Sorry, my sarcasm detector overloaded somewhere around page 16 so I'll just take your post at face value.
piku wrote:The plane... sitting on this mystic conveyor... needs to move forwards in order to take off.
Yes, relatively to the air to be precise.
To do this it has to roll forwards using its wheels
Not necessarily, but no wind was mentioned in the original riddle so we'll assume the air stands (more or less) still relatively to the ground. Yes, the plane (including its wheels) has to move forwards relatively to the ground.
(indirectly powered by the engines).
Huh? The wheels are pushed forward by the plane, which is pushed by its engines so yes, in a sense the wheels are powered by the engines.
Like the original evil post says, this mystic conveyor matches the speed of the wheels exactly.
Which is meaningless unless you define "the speed of the wheels". Fortunately, as we will see the precise speed of the conveyor doesn't really matter.
So the pilot turns on the engines, takes off the brakes on the wheels and... the mystic conveyor starts up and immediately matches the wheel speed exactly down to the macro nano pico metre per quantibillisecond.
Whatever that speed is, but fine.
Bearings, friction and things like that aren't mentioned, so we can't reason about them. They're beyond the scope of this thought experiement - up there with what the pilot had for lunch and the number of terrorists on board and immigrants in the baggage hold.
Fine, they don't really matter. We introduced them because, if there were a lot of friction, the conveyor belt could possibly prevent the plane from taking off. Except, as Kalmar pointed out, that if a plane lost enough energy to friction in its wheel to prevent it from taking off, then horrible things would happen to said wheels. Now, if we are to assume that there is no friction (as you seem to imply), the question is very easy: whatever the belt's speed, it will just make the wheels spin madly, but not slow down the plane at all.
Therefore plane no worky...
Eh?