A plane is standing on a treadmill. As the plane moves forward, the treadmill moves in the opposite direction exactly countering the rotation of the wheels. Can the plane take off?
Of course it can. If you disagree with that then read on.
Most people tend to compare the airplane on the treadmill to a car on a dynamometer but that is a false comparison because the motive force of the airplane does not go through the wheels as it does in a car.
Let's go inside the airport, shall we? Imagine you are walking through the airport at a speed of two mph pulling your wheeled suitcase behind you. Now you walk onto a moving sidewalk going the wrong way. The sidewalk is moving toward you also at a speed of two mph. Even though you are walking, you aren't going anywhere, right? That's because the motive force (your feet) is directly counteracted by the treadmill. Also note at this time that the wheels on your suitcase are rotating at two mph. This experiment represents the car on the dynamometer.
Now exit the moving sidewalk and let's try the experiment again, only this time it'll be analogous to the airplane. I want you to stand alongside the moving sidewalk while reaching over the railing and holding your suitcase on the treadmill. You are not moving; the suitcase is not moving but the wheels are turning at two mph, right? Now start walking at two mph while pulling your suitcase. Now how fast are the wheels turning? That's a trick question! You won't be able to move at all! Your feet will be held in place by the mysterious force of this magical treadmill!
I'm sorry - that was snotty. You will walk at two mph, your suitcase will move at two mph and the wheels on your suitcase will turn at four mph. The motive force (your feet) are not stopped or hindered by the treadmill because they aren't in contact with it. Want to go faster? Okay, start walking at four mph and we'll crank up the treadmill to match your speed. Did your legs stop working? No. How about at ten mph? No.
In other words, we're on our takeoff roll.
Here's another quick way to look at it: Imagine a very large airplane that has three main trucks and a nose wheel. But our treadmill isn't wide enough to accommodate the entire plane and only the nose wheel and the center truck are on the treadmill. Can you see that the plane will take off normally with the only difference being that the wheels on the treadmill will be spinning twice as fast as the wheels on the tarmac?
Frankly, I find this puzzle to be rather boring. What I find fascinating, however, is the mental scotoma that many people have about this problem. These people say the plane won't take off, believing that the wheels will just spin in place. They know that the treadmill is only operating in reaction to the spinning wheels. And they know that the wheels on a plane aren't driven. They have seen airplanes before and they know that the only time the wheels spin on an airplane is when the plane is moving. Isn't the contradiction obvious? If the wheels are spinning it's only because they are being compelled by the linear movement of the axles. If the axles are moving that means the plane is moving. Under the terms of the stated problem, it is not possible for the wheels to be spinning but the plane to be stationary.
If you still don't see this then you may be in need of a cranial irrigation. Stick your head in the toilet and flush. While you're doing that, the plane on the treadmill will be taking off.