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Yes, this is my head canon.
Edit: Actually, maybe this should be the new "if dogs wore pants..." or "if you put an airplane on a treadmill...".
But the airplane has a verifiable answer. If the starting force is applied to the wheels, which push off the ground, then the airplane will remain stationary. But if it uses jet engines or props, which push off the air, then the plane will move forward.
Funnily enough, if the wind speed is the same and opposite as the speed of the props/engine, then the plane will remain stationary horizontally. That being said, it's wings would still generate lift, so the plane would start to raise while remaining in the same spot.
That's just the thing though. Everyone claims to have the objectively correct answer. But there's no broad agreement as to what it is.
Is that really even an "airplane," though? If that's true of the hypothetical in the question, I've never heard it mentioned, and it seems counter to the spirit of the question.
But the formulation I've heard of it says that "the treadmill goes at exactly the same speed as the wheels rotate." It follows that the plane cannot be moving either forward or backwards. (Assuming no slipping of the wheel against the treadmill, which I've heard explicitly stated in the question, If the plane is moving forward, then the wheels are necessarily rotating faster than the treadmill. If backwards, slower.)
The versions of the question I've heard specify there's no wind, so the only way I can see for that to be true is if the friction of the wheel against the axle produces sufficient backwards force on the airplane to keep it from moving forward.
But the versions of the question I've heard also specify no friction between the wheel and axle. So either there's something putting extra backwards force (I don't want to call it "drag" because I believe that term's usually reserved for air resistence) on the plane, which I definitely don't think is in the spirit of the question or the question itself is nonsensical.
So, the proper answer is neither "the plane takes off" nor "the plane does not take off." It's "either the question itself is self-contradictory and thus unanswerable or it leaves out important details, in which case if I'm allowed to make up more details, I can make the answer whatever I want." (If I decide the plane is tethered to something to keep it from moving forward, the answer is almost definitely "it doesn't take off", unless the propellers themselves produce enough wind to create lift. If I instead decide the treadmill itself is moving "forward" (from the plane's perspective) relative to the air, then the plane does achieve lift.)
Or it's possible you've heard a different version of the question than I have that, for instance, didn't specify that the treadmill goes exactly the same speed as the wheels. If that's the case, then yes, all else being equal, the plane would take off. If it instead didn't state no friction between the wheel and the axle, I think the most reasonable answer, though one still not in the spirit of the question, would be that the plane would not take off.
One more thing to mention. I know MythBusters tested this one and that the plane took off, but the wheels of the plane and the surface of the treadmill either didn't go the same speed or the wheels slipped against the treadmil. (As evidenced by the fact that the plane moved forward.)
The plane on a treadmill one has seemed straightforward to me. Lift is caused by the air moving over the wing and the obvious intent of the treadmill in the thought exercise is that the plane is only moving relative to the belt, so there's no opportunity for lift (no air flowing over the wings). Any other interpretation seems disingenuous.
The one that made me scratch my head for a while is the bicycle on the treadmill. People generally know that it's easy to stay balanced on a moving bicycle, but hard to balance on a stationary one, so is it easy or hard to balance on one that's on a treadmill?
A lot of people think that the stability of a moving bicycle is caused by the gyroscopic effect of the spinning wheels, but that turns out to contribute relatively little compared to the force of a body leaning one way or another.
What actually makes it easy to keep a moving bicycle going straight (or what contributes the most) is that the design has self-correcting steering. When you lean to the right, the shape of the forks causes the front wheel to point left, and vice versa, keeping you going relatively straight as long as you have some minimum amount of forward motion.
And since that forward motion is relative to the surface the tires are on, it works just fine on a treadmill. You can even find YouTube videos of people who ride their bikes on treadmills, and it's very evident.
Ooh yeah that's a good point. The question is ill-formulated, because the frictionless wheels/axle combo and the "treadmill moves as fast as the wheels" stipulation cannot both be true.
I guess the difference would be if they just mark up his face without using ink like if they were scars or something