# Crossing the desert

There are explorers that want to go as far in the desert as possible. One explorer can carry water for himself for 10 days. If they would all stick together, they would have enough water to go 5 days into the desert, then they would have to turn to have enough water to get back to the start. But if one or two went back earlier and left some of their water with others, those could get further. Have in mind, one explorer can't carry more than 10 days worth of water.

1. question: How far into the desert could 1 explorer go, if we have 4 explorers all together, and they can only give water to each other at the end of the day?

2. question: How far into the desert could 1 explorer go, if we have 3 explorers all together, and they can only give water to each other at the end of the day?

3. question: How far into the desert could 1 explorer go, if we have 2 explorers all together, and they can only give water to each other at the end of the day?

You can't leave water on the ground, just give it to another person. But you can go back to the camp unlimited times.

• Is it allowed to leave water on the ground, such that it can be collected by a different explorer on their return trip? Commented Jan 6, 2020 at 23:46
• Also, is someone able to return to the starting position, collect more water (up to 10 rations) and then leave again? Commented Jan 7, 2020 at 0:00
• If you can leave water on the ground, you could simply move an unbounded amount of water to a position one day out (by taking trips every day, each explorer leaving 8 day's worth of water there for as many trips as desired). This effectively establishes a new base camp one day out. Since there is no bound on the amount of water we can put in this new camp, it seems like we can repeat the process as many times as we want. Commented Jan 7, 2020 at 0:01
• @hdsdv - funny, I interpreted it as you can leave water on the ground, but you can't leave camp more than 1 time. But the other way around would make sense too (you can leave camp multiple times but not leave water on the ground). We just can't have both be allowed.
– TTT
Commented Jan 7, 2020 at 0:03
• @hdsdv Indeed, that would be the solution if both restrictions were lifted; however, there are three other possibilities, depending on whether one or none of these actions are allowed, each a unique puzzle with a different solution. Commented Jan 7, 2020 at 0:08

## 5 Answers

Improvement on the 2-explorer case:

8 Days out, if one of the explorers returns to camp, refills, and goes back out. Here's a chart showing everyone's location and water reserves. Location 0 is the base camp.

First attempt at 3-explorer case, using the same strategy and formatting:

10 days out (I suspect this can be improved by a day, there are a number of small inefficiencies that might be enough to combine into an extra day for everyone):

• Nice chart. We'll have to see which way OP has in mind as between the two of us I think we have both sides of the 2 explorers problem covered. (Though ZanyG points out maybe neither is allowed!)
– TTT
Commented Jan 7, 2020 at 0:07

Here a possible a possible solution to the 4 explorer case, inspired by hdsdv's answer:

We can reach 12 days out, in the following manner:

Question 3 (2 Explorers):

7 days in. They both walk in 2 days (and they each have eight days worth left). Explorer 1 gives two days worth to Explorer 2 and leaves two days worth on the ground. Explorer 1 has four days left and takes the 2 day trip home alone. Explorer 2 can travel out 5 more days. Note Explorer 1 has 2 days worth extra when arriving home but that isn't enough to go an extra day for both explorers.

Question 2 (3 Explorers):

9 days in. All 3 explorers walk together for two days, then Explorer 3 hands two days worth to each of the other two explorers, leaves two days worth on the ground and has just enough to get back home. This reduces to Question 3 so the remaining two explorers can go 7 more days. Explorer 1 uses his extra two days to get home and Explorer 2 picks up the two left on the ground to get home.

Question 1 (4 Explorers):

10 days in? All 4 explorers walk together for 1 day. Explorer 4 trues up the others by giving a day to the other 3, leaves 5 on the ground, and goes home. Now this reduces to Question 2 (9 more days), and there is enough for the 3 explorers to get home when they get back to this position. Note there is an extra 2 days left though which makes me believe this is probably not optimal.

Assuming that water cannot be dropped, and return trips cannot be made, which leaves the puzzle as the simplest form described in the question:

1. 8 days out

2. 7 days out

3. 6 days out

### Reasoning:

If everyone drinks from one explorer's supply each day, until that explorer has enough water to get home on their own, but not enough to supply everyone for another day as well as get home the following day. Then that explorer starts heading home and then everyone drinks from the next explorer's supply, until one explorer is left, then they head home when their supply equals the distance away from home they are (or one greater, they would just have spare).

I will answer this puzzle based on the assumption that when the travelers need one day of water to go from point A to point B they also need the same amount of water to go from point B to point A. Let's have the travelers a,b,c,d . The travelers go 2 days into the desert then traveler a gives 2 days of water to travelers b,c,d and goes back to the beginning point A. Now travelers b,c,d have 10 days of water each. Travelers b,c,d go 3 days into the desert and traveler b gives one day of water to travelers c,d and goes back. Now travelers c,d each have 8 days of water. Travelers c,d go 1 more day into the desert. Traveler c gives traveler d 1 day of water and returns. Now traveler d has 8 days of water. Traveler d goes 1 more day into the desert and returns to point A. So the furthest one traveler has gone is 7 days into the desert. If there are 2 travelers, the farthest one can go is 6 days into the desert. If there are 3 travelers, the farthest one can go is 7 days into the desert.