# On a schoolchild puzzle of V.I. Arnold (re: toric varieties)

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?

• While we are at it, what do toric varieties have to do with dimensional analysis? – Matt Noonan Jan 20 '11 at 7:03
• Units can be seen as monomials... – Mariano Suárez-Álvarez Jan 20 '11 at 7:16
• 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
• 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
• 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