Cross the Desert Puzzle.

[poohbear_nc]

Was build a dune buggy ruled out?

[Dylrob]

Ugh, fractions give me such a headache.

[happy_terd]

Maybe if you drink two days of water before you go out.

[dreams]

Whoohoo! I think I got my guy out!

Spoiler:

On day 1 and 3 travel to Point 1 [=1 day's travel] and leave half the water carried. {take 4, use 1, and leave 2}

On Day 2 and 4 travel back to the start {using the 1 taken for 1 day's travel}

On Day 5 travel to Point 1 {use 1 and have 3 to leave}

On Day 6 you are Point 1 with 7 days worth of water. Take 4 and travel to Point 2. {Take 4, use 1, and leave 2}

On Day 7, return to Point 1{Take 1, use 1}.

On Day 8, travel to Point 2 {Take 3, use 1}

On Day 9 you are at Point 2 with 4 day's worth of water to get out on Day 12.

Did I do it? Yes, yes?

[omk3]

Will you try for the faster way now?

[dreams]

Originally Posted by omk3

Will you try for the faster way now?

Do you know how many times my guy died already?

Now, that I think I understand it, I'm going to see if I can get him out faster...

[pdurrant]

Originally Posted by dreams

Whoohoo! I think I got my guy out!

Spoiler:

On day 1 and 3 travel to Point 1 [=1 day's travel] and leave half the water carried. {take 4, use 1, and leave 2}

On Day 2 and 4 travel back to the start {using the 1 taken for 1 day's travel}

On Day 5 travel to Point 1 {use 1 and have 3 to leave}

On Day 6 you are Point 1 with 7 days worth of water. Take 4 and travel to Point 2. {Take 4, use 1, and leave 2}

On Day 7, return to Point 1{Take 1, use 1}.

On Day 8, travel to Point 2 {Take 3, use 1}

On Day 9 you are at Point 2 with 4 day's worth of water to get out on Day 12.

Did I do it? Yes, yes?

Yes, well done — that's a way to do it. Now, see my spoiler tag here: https://www.mobileread.com/forums/sho...5&postcount=19

[pdurrant]

Answer time! For the first part anyway.

Spoiler:

Our hero can, of course, escape the desert. He just needs to arrange to get to a point two days walk away from the oasis with four days' water.

Since he can only carry four days' water at a time, that means he'll have to have put some water at the two day point to pick up when he gets there to replenish his stores.

Now, he can't just carry water to the two day point directly from the oasis, as when he gets to the two day point, he'll have used up two days' water, and will only have two days' water left, which he'll need to get back to the oasis.

So he must make a cache of water between the oasis and the two day point. The obvious first place to try is the one day point. He can start off with four days' water, walk for a day, leave two days' water there, and walk back to the oasis, using up his last day's water.

Now - how much water will he need? Well, he needs to carry sufficient water from the oasis to supply all the days he's in the desert. That's four days getting out after he's set up the caches of water he needs. At least two days setting up the intermediate cache and at least three days setting up the two mile cache. That's an absolute minimum of nine days of water. But if he only goes back to the oasis once, he can only obtain eight days of water.

So we can see that he must re-visit the oasis at least twice, getting him a total of twelve days of water.

Since we've found he must re-visit twice, let's get all that done first. He goes back and forth between the oasis and the one day mark, carrying out water each time. At the end of the fifth day, he's at the one day mark with seven days water. He can now go out to the two day mark, leave two days water behind, and return to the one day mark at the end of the seventh day. At the one day mark, he now has only three days water. On the eighth day, he takes this water, collects the two days' water that he stored there, and now has four days' water with him, allowing him to reach the edge of the desert at the end of the twelfth day. Setting out the timings, it looks like this. I don't mention it in the table, but every day a day's water is used up.

Day 1: Take four days' water from the oasis. Walk out for one day
Day 2: Leave two day's water. Walk back to the oasis.
Day 3: Take four days' water from the oasis. Walk out for one day
Day 4: Leave two day's water. (Now four here in total.) Walk back to the oasis.
Day 5: Take four days' water from the oasis. Walk out for one day. (Now seven here in total.)
Day 6: Take four days' water from the seven you have, and walk out for one day.
Day 7: Leave two day's' water. Walk back to the one day mark.
Day 8: Take the three days' water you have and walk out one day.
Day 9-12: Take the two days' water you brought with you, and the two days' water you left here on day 7, and walk out of the desert, escaping the desert at the end of the twelfth day.

Unfortunately, I ‘forgot’ to mention that there's a literal deadline. The oasis is due to be the site of a nuclear test in 285 hours. (12 days less three hours.) There's an observation bunker at the edge of the desert where he'll be safe, but he must get there before the bomb goes off, or he will be killed by the blast*. The solution above doesn't get him there in time. Can you get him there in time?

Big Clue:

Spoiler:

You must have an intermediate cache of water. You must go back to the oasis twice. But it's worth pondering that although you have to walk between the oasis and the first cache of water five times, you only need to walk between the first cache of water and the two day point three times....

