Jeep problem facts for kids

The jeep problem is a cool math puzzle. Imagine a jeep that needs to travel as far as it can into a huge desert. The jeep can only carry a certain amount of fuel at one time. But here's the trick: it can drop off fuel at special spots called "fuel dumps" and pick it up later. The main goal is to figure out how to use the fuel best to go the longest distance.

This puzzle is very old! It first appeared in a book called Problems to Sharpen the Young around the 9th century. A smart person named Alcuin is often given credit for it. Later, in the 1500s, Luca Pacioli also wrote about it. A more modern look at the problem came from N. J. Fine in 1947.

This jeep problem is like a few other brain teasers:

• The camel and bananas problem is about a merchant trying to get as many bananas as possible to a market. The camel needs to eat bananas to move! It can only carry a limited amount, but the merchant can leave bananas at "camp posts" along the way.
• The travelers across the desert problem asks how many helpers you need to reach a far-off base. Everyone carries supplies, and they can share them. But they can't leave supplies behind. Everyone must have enough to get back safely.
• The cars across the desert problem is similar to the travelers' one. It asks how many cars you need to reach a far base. Cars can share fuel, but they can't leave fuel dumps. Empty cars are left behind. This problem is a bit like how multistage rockets work!

The Main Puzzle

Let's look closer at the jeep problem. Imagine you have a certain amount of fuel, let's say n units, at your starting base. The jeep can carry only 1 unit of fuel at a time. It uses 1 unit of fuel to travel 1 unit of distance. This means if it carries 1 unit of fuel, it can go 1 unit of distance.

The jeep can drop off any fuel it's carrying at a fuel dump. It can also pick up fuel from a dump. But it can never carry more than 1 unit of fuel at a time.

There are two main versions of this puzzle:

• Exploring the desert: In this version, the jeep must always return to the starting base after every trip.
• Crossing the desert: Here, the jeep must return to the base after most trips. But on the very last trip, it just goes as far as it can until it runs out of fuel. It doesn't have to come back.

In both cases, the main goal is to make the jeep travel the farthest distance possible on its final trip. Or, sometimes, the goal is to find the least amount of fuel needed to reach a certain distance.

Usually, we imagine fuel as a "continuous" amount, like water. This means you can have any tiny amount of fuel. But some harder versions of the problem only let you leave or pick up fuel in specific, fixed amounts.

How to Solve It

Let's think about how to solve the "exploring the desert" puzzle. The jeep wants to go as far as it can and still return to base.

Solution for "exploring the desert" when you start with 3 units of fuel. This shows how much fuel is in the jeep and at the dumps at different times.

Here's a smart way to do it:

• The jeep makes n trips. Each time, it starts from the base with 1 unit of fuel.
• On the first trip, the jeep travels a short distance. Let's say it goes 1/(2n) units of distance. It then leaves most of its fuel at a new fuel dump. It leaves (n − 1)/n units of fuel there. The jeep has just enough fuel left (1/(2n) units) to get back to the base.
• On the next n − 1 trips, the jeep goes to this first fuel dump. It picks up fuel so that it has a full tank (1 unit) when it leaves the dump. It also picks up fuel on the way back to make sure it can reach the base.
• The jeep keeps doing this. On each trip, it goes a bit farther and sets up a new fuel dump. It leaves fuel at each dump for later trips.
• When the jeep starts its final trip, it has set up n − 1 fuel dumps. The farthest dump has 1/2 unit of fuel left. The next one has 1/3 unit, and so on. The closest dump has 1/n units of fuel.
• On this final trip, the jeep starts with 1 unit of fuel from the base. It picks up half the fuel from each dump on its way out. This keeps its tank full. After the farthest dump, it travels a bit more and then turns around. It picks up the rest of the fuel from each dump on the way back. This gives it just enough fuel to reach the next dump or, finally, to get back to the base.

The total distance the jeep can travel on this final round trip is found by adding up a series of fractions: 1 + 1/2 + 1/3 + ... + 1/n. This is called the nth harmonic number. The farthest distance the jeep can go into the desert is half of this total.

This means that if you have enough fuel at the base, you can go any distance! But the more distance you want to cover, the more fuel you need. The number of fuel dumps also grows very quickly.

Now, for the "crossing the desert" puzzle, the strategy is similar. But on the last trip, the jeep doesn't need to come back.

Solution for "crossing the desert" when you start with 3 units of fuel. This shows fuel levels and where the jeep turns around.

• The jeep still sets up fuel dumps. But the amounts of fuel left at each dump are a bit different.
• When the jeep starts its final trip, it has n − 1 fuel dumps. The farthest dump has 1/3 unit of fuel. The next has 1/5 unit, and so on. The closest dump has 1/(2n − 1) units of fuel.
• On this final trip, the jeep starts with 1 unit of fuel from the base. It collects all the remaining fuel from each dump on its way out. This keeps its tank full. After leaving the farthest dump, it travels 1 more unit of distance. It doesn't need to return.

The total distance the jeep can travel on this final trip is 1 + 1/3 + 1/5 + ... + 1/(2n − 1). This distance is always greater than the distance for "exploring the desert." So, in theory, you can cross a desert of any size if you have enough fuel at the base! Again, the fuel and number of dumps needed grow very fast.

Other Puzzle Solutions

For the camel and bananas problem, the solution is similar to "crossing the desert." The goal is to get the most bananas to the market. The amount of bananas you can get there depends on how far the market is.

In the travelers across the desert problem, imagine n travelers start with n units of supplies. They travel a short distance. Then, one traveler gives their extra supplies to the others and returns to base. The remaining travelers continue. They keep dropping off travelers who return to base until only one traveler is left. This last traveler then goes as far as possible. The maximum distance n travelers can reach is 2 − 2/(n+1). This problem is tricky because travelers can't leave supplies behind.

For the cars across the desert problem, n cars start with n units of fuel. They travel a short distance, share fuel, and leave one empty car behind. The remaining cars continue. This happens until only one car is left. This last car then travels as far as it can. The longest distance n cars can reach is the nth harmonic number.

Order of Trips

It's interesting that the order of the jeep's trips doesn't really change the final maximum distance. For example, in the "exploring the desert" version, the jeep could make many small trips to the first fuel dump to build up fuel there. Then it could use that fuel to move more fuel to the second dump, and so on. The final distance reached would still be the same.

Fuel as a Continuous Amount

The math for these problems can get quite complex, especially when you think about fuel as a continuous amount (meaning you can have any fraction of a unit). There are special math formulas that use integrals to calculate the exact maximum distance for any amount of fuel.

Real-World Uses

While these are math puzzles, they can be useful in real life! For example, during wartime, armies might need to move supplies like fuel across long distances in deserts. Understanding these problems can help them plan the most efficient way to do it. It's all about making the most of limited resources!