Crossing the Desert with a Camel
I mentioned this in the Planes Around the World Puzzle, but this is the more classic version. You are travelling on a camel across the desert. The camel is called Ptolemy, but that has no bearing on the puzzle.
Ptolemy can carry enough water for the two of you to travel 500 miles, but unfortunately the desert is 800 miles wide. However, you have lots of containers with you (or Ptolemy has at least, since he is doing all the heavy lifting here, but he doesn't mind) with which you can set up little caches of water that you can come back to later. They have nice screw top lids, so their won't be an evaporation. Every time you return back to the start you can refill your water at the camp.
The question is: what is the minimum amount of water that you will need to cross the desert. As a bonus question: what is the widest desert you can cross.
This puzzle is quite hard, but it is probably easier than actually crossing a desert.
Hints and the answer below…
Hint 1 of 2: if one load of water would take you 600 miles rather 500, how could you get across with only 2 loads of water?
Hint 2 of 2: So 2 loads of water brings you 500(1+1/3), how can you extend this?
Let’s call 500 miles 1 unit to make this easier. We've already established that 2 loads of water is enough to bring you 1+1/3 units by building a cache 1/3 of a unit in, but this isn't far enough by itself. Instead we are going to build a cache 1/5 of a unit in and fill it twice. Each time Ptolemy dumps 3/5s of his water and then has just enough to get home. This leaves the cache with 6/5 of a full load. Then when you next head of, you use 1/5 of your water getting to the cache leaving you with 4/5 on you plus another 6/5 in the container for a total of 2 full loads of water. We already know that 2 full loads is enough to travel 1+1/3 units, so now we have travelled 1+1/3+1/5 units which is 766.666… miles.
This is just short, but you might be able to see the pattern here. 3 full loads of water was enough to add an extra 1/5 to our answer, so the natural extension is to to think about 1/7 units. However that is overkill because it gets us to over 800 miles. Instead we should look at how much distance we are missing with our answer above. It turns out that it is only 1/15 of a unit. To build a cache 1/15 of a unit in only requires 7/15 of a load of water fill it (in 4 trips there and 3 back to camp), so the total amount of water needed it 3+7/15 loads.
To answer the bonus question. Each load takes us 1 then 1/3 then 1/5 and so on. This sums to infinity and so with an unlimited amount of water at base camp you can cross a desert of infinity width.