*Yes, it's an unfeasibly large nuclear test

[omk3]

I was going to complain (once more) that I couldn't work it out, when it finally hit me! (I think). It only took me 2 days : (I mean 2 days to work it out, not 2 days to walk out of the desert!)

Spoiler:

I was typing how it didn't make sense, because if I left water before the 1 day mark I was leaving too much to carry further in only two trips, and if I left water further than the 1 day mark it was not enough.

And then it hit me, that I had to find the optimum amount of water, ie calculate that distance, in which I could leave exactly 8 day's worth of water, so I could carry it with only two trips and not start walking with less than the maximum amount.

This formula gives how much water I can leave at any distance, starting from the oasis and going back two times (walking the distance 5 times):
water left= 2(4-2x)+(4-x)=12-5x
So if I want to have water for 8 days left, 12-5x=8 => x=4/5

So when I have walked to the 4/5 (of 1 day) mark 3 times (and 2 back to the oasis), I've spent 5*4/5=4 days, and will have 8 days' worth of water.

From then on I can carry the water to the 2 day mark in two trips (3 with the one trip back), and it will take me 18/5 of a day, or 3,6 days. I will then have 22/5 (4.4) days' worth of water. I need 4 more days to get out of the desert, so 4+4+3.6=11.6 days, and with a little water to spare too.

Please tell me I got it right, I can't bear to think about this any longer!

[pdurrant]

Originally Posted by omk3

I was going to complain (once more) that I couldn't work it out, when it finally hit me! (I think). It only took me 2 days : (I mean 2 days to work it out, not 2 days to walk out of the desert!)

Spoiler:

I was typing how it didn't make sense, because if I left water before the 1 day mark I was leaving too much to carry further in only two trips, and if I left water further than the 1 day mark it was not enough.

And then it hit me, that I had to find the optimum amount of water, ie calculate that distance, in which I could leave exactly 8 day's worth of water, so I could carry it with only two trips and not start walking with less than the maximum amount.

This formula gives how much water I can leave at any distance, starting from the oasis and going back two times (walking the distance 5 times):
water left= 2(4-2x)+(4-x)=12-5x
So if I want to have water for 8 days left, 12-5x=8 => x=4/5

So when I have walked to the 4/5 (of 1 day) mark 3 times (and 2 back to the oasis), I've spent 5*4/5=4 days, and will have 8 days' worth of water.

From then on I can carry the water to the 2 day mark in two trips (3 with the one trip back), and it will take me 18/5 of a day, or 3,6 days. I will then have 22/5 (4.4) days' worth of water. I need 4 more days to get out of the desert, so 4+4+3.6=11.6 days, and with a little water to spare too.

Please tell me I got it right, I can't bear to think about this any longer!

You have saved our hero. Congratulations!


But you're going to kill me....

Yes, you have a found a way to get him out before the bomb drops. But it's not the quickest he could get out of there. Can you find the shortest time it is possible for him to walk out of the desert?

And so as not to spoil your Sunday, here's the answer if you want to stop thinking about this poor man:

Spoiler:

Since he walks between the oasis and the first water cache five times and between the water cache and the two day point three times, the closer the first water cache is to the oasis, the quicker he'll get out.

But there must be a limit to how close to the oasis we can place it — too close and he won't be able to leave enough water at the two day point.

The cache of water at the two day point must be at least as much as he uses in walking from the first water cache to the two day point, as otherwise he'll end up at the two day point (the second time) with less than four days water. This means that the distance from the first water cache to the two day point can be no more than 4 (the number of days' water he can carry) divided by 3 (two journeys and the water he leaves) = 1 and 1/3.

So there's our answer. We want the first cache as close to the oasis as possible, but it can't be further away from the two day point than 1 and 1/3 days walk. So the optimum solution is to place it 2/3rds of a days walk from the oasis.

The time take to escape is then 5*(2/3) + 3*(4/3) + 4 = 11 and 1/3 or 11 days and eight hours.

[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

[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.

[omk3]

Yes, I see now. I should have figured it out sooner.

Spoiler:

The cache at day two must be 4 or larger.
The cache can be calculated like this:
4-2x (the water we take from cache 1 minus the trip to day 2 and back, x)
+4-x (the water we take from cache 1 the second time, minus the trip to day 2)
And to get a cache of 4, x=3/4. This only works if there is 8 days' water or more in cache1.
My problem was that I wasn't sure that I would have 4 full days of water left to take from cache1 to cache2 the second time. I thought that this second load of water was unknown. In the 12 day solution, the total cache of water in day one is only 7 after all, so you only take 3 on your second trip.

[Moejoe]

I stare at these threads with the wide-eyed wonder of a child whose wandered into an advanced mathematics class at a university. I just can't get my head around them. I am, I'm afraid, King Duffer of Duff mountain.

[omk3]

Oh, I just pretend to get my head around them, myself
Don't let them intimidate you. They smell your fear and go for the ankle.