Problem 164 Solution

First solve for x. Assume that the distance between Las Vegas and Reno is 1. Call r the pigeon that starts in Reno and v the pigeon that starts in Las Vegas. For any given amount of time the ratio of the distance traveled by r to v will be the same. This ratio at the first meeting is (1-2x)/2x. At the second meeting this ratio is (2-x)/(1+x). Equating these two ratios: (1-2x)/2x = (2-x)/(1+x).

Then cross multiply...

(1-2x)(1+x) = 2x(2-x)

1-x-2x² = 4x - 2x²

1-x = 4x

5x = 1

x = 1/5

So the first time they meet is 2/5 of the way from Vegas and the second time is 1/5 of the way to Reno.

I find the easiest approach at this point is to just follow the two paths as the birds continue to fly and see where they meet. When v has traveled 2/5 of the distance between Vegas and Reno to a point 3x the distance from Reno r will have traveled 3/5 of this distance to 2x the distance from Reno. When v has traveled another 2/5 he will be exactly in Vegas. Conventiently this is also where r will be.

Thanks to Alan Goldberg and Contingencies magazine for this problem.

Michael Shackleford, ASA - June 28, 2001