#2             91. NOT MUCH TO GO ON             

Here is a delightful problem which supplies remarkably little information.

A car needs a certain quantity of fuel to travel around a certain circular circuit.

The exact quantity of fuel is distributed among a number of refuelling stations at fixed positions around the track.

Our problem is this. Suppose the car begins with tank empty. Can it begin at some one of the refuelling stations and obtain enough fuel as it passes the remaining stations on the way round to complete the circuit?

Obviously the total amount of fuel is sufficient. The danger is that the car may be stranded between stations.

To avoid complications, we assume that the car’s tank can hold enough fuel for the car to complete a circuit.

Hints and strategies

Hint 1                Hint 2              Solution               Extensions
HINT 1

Try some special cases. At which station do you think the car should start? Is there a pattern here?
HINT 2

You probably have some idea of what is happening npow. Can you prove it?
SOLUTION

Let us consider a slightly different problem. Suppose we have a car with a large reserve Q of fuel. This car begins at a refuelling station, and does the circuit, refuelling along the way. The car finishes with the amount of fuel it starts with.  At some refuelling station X, the car’s fuel reaches a (non-negative) minimum M.    This means that it carries quantity M of fuel right around the track.   Alternatively, there is always at least M amount of fuel in the tank the whole time.

Now in our given problem, suppose our car starts at station X with an empty tank.  As the car proceeds arond the circuit, the guage will never read below zero, as we have the above situation, less quantity M of fuel.  
  
EXTENSIONS

You might think our solution to this problem is a bit sophisticated.

Suppose F is the amount of fuel required to do a circuit. Try these examples.
• Three equally spaced stops each with F/3 amount of fuel.
• Two equally spaced stops with F/4 and 3F/4 amount of fuel.

In each case, graph the fuel consumption against the distance travelled for the two solutions of the problem. How are your graphs related? Try some other examples.