Airplane riddle: How to make one plane fly around the world using least number of planes?
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...
The riddle
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: 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 midair, 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: Trial run 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 midair 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 onethird of its fulltank fuel when the two are at onesixth around the world's distance from base. Both planes burned onethird their fulltank fuel to reach this onesixth point with twothird of fulltank fuel left in each tank.
With the extra onethird tank of fuel, Plane 1 can travel an extra onesixth around the world distance. It is onehalf plus onesixth, equal to twothird around the world's distance.
Onethird distance will stay uncovered for the dream to be fulfilled with only two planes used.
What you have learned from the trial run with 2 planes
 You cannot cover the full aroundtheworld distance with 2 planes.
So it is clear,
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: Use the third plane, face the barriers and find ways to get over them
Meet the primary barrier when using 3 planes. Discover the inventive solution to get over it and face the second barrier.
Start analysis of three planes flying together from their base.
The three airplanes can fly up to the onesixth point together, burning onethird of fulltank capacity each. At the onesixth point, Airplane 2 refuels Airplane 1 with onethird tank fuel and returns to base.
But what about the third plane?
 At onesixth 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.
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 important points:
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 onefourth tank capacity distance when the three planes have onefourth tank empty, Plane 3 refuels both Plane 1 and Plane 2 sharing half tank fuel in equal amounts of onefourth tank fuel for each, leaving onefourth tank fuel for itself to return to base.
Onefourth tank capacity distance is the oneeighth distance around the world.
Collecting the new effective ideas together,
Conclusion 7: Three planes will fly together up to oneeighth distance point. Plane 3 will refuel both Plane 1 and Plane 2 with onefourth tank fuel, thus filling up their tanks to the full and leaving onefourth tank of fuel for itself to return to base.
Plane 1 and Plane 2 can fly together for another oneeighth distance. It is the onefourth point, when Plane 2 will refuel Plane 1 with onefourth tank of fuel it can spare, leaving halftank fuel for itself to fly back to the base safe.
Overall, Plane 1 gets onefourth plus onefourth, onehalf 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 onefourth around the world distance. It's a total distance of threefourths of around the world distance. And it can fly no further because its tank will become empty at the threefourth 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: Discovering the key idea and the breakthrough—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. Outofthebox thinking is needed to get over the tough barrier.
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 takenforgranted assumptions about the key entity or action—an integral part of the powerful property change analysis technique—we have assumed all along all the planes will fly only in one direction.
This also is a direct example of use of the powerful Other Way Round inventive principle.
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 onefourth 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 THREEFOURTH POINT IN THE FORWARD DIRECTION, THAT IS THE ONEFOURTH 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 onefourth 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 halftank 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 oneeighth point in FORWARD DIRECTION, Plane 3 refuels Plane 1 and Plane 2 with onefourth tank fuel to make tanks of Plane 1 and Plane 2 full.
 With onefourth tank fuel left, Plane 3 will fly back to the base safe while Plane 1 and Plane 2 will fly onward.
 At oneeighth distance more, at onefourth around the world distance, Plane 2 refuels Plane 1 with onefourth tank fuel more, again filling the tank of Plane 1 to the full and leaving halftank fuel for itself to return to base.
 With full tank fuel at onefourth point, Plane 1 could fly on for half the entire distance more, up to threefourth 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 oneeighth point, and lands at base when Plane 1 and Plane 2 reach the onefourth point in the forward direction.
 Plane 2 refuels Plane 1 in forward direction flight at onefourth 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 onefourth around the world distance more and meet at threefourth forward direction point. This is the onefourth opposite direction point.
 Plane 3 refuels Plane 1 with onefourth tank fuel that leaves onefourth 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 onefourth 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 onefourth tank fuel. Both fly on oneeighth distance more when their tanks become empty. But just when this happens, Plane 2 catches up with them and refuels both with onefourth tank fuel. Plane 2 turns back after refueling. Plane 1, Plane 2 and Plane 3 each with onefourth 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 twobarrier 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.
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