Years ago I remember problems about crossing a desert or or desolate area. I do not remember if the problems were solved by trial and error or some sort of analysis. The following is the general idea.

Say you had a vehicle which could carry enough fuel to travel 600 units (miles or kilometers). You want to cross a desolate area which is say 1500 units (miles or kilometers). How do you do this using a minimum amount of fuel?

It seems obvious that this could be solved by trial and error. Certainly you could write a program to assist the trial and error process.

Is there some analytical approach or successive approximations method which would be certain of obtaining an optimum solution?