There is a desert which can be crossed only by walking and this takes six days. The explorers of the desert can carry at most four days' water and food. If they want, they can take helpers with them. Each helper can also carry at most four days' supply. What is the minimum number of helpers that an explorer need to cross the desert?
"If the explorer takes one helper with him, after two days both are left with two days' food and water. Then the explorer could take the helper's food and water and cross the desert easily. But the helper would die in the desert."- This kind of cruel solution is not acceptable. No helpers should starve at any point.
Taken from the book Neurone Abaro Onuronon by Muhammad Zafar Iqbal.
The explorer heads 1 day into the desert with 4 rations of food. He eats 1 ration that day, leaves 2 rations, and returns the 2nd day, eating his final ration.
The explorer then heads out again with 4 rations of food. He travels 1 day, eating his first ration, and arrives at the cache, where he retrieves 1 ration, leaving 1 there and now has 4 on hand. He travels another day; he’s now 2 days into the desert with 3 rations of food. He leaves 2 rations there and returns to the first cache, eating his last ration. At the first cache, he retrieves the final ration and uses it to get to the beginning.
Then he heads out for the full crossing, with 4 rations of food. He travels 2 days, eating his first 2 rations, and then gets to his 2nd cache, where he retrieves 2 rations, bringing his total back up to 4. He then travels the remaining 4 days across the desert with the remaining 4 rations.
Here is the total number of rations and helpers required
This requires a total of 12 rations of food, the same amount as the previous solutions, but no helpers.
The explorer and the helpers walk 1 day into the desert. One of the helpers then gives two of his remaining portions to the other helper and the explorer, leaving him with 1 and 4 for the other two people. He then walks back home. The explorer and remaining helper walk one more day and the remaining helper gives the explorer one portion giving the explorer 4 and the helper 2. The helper then returns home and the explorer continues onward with enough to complete the trip.
You must start day 3 with 4 days of food to last you through days 3,4,5,6. Since you start with 4 days of food and you've used 2 by then, you have 2 remaining and must get 2 more rations from your helpers. Any helper that has travelled with you will have eaten 2 days of food and will need 2 more to get back, so some form of help has to occur at the end of day 1 to give you and your helpers enough food.
If you and a helper each have 3 days worth of food at the end of day 2, that works to give you enough food (4) to complete the journey and your helper enough food (2) to make it back. So you and a helper can begin day 2 with full rations and that would work. Since you've used 2 rations you need another helper to top you off after day 1. That helper starts with 4, uses 1 on day 1 and gives 2 to you and the other helper, and has one left to make it back on day 2. Then you and your helper start day 2 with 8 total rations and have 6 remaining after day 2. Your helper gives you 1 ration so you can then complete the journey and has 2 remaining to make it back to the start.
So I think 2 is the minimum number of helpers you need to cross the desert in 6 days.
Minimum food & helper time wasted, but requires reliable communication across the desert:
1. You arrange for a helper from the other side to start at the end of day 4 with 3 days worth of food towards you.
2. You start with 4 days worth of food and a helper with 3, just like the other one.
3. At the end of day 1, you get a 1 day pack from your helper and send him home with 1 day of food, just enough for his 1 day return. You now have 4 days food and 4 days to travel alone.
4. You meet the second helper at the end of day 5, with no food, but he has 2 man-days of food left, just enough to get you both where you want.
4 man-days (both food and time) wasted for helpers.
(If both helpers start with you, the minimum is 6 wasted man-days. The same if both helpers start from the opposite side.)
[wrong answer] I don’t seem to have option to comment on the question or answer (or atleast I am not sure how to do that) but can’t it be done by using just 1 helper?
On, days 1, helper and explorer both start with 4+4 days worth of supplies. On day 2, they are left with 3+3 days worth of supplies, today helper takes 1 day worth of supplies and returns back leaving explorer with 5 days of supplies for 5 days of journey.
This seems way simple so probably this answer is wrong but thanks in advance for pointing out why.
There is abundant food and water on either side of the desert.
The explorer eats 3 meals a day.
The explorer starts travelling each day after breakfast and stops before dinner, then sleeps until breakfast the next day. 'Breakfast' and 'dinner' times can be recalibrated to suit whatever time of day is good for travel in the desert.
1. Travelers (explorers and helpers) eat the first breakfast before leaving, so no journey provisions are required for the first breakfast.
2. The explorer arrives at dinner on the 6th day, so eats dinner at the other side of the desert (not from journey provisions). Likewise for helpers.
3. A total of 2 + 4x3 + 2 = 16 meals are required for the explorer for the journey, plus whatever else the helpers require.
One helper is commissioned to carry a full set of 4x3 meals, starting with the explorer after breakfast on day 1.
1. Both eat the helper's provisions for 3 meals (day 1 lunch & dinner, day 2 breakfast), then have lunch together on day 2; 8 meals total so far. The helper returns after lunch and requires another 3 meals (with a spare meal to bring home).
2. The explorer then has a full set of 4x3 meals for the rest of the journey (day 2 D, days 3-5 B+L+D, day 6 B+L).
3. At dinnertime on day 6, the explorer arrives with no remaining provisions, but having arrived, enjoys the cuisine available at the other side of the desert.