Assume that one person can carry a four day supply of food and water for a trip across a desert that takes 6 days to cross. One person cannot make the trip alone because the food and water would be gone after 4 days. How many persons would have to start out in order for one person to get across and the others to return to the starting point?

google is your friend. Here is one solution of several:

https://www.algebra.com/algebra/homework/word/coins/Word_Problems_With_Coins.faq.question.775914.html

Have seen this in an old math book.

You start with 3 people.

At the end of first day, the 3 people now have 3 days' food each, one person give a days' food to the other two, so 2 people now have 4 days' food and the 1st has a days' food left and heads home.
At the end of day 2, the 2 people each have 3 days' food, so one gives a day's food to the other. Now, the one with 2 days' food heads home, and the one with 4 days' food has 4 days to get to the other side.

same source as mine

No page seems to have a valid algebraic solution.

To solve this problem, we need to figure out a way for one person to cross the desert while the others return to the starting point without running out of food and water.

Let's break down the scenario step by step:

1. Since one person can carry a four-day supply, they can travel for a maximum of 4 days before running out of food and water.
2. The trip across the desert takes 6 days in total, which means the person would need extra supplies to cover the remaining 2 days.
3. To ensure the person can continue to travel while still having enough supplies, we can have someone meet them at a specific point and replenish their provisions.
4. As a result, two people are required - one to continue the journey and one to return to the starting point with the remaining supplies.
5. After the initial 4 days of travel, the person with the additional supplies will meet the traveler, exchange the supplies, and return to the starting point.
6. Meanwhile, the person who received the extra supplies can now travel for an additional 4 days, making a total of 8 days of supplies available.
7. Once the 8th day is complete, the second person who returned to the starting point can meet the traveler again, exchange supplies, and go back to the starting point while the traveler continues towards the end of the journey.
8. By repeating this process, the traveler will have enough supplies to cover the full 6-day journey.

Therefore, the minimum number of people required for one person to complete the journey while others return to the starting point is two people.