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When reading the interview with Vladimir Arnold in the April 1997 edition of the Notices, I came across the following anecdote.

Many Russian families have the tradition of giving hundreds of such problems to their children, and mine was no exception. The first real mathematical experience I had was when our schoolteacher I. V. Morozkin gave us the following problem: Two old women started at sunrise and each walked at a constant velocity. One went from A to B and the other from B to A. They met at noon and, continuing with no stop, arrived respectively at B at 4 p.m. and at A at 9 p.m. At what time was the sunrise on this day?

I spent a whole day thinking on this oldie, and the solution (based on what is now called scaling arguments, dimensional analysis, or toric variety theory, depending on your taste) came as a revelation.

I found the solution in a rather straightfoward fashion, but I was curious as to the parenthetic remark. So, can anybody tell me (as a total outsider to algebraic geometry), what does this problem have to do with toric varieties?

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While we are at it, what do toric varieties have to do with dimensional analysis? –  Matt Noonan Jan 20 '11 at 7:03
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Units can be seen as monomials... –  Mariano Suárez-Alvarez Jan 20 '11 at 7:16
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Dimensional analysis is abstractly the study of the action of (K*)^n on K[x_1, ... x_n]. One should think of each of the x_i as having different units and (K*)^n as describing what happens to quantities (homogeneous elements) as we change units. And I am told that actions of tori have something to do with toric varieties. –  Qiaochu Yuan Jan 20 '11 at 9:16
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It seems to me that this problem has nothing whatsoever to do with "scaling arguments, dimensional analysis, or toric variety theory". On the other hand the underlying proportion x:y=y:z is the equation of my favorite toric singularity. PS I am now sorely tempted to ask this question to my daughter this evening :-). –  Barbara Jan 21 '11 at 13:47
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If you scale the velocities of both women and the distance between A and B all by the same amount, of course you get the same answer. That makes it possible to arrange any value you like for one velocity, or for the distance from A to B. –  Ben McKay Jan 6 '12 at 12:30
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1 Answer

I think the problem cannot be solved because the sun rises at different times at A and B in general (say A = Vladivostok and B = Moscow). All what can be said is that (12-tA)(12-tB) = 4x9 = 36.
If by chance the sun rise times are identical then tA = tB = 6.
This is obtained by using the similitude of triangles in the space-time diagram.

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An old woman walking from Vladivostok to Moscow in one day is certainly in very good shape. –  Niemi Jan 6 '12 at 10:46
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