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I think I once saw a sentence in an article by V.I. Arnol'd saying something like: here is a problem that every Russian schoolchild can solve, but no western mathematician can solve. But I can't find it now.

Does anyone know the quote and where it was written?

I ask because I'm preparing a talk on research by high school students, and I might include this quote if the actual wording isn't too inflammatory.

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    $\begingroup$ I imagine you are thinking of this math.stackexchange.com/questions/1594740/… . I assume the book is springer.com/gp/book/9783540206149 but I haven't checked. $\endgroup$
    – David E Speyer
    Commented Jan 18, 2017 at 12:20
  • $\begingroup$ No, that's definitely not what I was thinking of. It was a comparison of Russian children to western mathematicians. I saw it maybe 20 years ago. $\endgroup$
    – Michael Zieve
    Commented Jan 18, 2017 at 12:22
  • $\begingroup$ Anyway, the example with the triangle is typical of Arnold provocative (and stimulating) personality. I have to confess that I feel sympathetic towards the poor american students. After all, you trust your teacher, right? So if he gives to you a triangle you suppose that it actually exists, not that he's checking your smartness with some kind of mischievous joke :-) $\endgroup$
    – Francesco Polizzi
    Commented Jan 18, 2017 at 12:27
  • $\begingroup$ I received an email from a colleague saying he believed he had once heard Faltings mention an Arnol'd quote about a problem that could be solved by every Russian schoolchild but no western mathematician, and how Faltings had solved the problem just to show he could. So maybe if someone reading this is a close enough friend of Faltings' to make it ok to ask him about this trivial issue, then an answer could be found. But please don't email him unless you are indeed a close personal friend of his, I wouldn't want him to get spammed for this. $\endgroup$
    – Michael Zieve
    Commented Jan 22, 2017 at 14:34
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    $\begingroup$ @Michael Zieve: But the Arnold-Faltings anecdote was about a "problem that people like Barrow, Newton and Huygens would have solved in a few minutes and which present-day mathematicians are not, in my opinion [i.e., Arnold's], capable of solving quickly...": mathoverflow.net/questions/20696/… $\endgroup$
    – José Hdz. Stgo.
    Commented Jan 26, 2017 at 7:10

3 Answers 3

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You possibly mean this 'two volumes' problem from the book ``Problems for children from 5 to 15'':

Russian original:

  1. На книжной полке рядом стоят два тома Пушкина: первый и второй. Страницы каждого тома имеют вместе толщину 2 см, а обложка –– каждая –– 2 мм. Червь прогрыз (перпендикулярно страницам) от первой страницы первого тома до последней страницы второго тома. Какой путь он прогрыз? [Эта топологическая задача с невероятным ответом –– 4 мм –– совершенно недоступна академикам, но некоторые дошкольники легко справляются с ней.]

English translation:

  1. Two volumes of Pushkin, the first and the second, are side-by-side on a bookshelf. The pages of each volume are 2 cm thick, and the cover – front and back each – is 2 mm. A bookworm has gnawed through (perpendicular to the pages) from the first page of volume 1 to the last page of volume 2. How long is the bookworm’s track? [This topological problem with an incredible answer – 4 mm – is absolutely impossible for academicians, but some preschoolers handle it with ease.]
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    $\begingroup$ The link to the book is given by Martin Peters. There is a comparision of American and Russian school students (in problem 6, mentioned by David Speyer), but not of Russian school students and Western professors. I personally also do not remember this specific comparision, though I read probably all Arnold's publicism. $\endgroup$
    – Fedor Petrov
    Commented Jan 18, 2017 at 12:55
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    $\begingroup$ That might actually become an unsolvable problem if you (1) omit the 2cm part (2) give it to someone who has a different convention on the order in which one should put books on a shelf, or pages in a book. Maybe Russians and American had different conventions? For instance, it would be interesting to hear what the common convention is for books in Arabic (right-to-left writing) or Japanese (opposite page ordering). $\endgroup$
    – Federico Poloni
    Commented Jan 18, 2017 at 12:57
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    $\begingroup$ Would you elaborate on the correct answer? I see 4mm possible only if the books are arranged such that the back cover of volume 2 touches the front cover of volume 1 and the bookworm travels away from the pages of volume 1. Perhaps the question might be better written/more understandable as "What is the shortest path the bookworm can travel?" This is not the natural ordering of books in most Western cultures (I have no idea about how it's done in Russia though). $\endgroup$
    – par
    Commented Jan 18, 2017 at 18:05
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    $\begingroup$ @par It actually is the standard ordering. If you take a book and look at its front cover, then turn it to place it on the shelf, the spine will now be facing you, the front cover will be on your right and the back cover on your left. Now, if you take two such books and do it, the front cover of the one on the left is touching the back cover of the one on the right. In every library I've seen, Volume I is placed "first" (that is, to the left in Western convention) of volume 2, so its front cover touches the back cover of volume 2. $\endgroup$
    – JKreft
    Commented Jan 18, 2017 at 18:22
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    $\begingroup$ @JKreft Of course, this assumes that your books are read by flipping pages from left to right, and so that text is read left to right; languages that are innately read the other way (e.g. Japanese) will have books bound the other way - but will often still be sorted in increasing order from left to right. $\endgroup$
    – Steven Stadnicki
    Commented Jan 19, 2017 at 7:47
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Here's Arnold's essay on teaching mathematics.

http://pauli.uni-muenster.de/~munsteg/arnold.html

"For example, these students have never seen a paraboloid and a question on the form of the surface given by the equation $xy = z^2$ puts the mathematicians studying at ENS into a stupor. Drawing a curve given by parametric equations (like $x = t^3 - 3t$, $y = t^4 - 2t^2$) on a plane is a totally impossible problem for students (and, probably, even for most French professors of mathematics)."

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    $\begingroup$ The surface $xy=z^2$ is a cone. Should it read $xy=z$? Or is it because I am one of those western mathematicians? $\endgroup$
    – Michael Renardy
    Commented Jan 19, 2017 at 9:02
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Maybe you are referring to his book with problems for schoolchildren? See problem 13 there, and also a remark in the preface.

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  • $\begingroup$ The comment in the preface might be what I was looking for. In my memory the quote was more aggressive than that though, so I wonder whether he said something similar elsewhere. $\endgroup$
    – Michael Zieve
    Commented Jan 18, 2017 at 12:53
  • $\begingroup$ I also thought I had seen the quote some years before Arnol'd wrote that book. $\endgroup$
    – Michael Zieve
    Commented Jan 18, 2017 at 13:01
  • $\begingroup$ Hm, yes, could be; I recall him making such a comment in a lecture at the Fields Institute in the year 1997. You could search further inside his following books: springer.com/gp/book/9783540548119 , springer.com/gp/book/9783540287346#otherversion=9783642066863 , bookstore.ams.org/mcl-17 , and several of the small books published by МЦНМО, see biblio.mccme.ru/… . I do not have all of them here and cannot check it now. $\endgroup$
    – Martin Peters
    Commented Jan 18, 2017 at 13:35

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