I don't know of a more suitable forum to post this so I'm going to start here with a rough draft.
Myth: If an airplane is placed on a conveyor belt that spins in the opposite direction from which the plane is trying to take off and the conveyor belt matches the planes speed, will the plane ever take off?
Authors answer? No, because the plane will not be moving relative to the air and air under the planes wings creates lift, not speed relative to the ground beneath it.
Commonly accepted answer? Yes, because when the plane is going 100mph and the belt is going 100mph the plains wheels will just be going 200mph and the plane will still move down the conveyor belt and take off.
I propose that mythbusters and the commonly accepted answer are wrong. The plane does not take off if you remain true to the authors intent. The fact that the author was knowledgeable enough to create a puzzle that's meant to teach the concepts of aerodynamics and lift tell me that he was smart enough to know that if the conveyor belt is matching the planes speeds relative to the ground around the conveyor belt it would still take off.
So then why did the author say No, it doesn't take off?: Because his intention was that the conveyor belt was matching the speed of the plane relative to itself. If the plane is going 100mph relative to the ground and the conveyor belt is moving 100mph against the plane then the plane is going 200mph relative to the conveyor belt and the belt is not matching its speed. If we assume that the conveyor belt is capable of matching the planes speed relative to itself, as opposed to the ground, then for every millimeter the plane moves forward the plane is now moving faster than the belt and the belts speed increases. The plane can never truly move forward unless the belt is failing to match its speed. There is no reason why a hypothetical belt that's given one purpose should be assumed to fail that purpose. In order for the wheels to turn they must overcome the friction being created by gravity pushing the plane down. The faster the wheels spin the more friction is created and the more force is required to increase the rate of spin. The amount of friction on the axle is limited only by the belts speed. With enough speed we create enough friction to match the engines ability to pull the plane forward. If the belt reacts fast enough it will accelerate to incredible speeds instantly in order to create enough friction on the wheels. Once again there is no reason to assume that the hypothetical belt is slow and fails to match the planes speed and fulfill its only role in this hypothetical scenario.
If the plane is sitting on a conveyor belt and the belt starts moving backwards before the plane starts then the plane moves backwards with it. If, as most people state, the wheels are irrelevant, why doesn't the plane remain still while the wheels simply turn? Because of Friction. In order for the plane to remain still the engines would have to create enough force to move the plane forward. Its true that the plane behaves very differently in this situation than a car which is often used as an example in this debate. The car is pushing off of the ground and overcoming friction and air speed. Since a car on a belt that's matching its speed isn't encountering any air speed its only overcoming the friction of the wheels. The wheel speed is practically limited by gearing but theoretically only limited by friction. The plane also has to overcome the friction on the wheels but since its pulling itself forward with its propellers as opposed to pushing off the ground it can overcome that friction much easier than the car. Just because its easier doesn't mean there is no limit. If the belt is moving backwards at 5 mph and the planes engines turn very slowly they could match the friction of the wheels and cause the plane to remain still indefinitely. Now if the belt increases its speed the plane will start moving backwards. It won't move backwards at a speed equal to the increase in speed of the belt but the friction will increase so without an increase in thrust the plane will start moving backwards ever so slightly. The amount of additional force the plane needs to produce to remain stationary or start moving forward is extremely small because the increase in friction on the wheels is almost insignificant but its still there. With a huge increase in belt speed you need a tiny increase in propeller speed to compensate for the friction on the axle. So if the speed of the belt has no upper bound but the force of the engine does then incredible speeds could produce incredible amounts of friction and overcome the power of the engine.
Some of you will say that no known conveyor belt can move that fast. Remember: the conveyor belt is not the point of the puzzle. The point is whether the plane takes off or not. The belt is just a hypothetical prop with one function: to match the planes speed. If you want to get really technical the belt would increase to such an incredible speed that the tires of the plane would blow out and the rims would make contact with the belt which is traveling over a thousand mph and the plane would flip. This still does not result in a take off. Once again, the tires are not the point though. The point is that if a belt could match the planes speed and never allow it to move forward, would the plane take off?
The author understood physics enough to try to teach others the lesson that speed relative to the air creates lift, not speed relative to the ground/belt beneath it. If the belt beneath the plane matches its speed relative to itself and prevents the plane from moving forward, will the plane take off? No, because there will be no air flowing under the planes wings. This is obviously the intent of the puzzle. The only way you can reach the conclusion that the plane takes off is if the belt is only supposed to match the planes speed relative to the ground(which obviously wasn't his intent given his answer and would make the entire puzzle pointless) or the belt is designed to match the planes speed relative to itself but it fails to do the one job it is given in the puzzle.
If you stay true to the authors intent and assume that all of the elements of the puzzle do what they're supposed to do the plane does not take off.