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Jeep problem

Jeep problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Jeep problem rather than just read about it. In short: The jeep problem, desert crossing problem or exploration problem is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The jeep can only carry a fixed and limited amount of fuel, but it can leave fuel and collect fuel at fuel dumps anywhere in the desert.

Jeep problem — main illustration
Jeep problem — illustration

Key takeaways

  • Jeep problem belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Jeep problem to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Jeep problem from memory before moving on to harder problems.

Reference excerpt

The jeep problem, desert crossing problem or exploration problem is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The jeep can only carry a fixed and limited amount of fuel, but it can leave fuel and collect fuel at fuel dumps anywhere in the desert. The problem first appeared in the 9th-century collection Propositiones ad Acuendos Juvenes (Problems to Sharpen the Young), attributed to Alcuin, with the puzzle being about a travelling camel eating grain. The De viribus quantitatis (c. 1500) of Luca Pacioli also discusses the problem. A modern treatment was given by N. J. Fine in 1947.

Problem

There are n units of fuel stored at a fixed base. The jeep can carry at most 1 unit of fuel at any time, and can travel 1 unit of distance on 1 unit of fuel (the jeep's fuel consumption is assumed to be constant). At any point in a trip the jeep may leave any amount of fuel that it is carrying at a fuel dump, or may collect any amount of fuel that was left at a fuel dump on a previous trip, as long as its fuel load never exceeds 1 unit. There are two variants of the problem:

Exploring the desert – the jeep must return to the base at the end of every trip. Crossing the desert – the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel. In either case the objective is to maximize the distance traveled by the jeep on its final trip. Alternatively, the objective may be to find the least amount of fuel required to produce a final trip of a given distance.

Variations In the classic problem the fuel in the jeep and at fuel dumps is treated as a continuous quantity. More complex variations on the problem have been proposed in which the fuel can only be left or collected in discrete amounts.

Solution

A strategy that maximizes the distance traveled on the final trip for the "exploring the desert" variant is as follows:

The jeep makes n trips. On each trip it starts from base with 1 unit of fuel. On the first trip the jeep travels a distance of 1/(2n) units and leaves (n − 1)/n units of fuel at a fuel dump. The jeep still has 1/(2n) units of fuel – just enough to return to base. On each of the subsequent n − 1 trips the jeep collects 1/(2n) units of fuel from this first fuel dump on the way out, so that it leaves the fuel dump carrying 1 unit of fuel. It also collects 1/(2n) units of fuel from this first fuel dump on the way back, which is just enough fuel to return to base. On the second trip the jeep travels to the first fuel dump and refuels. It then travels a distance of 1/(2n − 2) units and leaves (n − 2)/(n − 1) units of fuel at a second fuel dump. The jeep still has 1/(2n − 2) units of fuel, which is just enough to return to the first fuel dump. Here it collects 1/(2n) units of fuel, which is just enough fuel to return to base. On each of the subsequent n − 2 trips the jeep collects 1/(2n − 2) units of fuel from this second fuel dump on the way out, so that it leaves the fuel dump carrying 1 unit of fuel. It also collects 1/(2n − 2) units of fuel from the second fuel dump on the way back, which is just enough fuel to return to the first fuel dump. The jeep continues in this way, so that on trip k it establishes a new kth fuel dump at a distance of 1/(2n − 2k + 2) units from the previous fuel dump and leaves (n − k)/(n − k + 1) units of fuel there. On each of the subsequent n − k trips it collects 1/(2n − 2k + 2) units of fuel from the kth dump on its way out and another 1/(2n − 2k + 2) units of fuel on its way back. When the jeep starts its final trip, there are n − 1 fuel dumps. The farthest contains 1/2 of a unit of fuel, the next farthest contain 1/3 of a unit of fuel, and so on, and the nearest fuel dump has just 1/n units of fuel left. Together with 1 unit of fuel with which it starts from base, this means that the jeep can travel a total round trip distance of

1 + 1 2 + 1 3 + ⋯ + 1 n = ∑ k = 1 n 1 k ≡ 2 × e x p l o r e ( n ) {\displaystyle 1+{\frac {1}{2}}+{\frac {1}{3}}+\cdots +{\frac {1}{n}}=\sum _{k=1}^{n}{\frac {1}{k}}\equiv 2\times \mathrm {explore} (n)}

units on its final trip (the maximum distance traveled into the desert is half of this). It collects half of the remaining fuel at each dump on the way out, which fills its tank. After leaving the farthest fuel dump it travels 1/2 a unit further into the desert and then returns to the farthest fuel dump. It collects the remaining fuel from each fuel dump on the way back, which is just enough to reach the next fuel dump or, in the final step, to return to base.

