0

This question I am feeling very difficult to solve. It is said to be a problem on mathematical induction:

On a circular path, there are are $n$ cars and among them they have enough fuel to cover the entire circumference of the circular path.
I have to prove that there exists a car which can cover the road by collecting fuel from other cars on the way.

Please help me.

Suraj Mandal
  • 1

1 Answers1

3

First, suppose that the cars can run with a negative amount of fuel. Take any car and let it run. Consider the graph that shows the amount of fuel it has along its way. It will be a piecewise decreasing function. The discontinuities are the points where the other cars are. Since the cars have exactly the fuel needed for the path, the car that is running ends with fuel $0$.

Now, look for the minimum point of your graph, that is, the point at which the car had the least amount of fuel. This will probably be negative, and this is for sure at the point where some other car is stopped.

Think about what would have happen if this other car had run.

ajotatxe
  • 65,084
  • I'm honestly not sure if this is a joke or what? "Suppose that the cars an run with a negative amount of fuel." How is this in any way sensible? Is there something obvious going on that I'm missing? – Squirtle Aug 01 '14 at 20:49
  • You're only considering the option of them going to negative fuel so you can draw the graph. Once you've got the graph, going from a minimum point will still be modeled in the same way and it will never go negative. – Dan Rust Aug 01 '14 at 21:01