[pdurrant]

Originally Posted by omk3

I don't know whether to feel proud or rather slow

They funny thing is

Spoiler:

I almost got the right answer by accident, because after experimenting with halfs and failing, I started to experiment with 1/3s but gave up.
I had thought of course that the smallest the first distance the better, but I couldn't arrive at a formula. I still don't entirely get how you end up with the 4/3 distance from the first cache to the second. I'm sure I'll get it when my head is a bit clearer, it now needs some rest. I really should have progressed more quickly on this one.

Oh, well, many thanks for the puzzles and congratulations on making me obsess about them

I'll try to expand on the point you raise.

Spoiler:

I think we agree that the second cache /must/ be at the two day point.

(It can't be any closer to the oasis, and making it further away will increase the length of the journeys to that point to supply the water for the cache. I must admit I haven't fully analysed whether making it a little further away would allow the first cache to be a little closer to the oasis, but I don't think it would.)

Taking the second cache to be fixed, we need to look at the first cache position. It should be as close to the oasis as possible, to reduce the time taken to supply it, but there must be a limit on how close it can get, as it obviously can't be at the oasis itself.

We make one journey from the first cache to the second cache, to leave the water there, and return to the first cache. We then fill up to maximum capacity, journey to the second cache, and we must have enough water at the second cache to replenish the water we used up walking to it from the first cache.

Say that we use up x amount of water walking from the first cache to the second. We need to find at least x amount of water at the second cache, to get our water back up to the full four days to be able to get out of the desert.

Let's look at the journey to supply the second cache. We start off from the first cache with four days supply of water. We use up x amount of water getting to the cache position, we must leave x amount of water as the cache, and we use x amount of water getting back to the first cache. So on the entire trip we used up x amount of water three times. We started off with fours days of water, so the most that x can be is 4/3.