The distance travelled on the last trip is the nth harmonic number, Hn. As the harmonic numbers are unbounded, it is possible to exceed any given distance on the final trip, as along as sufficient fuel is available at the base. However, the amount of fuel required and the number of fuel dumps both increase exponentially with the distance to be traveled. The "crossing the desert" variant can be solved with a similar strategy, except that there is now no requirement to collect fuel on the way back on the final trip. So on trip k the jeep establishes a new kth fuel dump at a distance of 1/(2n − 2k + 1) units from the previous fuel dump and leaves (2n − 2k − 1)/(2n − 2k + 1) units of fuel there. On each of the next n − k − 1 trips it collects 1/(2n − 2k + 1) units of fuel from the kth dump on its way out and another 1/(2n − 2k + 1) units of fuel on its way back. Now when the jeep starts its final trip, there are n − 1 fuel dumps. The farthest contains 1/3 of a unit of fuel, the next farthest contain 1/5 of a unit of fuel, and so on, and the nearest fuel dump has just 1/(2n − 1) units of fuel left. Together with 1 unit of fuel with which it starts from base, this means that the jeep can travel a total distance of

… excerpt ends here. Continue reading the full article.

Illustrations

Jeep problem: Plot of amount of fuel f vs distance from origin d for exploring (1–3) and crossing (I–III) versions of the jeep problem for three units of fuel – coloured arrows denote depots, diagonal segments denote travel and vertical segments denote fuel transfer
Plot of amount of fuel f vs distance from origin d for exploring (1–3) and crossing (I–III) versions of the jeep problem for three units of fuel – coloured arrows denote depots, diagonal segments denote travel and vertical segments denote fuel transfer
Jeep problem: Plot to scale of the Exploring (top) and Crossing (bottom) versions of the jeep problem for three units of fuel. The horizontal axis denotes distance and vertical axis denotes time. Vertical coloured line segments denote stashing fuel and horizontal ones denote travel using withdrawn fuel. Coloured numbers denote units of fuel stashed at that moment.
Plot to scale of the Exploring (top) and Crossing (bottom) versions of the jeep problem for three units of fuel. The horizontal axis denotes distance and vertical axis denotes time. Vertical coloured line segments denote stashing fuel and horizontal ones denote travel using withdrawn fuel. Coloured numbers denote units of fuel stashed at that moment.
Jeep problem: Solution to "exploring the desert" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip.
Solution to "exploring the desert" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip.
Jeep problem: Solution to "crossing the desert" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip, at turnaround point on first two trips, and at end of final trip.
Solution to "crossing the desert" variant for n = 3, showing fuel contents of jeep and fuel dumps at start of each trip, at turnaround point on first two trips, and at end of final trip.
Jeep problem: In Operation Black Buck One, the attacking Vulcan was refuelled seven times on the outward journey and once on the return journey. Grey lines indicate reserve aircraft to replace casualties.
In Operation Black Buck One, the attacking Vulcan was refuelled seven times on the outward journey and once on the return journey. Grey lines indicate reserve aircraft to replace casualties.

Worked examples

Example 1 — a first encounter with Jeep problem

Start with the simplest possible case. Write down what Jeep problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Jeep problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Jeep problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Jeep problem

In research
Jeep problem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Jeep problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Jeep problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Mathematical optimization, Recreational mathematics, so understanding it makes those chapters shorter.
In everyday life
Look for Jeep problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Jeep problem in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Jeep problem means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Jeep problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jeep problem in simple terms?

The jeep problem, desert crossing problem or exploration problem is a mathematics problem in which a jeep must maximize the distance it can travel into a desert with a given quantity of fuel. The jeep can only carry a fixed and limited amount of fuel, but it can leave fuel and collect fuel at fuel…

Why does Jeep problem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Jeep problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Jeep problem.

Tags

  • Mathematical optimization
  • Recreational mathematics

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