Maths prize for 1 July 2016: Crossing the desert
You want to cross a huge desert. You have some Toyota pick-up trucks and drivers, and an unlimited supply of fuel. But each pickup can travel only 360km on a full tank, and the desert is too dangerous to leave fuel dumps or for pickups or drivers to wait or stay in the desert. You can extend the range by, for example, sending out two pickups together to ⅓ their individual range, siphon half the remaining fuel (i.e. ⅓ tank) from pickup A into pickup B, and then have pickup A return to base and pickup B to continue to 1⅓ of range, 480km. How much better than that can you do with an unlimited number of pickups and drivers?
Alex On worked out the answer – that with an unlimited number of pickups and drivers, you have an infinite range – but he didn’t write it down, so no prize-winner this time.
If we use 3 pickups, then we can send them out to ¼ of their individual range, siphon ¼: from the tank of one of the pickups into the tank of each of the other two (so they now have full tanks again), and leave the siphoned-out pickup with enough fuel to get back to base.
So the range is now (1+⅓+¼) times the original range.
Continue the same way with the 4th, 5th, 6th, etc. pickups, and the range increases to:
original range × (1+⅓+¼+1⁄5+1⁄6+….)
By the same argument as in the book-stacking problem, the total in the brackets can get as big as we want, just by adding more pickups. ∎