Airplane riddle: How many airplanes needed for one plane to fly around the world?
Airplane Riddle of Flying Around the World: One airport can be used by airplanes. An airplane carries enough fuel to fly halfway across the globe...
Several airplanes are available for instant flight on a solitary airport. Airplanes take flight from and land only on this solitary airport. Landing time is zero.
Each plane can hold fuel to take it halfway across the world. You may refuel a plane from a second plane in flight as much as needed.
Refueling in flight or at ground is instant. Turning back of a plane in flight takes no time.
Source of fuel only on the island.
Dream to fulfill: One plane is to fly around the world in a great circle.
Find the least number of planes to fulfill the dream. All planes used must return to base.
Fuel use must be least, with no wastage.
Time to solve: 30 minutes.
Try it before going through the solution.
Solution stage 1: Riddle of Airplane to fly round the world: Problem definition
In a project of problem solving you will
- Make an exhaustive list of stated facts and information UNSTATED in the problem, and,
- Clearly describe the requirements to solve the problem.
For this problem the list of facts stated is,
- Airport for starting flight and landing is ONLY ONE.
- You can get enough number of planes and amount of fuel in the single airport base for successful completion of the project.
- Each plane can hold fuel to fly HALF the distance around the world.
- A plane can take flight, land, turn around or refuel mid-air, taking zero time.
Dream to fulfill: Fly a plane around the world using least number of planes with no fuel wastage, least fuel use and all planes returning to base.
This is the problem definition at this stage.
Solution stage 2: Riddle of Airplane to fly round the world: Trial trip with 2 planes
To know the difficulties, make a virtual TRIAL TRIP using two planes first.
The idea is to know,
How much fuel Airplane 2 can give to Airplane 1 for a mid-air refueling, and,
The distance the two airplanes can fly together before Airplane 2 must refuel Airplane 1 and start its return flight to base.
Immediate outcome is,
Conclusion 1: To return to base after refueling Airplane 1, Airplane 2 can give Airplane 1 one-third of its full-tank fuel when the two are at one-sixth around the world's distance from base. Both planes burned one-third their full-tank fuel to reach this one-sixth point with two-third of full-tank fuel left in each tank.
With the extra one-third tank of fuel, Plane 1 can travel an extra one-sixth around the world distance. It is one-half plus one-sixth, equal to two-third around the world's distance.
One-third distance will stay uncovered for the dream to be fulfilled with only two planes used.
Conclusion 2: You must use at least three planes to make one plane fly around the world satisfying the conditions.
Question is: How would it be possible with three planes?
Solution stage 3: Riddle of Airplane to fly round the world: Crossing barriers
Meet the primary barrier when using 3 planes and create the inventive solution to get over it. Face the second barrier.
Start analysis of three planes flying together from their base.
The three airplanes can fly up to the one-sixth point together, burning one-third of full-tank capacity each. At the one-sixth point, Airplane 2 refuels Airplane 1 with one-third tank fuel and returns to base.
But what about the third plane?
- At one-sixth point, it CANNOT transfer ANY FUEL to Plane 1 as the tank of Plane 1 is full.
- It cannot fly further on, as in that case, its fuel would run out before returning to base.
It's the first real barrier you face. If you increase the number of planes, the barrier remains unbroken.
Thinking this way won't do.
Solving a challenging riddle needs innovative thinking. Thinking with reason helps to find an innovative solution.
You continue to reason based on the experience until now and make conclusions.
Conclusion 3: In case of 3 planes used, the two other planes CANNOT REFUEL THE CIRCUMNAVIGATING PLANE 1 AT THE SAME POINT OF FLIGHT.
The crucial conclusions follow,
Conclusion 4: Plane 2 and Plane 3 must refuel Plane 1 at different points of flight.
Conclusion 5: To fulfill condition 4, after the three planes fly together up to a point, the best course of action would be for Plane 3 to refuel both Plane 1 and Plane 2 by equal amounts leaving just enough fuel for itself to fly back to base.
Conclusion 6: Plane 3 will return to base safe and Plane 1 and Plane 2 will fly onward until a point where Plane 2 now refuels Plane 1, leaving enough fuel to return to the base.
The distance from the base at which Plane 3 must refuel the two other planes
It must be the one-fourth tank capacity distance when the three planes have one-fourth tank empty, Plane 3 refuels both Plane 1 and Plane 2 sharing half tank fuel in equal amounts of one-fourth tank fuel for each, leaving one-fourth tank fuel for itself to return to base.
One-fourth tank capacity distance is the one-eighth distance around the world.
Collecting the new effective ideas together,
Conclusion 7: Three planes will fly together up to one-eighth distance point. Plane 3 will refuel both Plane 1 and Plane 2 with one-fourth tank fuel, thus filling up their tanks to the full and leaving one-fourth tank of fuel for itself to return to base.
Plane 1 and Plane 2 can fly together for another one-eighth distance. It is the one-fourth point, when Plane 2 will refuel Plane 1 with one-fourth tank of fuel it can spare, leaving half-tank fuel for itself to fly back to the base safe.
Overall, Plane 1 gets one-fourth plus one-fourth, one-half tank of fuel from the two planes.
So even with this ingenious plan,
Conclusion 8: With three planes used for refueling and round the world trip, Plane 1 can fly on for an extra one-fourth around the world distance. It's a total distance of three-fourths of around the world distance. And it can fly no further because its tank will become empty at the three-fourth point.
This is the most that you can achieve when you use three planes.
