The connundrum above has been doing the rounds for a while now, most recently on a message board that appears to be related to the New York Times, so I decided to have a go at answering it myself. Here is my own, definitive answer, in vague non-scientific terms.
With very little thought, what the question appears to be asking is "could a plane take off without it actually moving relative to the ground if its wheels were spinning fast enough?" Of course, anybody familiar with the way flight works will tell you that the movement of the wheels is for all intents and purposes irrelevant, and that it's the plane's movement relative to the air that determines whether or not take-off is possible. If we were to assume a very strong and consistent headwind, so the plane had a high airspeed but zero groundspeed, then it's certainly possible in theory that a plane could take off without moving relative to the ground. Science museums, and museums specialising in flight often demonstrate this principle using wind tunnels. In practice however it's very unlikely and would be reliant on additional circumstances not specified by the question. Indeed, if the headwind was sufficient that the plane could take off with zero groundspeed, the wheels would not need to turn and the runway would not need to be a treadmill.
Also not specified by the question is whether or not the plane is equipped with VTOL (vertical take off and landing) capabilities, like a Harrier jumpjet. Such aeroplanes specialise in taking off with both zero ground AND airspeed.
However, upon closer inspection, the question is not in fact asking that at all. Whether or not the plane could take off is something of a red herring to disguise the flawed reasoning that makes the question itself a paradox. The paradox lies in the use of a false relation between cause and effect. Let's break it down into the order of causes.
- The plane is pushed through the air by the power of the engines, which act like rockets sticking out the back. This causes the plane to also move relative to the ground.
- The movement of the plane relative to the ground causes the free-spinning wheels to rotate, like pushing a supermarket trolly.
- According to the original premise, the treadmill counters the rotation of the wheels EXACTLY, cancelling their rotation and causing the plane not to move.
- The plane is not moving, so the wheels do not turn.
- The wheels are not turning, so the treadmill does not move to counter them.
- The treadmill does not move, so there's nothing stopping the plane from being pushed through the air by the XX kilograms of force produced by its engines.
- We find outselves back at (1) - the plane is pushed through the air by the power of the engines, causing the plane to also move relative to the ground.
Clearly this cannot be right, and the flaw in the reasoning can be traced to part 3, where it states the treadmill EXACTLY counters the rotation of the wheels. This cannot work in theory or reality.You have to think about it in terms of cause and effect. The wheels rotate, so the treadmill matches them. The rotation of the wheels is the cause. The movement of the treadmill matching them is the effect. The wheels turn first. The treadmill moves to match them second. The wheels have to be turning BEFORE the treadmill attempts to catch up. And of course, the treadmill's attempt to catch up would itself cause the wheels to rotate faster, which cause the treadmill to move faster, which cause the wheels to rotate faster, which cause the treadmill to move faster, so on and so forth . It's the classic scenario of a dog chasing its own tail, or Homer Simpson looking around to see what Bart had written on the back of his head. All the time the treadmill attempts to copy the wheels' rotation, but in doing so affects the wheels' rotation, there will remain a difference between the speed the wheels rotate and the speed the treadmill moves. That speed difference matches the speed of the plane as it moves down the runway.
Of course, the wheels may be DIRECTLY attached to the treadmill by a chain and sprocket with zero slack, but then it wouldn't be a plane on a treadmill anymore, so much as a plane tethered to the spot. Only enough engine power would be required to break the chain before the plane is off on its merry way.
Now in reality, were such an experiment to be tried, the engine of the treadmill would probably explode, the wheel bearings would probably melt, the tires would probably burst, and the plane would probably end up sliding down the runway on its belly before exploding, killing everyone. Put another way, it WOULD find a way to move under the power of the engines. Whether it is able to take off or not under those circumstances would involve a number of factors that are not specified by the question, but as such circumstances would not match the initial premise, the answer would be moot. The actual circumstances of the question assume we are in a fantasy land with laws of physics quite quite different from that of the world we actually inhabit, so it would be impossible to answer without understanding the other physical laws of the universe. Does this crazy world where cause and effect happen together even have gravity? Is up really down? We just don't know, and I'm not hanging around to find out.
So if you want a simple answer, ask a science fiction writer.