The answers!

[theweaselking]

I posed three questions yesterday.

The answers are, in order:

1. Yes, the plane will take off. There is friction on the wheels, but the wheels spin freely and do not drive the plane, so as you accelerate the treadmill against the growing thrust of the jet engine, the wheels will simply start to spin faster.

The horizontal force that the treadmill applies to the plane through the wheels, while present, is trivial compared to the thrust from the jet engines. Even at ludicrous speed, the plane's wheels will spin much faster without allowing much more impact to the plane.


2. You should ALWAYS switch. This is the Monty Hall problem - and because you've got the initial choice before he removes a door, it does NOT reduce the problem to a coin flip between two doors. The entire setup is a deceitful way of asking the question "should you take the door you picked originally, or should you take the best of all other doors combined". And the answer to that, clearly, is "best of all other doors" - Monty just opens all the other doors EXCEPT the one he tells you is the best of all other doors.


3. Well, I fucked up the question. While "there could have been two lions" does technically meet the requirements, a more correct phrasing of the question would be "four of the five doors have nothing behind them. One door has a single totally unexpected lion, and nothing else."

And, given that:
A) There is only one lion.
and
B) The lion will be totally unexpected
... there's nothing wrong with his logic. He didn't make a mistake.

His logic is impeccable, and 100% correct. Since the lion has to be "unexpected", it can't be behind door #5, which means it can't be behind door #4, which means it can't be behind door #3, etc. His logic correctly deduces that there cannot be a lion who is unexpected - and because of his perfect logic, the lion *is* totally unexpected.

Comments

[publius1.livejournal.com]

Hey... that was my original guess! WOOT.

[cleodhna.livejournal.com]

Just shows how devious you have to be if you cannot lie.

Unexpected lion, redux

[jaffa-tamarin.livejournal.com]

The man's logic is not quite correct. Having established that A & B -> !A, he knows only that one of the propositions A or B is false, not that it must be A that is false.

Re: Unexpected lion, redux

[theweaselking.livejournal.com]

The king *cannot* lie. This is a law of reality, that he knows absolutely to be true. Should this not be true, the universe would vanish.

That kind of thing happens a lot in the setup for logic puzzles.

Re: Unexpected lion, redux

[jaffa-tamarin.livejournal.com]

If you have contradictory propositions you can prove anything.

If 'the king cannot lie' is a physical law, he doesn't need to bother with all the extra doors. He can have one door, tell the hero there's an unexpected lion behind it, and the universe will adjust by causing the hero to have a sudden attack of amnesia just as he's opening the door, or something.

Re: Unexpected lion, redux

[jagash.livejournal.com]

Actually that one would be easy. If there is a door then either side could be "behind" the door depending on your perspective. I suspect the hero wouldn't expect to be attacked by a lion from behind.

[moosl.livejournal.com]

I have some doubts about the first question. What lifts the plane off the ground is the negative pressure created by the airfoil of the wing. I don't see that the frictionless surface treadmill will create that.

[maskedretriever.livejournal.com]

The frictionless surface merely fails to slow the plane down.

This had me flummoxed for a moment because I was thinking that if the treadmill always kept pace with any thrust the airplane achieved, it'd keep that sucker grounded. Which it WOULD.

See, the treadmill doesn't go fast enough to cancel the thrust of the engine. It merely goes fast enough to keep the wheels from providing any traction. This question strikes me as a little cheap because it's not made clear that the treadmill doesn't move fast enough to prevent the plane from moving forward. If THAT were the case the plane would never have the remotest chance of taking off, no matter how furiously the engines pulled air through.

Replacing the treadmill with a frictionless surface (logically equivalent so long as the treadmill only stops the wheels from getting traction) makes the question easier, since it's clear enough that if you started a plane off on an ice flow it'd eventually push hard enough on the air to accelerate.

[corruptedjasper.livejournal.com]

After all, planes actually do take off from ice floes in Antarctica on a regular basis. But yes, the problem is in how the question is phrased. If it'd said "the treadmill is moving as fast as the plane would be moving over the runway in a normal takeoff at that time, but in the opposite direction, sure, yeah, chocks away. When it says "however fast you want", there is the possibility of taking it up to speeds fast enough to provide counterthrust through the rolling friction.

It is easily visualisable that rolling friction provides such a counterthrust -- if it didn't, we'd keep going forever on a bicycle in vacuum.

[siouxsyn.livejournal.com]

I took the question to mean that the treadmill prevents the plane from forward movement.

I still don't see how it takes off. The question then becomes can a plane take off if it has no wheels? Which it can if the surface is frictionless, but the wheel/treadmill set-up isn't.

Gah.

[theweaselking.livejournal.com]

I took the question to mean that the treadmill prevents the plane from forward movement.

The treadmill *tries* to prevent forward movement, but can't. That's the point - the wheels spin freely. They're not attached to the drive train, and the plane doesn't push off the ground to gain speed. It pushes off the air, and whatever it's on only needs to have a low enough coefficient of friction that it doesn't stick to the ground.

The wheel/treadmill is *nearly* frictionless.

[theweaselking.livejournal.com]

It won't create that. However, the frictionless surface is not important to the creation of the flow of air.

[lafinjack.livejournal.com]

If there's no forward movement because of the treadmill, there would be no airflow. Jet engines don't create airflow, just forward movement. When you cancel that out, you can't get a plane off the ground.

[theweaselking.livejournal.com]

The wheels spin freely, and are not connected to the drive train, so the treadmill cannot prevent movement. Since the engines are pushing against the air, not against the treadmill, they *can* create movement.

The treadmill is an amusing, but irrelevant, side effect. It cannot prevent the plane from moving due to engine thrust.

[lafinjack.livejournal.com]

The treadmill keeps the plane in dead air. It's moving forward in relation to the surface of the treadmill, but not the world at large. All the thrust in the world won't do a thing if the wings can't grab on to anything.

[dilickjm.livejournal.com]

The key is not the wheels, the key is the engines. If the wheels spin freely, the plane will not move backwards with its engines off, the wheels will merely spin in place. When the engines are lit off, the thrust will move the plane forward, causing the wheels to have to spin twice as fast to overcome the motion of the treadmill, but they are not connected to the drivetrain.

So the thrust will move the plane forward regardless of the motion of the treadmill, resulting in lift as the air passes over the wing.

[lafinjack.livejournal.com]

According to the rules of the game, I have complete control over the speed of the treadmill. I just crank up the (infinite) speed of the treadmill until I reach the (finite) speed the jet engines would move the jet forward. Wheel drivetrain and all of that crap have nothing to do with it. By the rules of the game, it might as well be a bare, greased fuselage.

Unless the jet engines can move the jet forward in relation to the air to generate lift with the wings, as opposed to the treadmill, the whole thing might as well be a landspeed record vehicle, with vestigial wings on the sides.

[lafinjack.livejournal.com]

OK, nevermind, I'm a fuckwit. I just thought it through again and you're right.

This is why I'm not allowed near Livejournal when I'm tired.

[leighdb.livejournal.com]

Yeah, but what the answer to #3 should have been was "It was a Spanish lion."

Q.E.D. (which is Latin for "ba dum-dum.")