Maximize the distance a Jeep can penetrate into the desert using a given quantity of fuel. The Jeep is allowed to go forward, unload some fuel, and then return to its base using the fuel remaining in its tank. At its base, it may refuel and set out again. When it reaches fuel it has previously stored, it may then use it to partially fill its tank. This problem is also called the exploration problem (Ball and Coxeter 1987).
Given 
(with 
)
drums of fuel at the edge of the desert and a Jeep capable of holding one drum (and
storing fuel in containers along the way), the maximum one-way distance which can
be traveled (assuming the Jeep travels one unit of distance per drum of fuel expended)
is
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(1)
|
 |  | ![f/(2n+1)+1/2[gamma+2ln2+psi_0(1/2+n)],](/images/equations/JeepProblem/Inline8.svg) |
(2)
|
where 
is the Euler-Mascheroni constant and
the polygamma
function.
For example, the farthest a Jeep with 
drum can travel is obviously 1 unit. However, with 
drums of gas, the maximum distance
is achieved by filling up the Jeep's tank with the first drum, traveling 1/3 of a
unit, storing 1/3 of a drum of fuel there, and then returning to base with the remaining
1/3 of a tank. At the base, the tank is filled with the second drum. The Jeep then
travels 1/3 of a unit (expending 1/3 of a drum of fuel), refills the tank using the
1/3 of a drum of fuel stored there, and continues an additional 1 unit of distance
on a full tank, giving a total distance of 4/3. The solutions for 
, 2, ... drums are 1, 4/3, 23/15, 176/105, 563/315, ...,
which can also be written as 
, where
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(3)
|
 |  |  |
(4)
|
