In chapter 7 of his book Number-Crunching Paul Nahin treats a "Leapfrog" problem posed by M. Schwartz from Ventura, CA, to Marylin von Savant in her "Ask Marylin" column:
A friend and I once went from his house to mine with one bike. I started walking and he rode the bike. When he got a couple of blocks ahead, he left the bike on the sidewalk and started walking. When I got to the bike, I started riding, passing him, and then left the bike a couple of blocks ahead. When he got to the bike. he started riding. We did this the whole way. At least one of us was always walking. At times, one was riding; at other times, we were both walking. I’m sure this was faster than if we had no bike. But some people insist that it was no faster because somebody was always walking. Who’s right?
The correct answer was:
The reader is right. It's true that someone was always walking. But neither friend walked the whole distance. Both biked part of the way. This increased their average speed, so they saved time.
In his book, Paul treats a more general problem replacing "a couple of blocks" with a variable parameter. He runs a simulation, finds several optimal values and observes that - to his surprise - the "a couple of blocks" is one of them.
Paul also asks for an analytic solution and promises to included such, if submitted, in a revised paperback edition.
- P. H. Nahin, Number-Crunching, Princeton University Press, 2011