click, there's no need to open Dev tools like the other comments are suggesting.
- mikeash 11y agoI see no such link to hide it. The only links are "Sign Up With Facebook" and "Sign Up With Google."
- marssaxman 11y agoI refuse to click such links on principle; if they want to hide their content, I will direct my eyeballs elsewhere.
- Lawtonfogle 11y agoChecked in IE 11, Chrome, and FF, and I don't see any hide link.
- Lawtonfogle 11y agoChecked in IE 11, Chrome, and FF, and I don't see any hide link.
- idlewords 11y agoYou may be the Chosen One. I don't see a hide link on the overlay that obscures comments.
- pgtan 11y agoThis website is the TeX/Web combination I ever dreamt of!
- Omnipresent 11y agoUnrelated to the published paper. Fermatslibrary is fascinating. Is there something similar for CS papers? I would love this for some distributed systems papers.
- noblethrasher 11y agoHere's something similar for CompSci: http://paperswelove.org/ http://paperswelove.org/ They had a presentatino on the Paxos paper about a month ago: http://www.meetup.com/Papers-We-Love-London/events/225736762/ http://www.meetup.com/Papers-We-Love-London/events/225736762...
- mbudde 11y agoFermat's Library is not only for math, it has a few CS papers already, for instance http://fermatslibrary.com/s/bitcoin 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 :)
- adenadel 11y agoIs there an answer to the question? Is an n^2 + 1 tiling possible? Is there a proof that it is not?
- evanpw 11y agoIt looks like proving that n^2 + 1 is impossible is still an open problem: http://www.wfnmc.org/mc20101.pdf http://www.wfnmc.org/mc20101.pdf.
- amluto 11y agoI'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. [1] http://web.mit.edu/linguistics/events/iap07/Nash-Eqm.pdf http://web.mit.edu/linguistics/events/iap07/Nash-Eqm.pdf
- kristopolous 11y agowhat 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...
- Someone 11y agoThe 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. 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-something-obvious http://math.stackexchange.com/questions/151782/when-is-somet...) and the trivial (https://en.m.wikipedia.org/wiki/Triviality_(mathematics)#Triviality_in_mathematical_reasoning https://en.m.wikipedia.org/wiki/Triviality_(mathematics)#Tri...)
- adevine 11y agoThe 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.
- colanderman 11y ago(the comments show how they overlap a little bit) OH. Thank you. This is what I was missing; I understood "cover" to mean "tile" and thought the bottom row was just misprinted slightly.
- outside1234 11y agoThis simultaneously proves that there is a lower bound on the number of words in an academic paper. :)
- apricot 11y agoAnother short paper published in the American Mathematical Monthly by Doron Zeilberger can be found here: http://i.imgur.com/PSigYcb.png http://i.imgur.com/PSigYcb.png
- peterevans 11y agoOne of the authors, John Conway, is the inventor of the Game of Life: https://en.wikipedia.org/wiki/John_Horton_Conway https://en.wikipedia.org/wiki/John_Horton_Conway.
- eru 11y agoHis `On Numbers and Games' (https://en.wikipedia.org/wiki/On_Numbers_and_Games https://en.wikipedia.org/wiki/On_Numbers_and_Games) is also worth a look.
- vacri 11y agoIf 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. The banner on their homepage is a [low res version of a] valid game-of-life set: http://golly.sourceforge.net/ http://golly.sourceforge.net/
- xigency 11y agoSome shorter proofs are available in the book "Proofs without Words." But then again, that book begins with a description that visual arguments aren't truly proofs.
- Tloewald 11y agoAs one of my favorite math professors once pointed out, a proof is merely an argument that convinces people, so whatever works works.
- eru 11y agoBefore the advent of computers we had two notions of proof in mathematics, the French school and the Russian school. 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.
- selimthegrim 11y agoAsher Peres has a pretty famous short paper in quantum information: http://arxiv.org/abs/quant-ph/0310035 http://arxiv.org/abs/quant-ph/0310035
- eps 11y agoWasn't there a single page PhD thesis that was a single sentence describing a counter-example for some long-standing conjecture? 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=PA40&dq=An+integral+function+never+0+or+1+is+a+constant&source=bl&ots=ov_dqTGKL1&sig=4yqDoeXwS_JKjcb8NqdpnHqQI-o&hl=en&sa=X&ved=0ahUKEwjv7Y66iarJAhXJaRQKHS-2AN0Q6AEIGzAA#v=onepage&f=false https://books.google.ch/books?id=MjVgeT7Laf8C&pg=PA40&lpg=PA...
- pervycreeper 11y agoIt 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!