## Frank Morgan's Math Chat |

October 7, 1999

**ANSWER.** Probably not, as best explained by Richard Ritter. The current
population of China is about 1.25 billion, with about 20 million births per
year. We'll assume that the birthrate stays about the same, as the
population grows a bit but the births per 1000 drops a bit, under the
current one child per family policy. The Chinese walk say 3 feet apart at 3
miles per hour, for a rate of 46 million Chinese per year. So even if no
one died in line, the line would shorten by 26 million per year and run out
in about 1250/26 = 48 years. (Different assumptions could lead to a
different conclusion.)

Incidentally, the UN Population Fund projects that the world population will hit 6 billion next week (around October 12).

**NEW CHALLENGE** with **$200 PRIZE** for best complete solution
(otherwise usual book award for best attempt).
A double bubble is three circular arcs meeting at 120 degrees, as
in the third figure.

Consider a circle of area A, a circle of area A+1, and a double bubble of
areas A and 1. Let H_{0}, H_{1}, H_{2} denote the curvatures (1/radius) of the
bottom of each. Prove that

Prove the same result for **R**^{n }(replacing area by volume and circles by
spheres).

This open problem appears as Conjecture 4.10 in "Component bounds for area-minimizing double bubbles," by Cory Heilmann, Yvonne Lai, Ben Reichardt, and Anita Spielman (NSF "SMALL" undergraduate research Geometry Group report, Williams College, 1999). It bears on proving the Double Bubble Conjecture (see Math Chat of October 25, 1996).

Any individual or group is welcome to submit a solution for receipt by October 31, 1999 to Prof. Frank Morgan, Department of Mathematics, Williams College, Williamstown, MA 01267. Even incomplete solutions may compete for the usual book award.

To allow extra time for this special prize challenge, the next Math
Chat will appear on November 4. Math Chat regularly appears on the first
and third Thursdays of each month. Prof. Morgan's homepage is at
www.williams.edu/Mathematics/fmorgan.

Frank.Morgan@williams.edu.

Copyright 1999, Frank Morgan.