Can we use four planes to push the circumnavigating plane to go a little further?
When you explore this possibility, you will soon know the futility of this approach.
This must be the second barrier. Have to think new. Look for possibilities you might have missed.
Solution stage 4: Riddle of Airplane to fly round the world: The world is round
Nothing new to discover about number of planes. Your goal is to fulfill the dream with 3 airplanes and you know two airplanes flew together forward while the third returned to base. And a little later, the second airplane also landed while the first plane flew onward.
You don’t have a way to refuel the first plane though you have two planes idle at the airport—the first plane is too far away from the airport.
Are we missing something here? What about the direction of flight and direction of refueling. This is the point when you realize, the first plane is actually approaching the single airport right from the moment it crossed the halfway mark. After all, the world is round and refueling must be feasible from the direction opposite to the direction of flight of the first plane.
The second breakthrough results from persistent questioning the taken-for-granted assumption about the key entity or action—an integral part of the powerful property change analysis technique.
It strikes you in a moment of revelation,
It is possible to refuel Plane 1 from the opposite direction, as the world is round.
A plane can start from the single airport BUT IN OPPOSITE DIRECTION to meet the Plane 1 at the one-fourth point in the opposite direction.
Now you know you have hit upon the core idea for the ultimate solution. It only remains to think of the further detailed steps.
The goal in this stage 4 of the solution is to,
Refuel the oncoming Plane 1 flying in the forward direction at the THREE-FOURTH POINT IN THE FORWARD DIRECTION, THAT IS THE ONE-FOURTH POINT IN THE OPPOSITE DIRECTION so that it can complete its around the world flight.
From earlier experience of forward direction refueling, you already know that refueling Plane 1 at one-fourth distance needs two more planes.
But refueling will be different in the opposite direction. In the opposite direction, two planes can start together and not three.
Knowing how to use two extra planes for refueling Plane 1 with an extra half-tank fuel, steps for a successful around the world trip by Plane 1 fall into place.
Forward direction flights and refueling
- Plane 1, Plane 2 and Plane 3 take flight together in the forward direction.
- At one-eighth point in FORWARD DIRECTION, Plane 3 refuels Plane 1 and Plane 2 with one-fourth tank fuel to make tanks of Plane 1 and Plane 2 full.
- With one-fourth tank fuel left, Plane 3 will fly back to the base safe while Plane 1 and Plane 2 will fly onward.
- At one-eighth distance more, at one-fourth around the world distance, Plane 2 refuels Plane 1 with one-fourth tank fuel more, again filling the tank of Plane 1 to the full and leaving half-tank fuel for itself to return to base.
- With full tank fuel at one-fourth point, Plane 1 could fly on for half the entire distance more, up to three-fourth around the world distance when its tank will become empty. To this point, it is FORWARD DIRECTION FLIGHTS AND REFUELING ACTIONS. Now it will be opposite direction flights and refueling actions.
Timings, Forward and Opposite direction flights and refueling
- Plane 3 refuels Plane 1 and Plane 2 at one-eighth point, and lands at base when Plane 1 and Plane 2 reach the one-fourth point in the forward direction.
- Plane 2 refuels Plane 1 in forward direction flight at one-fourth point and reaches back to base as Plane 1 reaches the halfway point.
- Plane 3 starts with full tank fuel from the OPPOSITE DIRECTION just as Plane 1 reaches the halfway point. Both Plane 1 and Plane 3 fly towards each other at constant speed for one-fourth around the world distance more and meet at three-fourth forward direction point. This is the one-fourth opposite direction point.
- Plane 3 refuels Plane 1 with one-fourth tank fuel that leaves one-fourth tank of fuel in both planes. Plane 3 turns back and, together with Plane 1, flies towards the base.
- Just at the time Plane 1 and Plane 3 meet at opposite direction one-fourth point and all refueling and other events take place, Plane 2 starts from base in an opposite direction flight towards the two planes with full tank fuel.
- Plane 3 refuels Plane 1 with one-fourth tank fuel. Both fly on one-eighth distance more when their tanks become empty. But just when this happens, Plane 2 catches up with them and refuels both with one-fourth tank fuel. Plane 2 turns back after refueling. Plane 1, Plane 2 and Plane 3 each with one-fourth tank of fuel, fly back to the base.
- Success! Plane 1 has flown around the world with the least number of planes used with full safety in the entire project. Least fuel used with no wastage.
- Plane 2 and Plane 3 used twice to refuel Plane 1, once in the forward direction and second time in the opposite direction.
This is a two barrier riddle where you have to find inventive solutions to break the two barriers.
Other way round inventive principle used for breaking through the second barrier.
A schematic of the successful flights is shown.
To solve the two-barrier riddle, you needed innovative ideas to break through the barriers.
Other way round inventive principle and property change analysis technique used for breaking through the second barrier.
Know how to solve difficult problems easily without wasting time on random attempts
Our ebook on puzzle solutions by innovative methods will show you just that.
Puzzles for Adults: 50 Brain Teasers with Step-by-Step Solutions: Boost Your Power of Problem Solving
Second book on Innovative Solutions to Matchstick Puzzles from Suresolv
BUY the paperback here.
Puzzles you may enjoy
Easy to hard brain teasers with systematic solutions
This will always be the most up-to-date full list with the brain teasers classified into categories that can be browsed separately.
You may also click on the category term link below to enjoy the brain teasers that are classified in the present category.
For example, if the category term link shown below is "Riddle", click on it to go through all the Riddles.