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We hope you are enjoying the puzzles we have been putting up. Though they may not be directly convertible to standardised test questions, they are great mathematical brain teasers!
Another variety of puzzles involves distributing fuel among vehicles to reach a destination. Let’s look at those today.
Puzzle 1: A military car carrying an important letter must cross a desert. There is no petrol station in the desert, and the car’s fuel tank is just enough to take it half way across. There are other cars with the same fuel capacity that can transfer their petrol to one another. There are no canisters or rope to tow the cars.
How can the letter be delivered?
Solution: A single car can only reach the mid point of the desert on its own tank. Since there are no canisters, it cannot carry extra fuel. Hence, it would need to be fuelled up by other cars travelling along with it.
We fill up 4 cars and get them to start crossing the desert together. By the time they cover a quarter of the desert, half of their fuel tanks would be empty. So we would have 4 cars with half tanks.
(0.5, 0.5, 0.5, 0.5)
We transfer the fuel from two cars into two other cars. Now it looks like this:
(1, 1, 0, 0)
The two cars start crossing the desert and cover another one fourth of it. Now both the cars have half tanks each and have reached the middle of the desert.
(0.5, 0.5, 0, 0)
Now one car transfers its entire fuel to the other car and that car has one full tank.
(1, 0, 0, 0)
That car carries the letter through the remaining half desert.
Here, we didn’t really care about the stalled cars in the middle of the desert since we are not required to bring them back. The only important thing is to get the letter across. Now, how do we handle a puzzle which asks us to get all the vehicles back too? Let’s look at a puzzle with those constraints.
Puzzle 2: A distant planet “X” has only one airport located at the planet’s North Pole. There are only 3 airplanes and lots of fuel at the airport. Each airplane has just enough fuel capacity to get to the South Pole (diametrically opposite). The airplanes can land anywhere and transfer their fuel to one another.
The mission is to fly around the globe above the South Pole with at least one airplane, and in the end, all the airplanes must return to the airport at the North Pole.
Solution: A plane with its tank full can only reach the South Pole, i.e. cover half the distance. We need it to take a full round from North Pole, to South pole and back to North Pole. Obviously, we will need more than one plane to fuel the plane which will take the full round.
Let’s divide the way from pole to pole into thirds (from the North Pole to the South Pole we have three thirds and from the South Pole to the North Pole, we have another three thirds).
First step – 2 aeroplanes fly to the first third. A third of their fuel is used. So the status of their fuel tank is this:
(2/3, 2/3)
One aeroplane fuels up the other plane and goes back to the airport. Now their tank status is this:
(3/3, 1/3)
Second step – 2 aeroplanes fly again from the airport to the first third, fuel up one aeroplane first aeroplane goes back to the airport. So the status of the two airplanes is this:
(3/3, 1/3)
Third step – Now there are two aeroplanes at one third with their tanks full. They fly to the two third point.
(2/3, 2/3)
One of them fuels up the second one (till its tank is full) and goes back to the first third, where it meets the third aeroplane which comes from the airport to support it with fuel so that they both can return to the airport.
In the meantime, the aeroplane at the second third, with a full tank, flies as far as it can (so over the South Pole to the last third before the airport).
Fourth step – One of the two aeroplanes from the airport goes to the first third (the opposite direction as before), shares its 1/3 fuel and both aeroplanes safely land back at the airport.
And that is how one full round of the globe is made and all aeroplanes arrive back safely!
