| The Trip around the Moon |
| 1. The Trip around the Moon
The year is 1984. A moon base has been established and an astronaut is to make an exploratory trip around the moon. Starting at the base, he is to follow a great circle and return to the base from the other side. The trip is to be made in a car built to travel over the satellite's surface and having a fuel tank that holds just enough fuel to take the car a fifth of the way around the moon. In addition the car can carry one sealed container that holds the same amount of fuel as the tank. This may be opened and used to fill the tank or it may be deposited, unopened, on the moon's surface. No fraction of the container's contents may be so deposited. The problem is to devise a way of making the round trip with a minimum consumption of fuel. As many preliminary trips as desired may be made, in either direction, to leave containers at strategic spots where they can be picked up and used later, but eventually a complete circuit must be made all the way around in one direction. Assume that there is an unlimited supply of containers at the base. The car can always be refueled at the base from a large tank. For example, if it arrives at the base with a partly empty tank, it can refill its tank without wasting the fuel remaining in its tank. To work on the problem, it is convenient to draw a circle and divide it into twentieths as shown on the next page in Figure 85. Fuel used in preliminary trips must of course be counted as part of the total amount consumed. For example, if the car carried a container to point '30, left it there and returned to base, the operation wouild [sic] consume one tank of fuel. (c) Martin Gardner, Sixth Book of Mathematical Diversions At first, I didn�t understand why you�d want to divide the circle into twentieths. I thought, �It can go a fifth of the way, you drop one, then you drop one another fifth of the way by using up the first one...oh...it can�t go that far and make it back if you do that. Crap.� So I got to thinking about it some more and after a few minutes, I determined how to solve it. I did, indeed, break the circle into twentieths like so. |
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| When using twentieths, the car can go from one location to another four times, whether in one direction or by going back and forth or a combination of the two.
I labeled the locations this way for a reason: initially, I figured the best way would be to hit all the numbered sections, but later I thought the then blank ones would be good, so instead of making them halves, I made them letters. You�ll thank me later, as the way I plan to explain it will use quite a bit of shorthand. Having halves or two-digit numbers or whatever else would be terrible. |