Puzzles,conundrums, riddles and thoughts?

[Sparta]

Yeah I get it. In rather less than 468 pages. Do I get a prize?

Being a member of CFC and a participant in this thread is a prize unto itself.


[Sorry that the guess [about you lurking SD] was incorrect, btw, Ziggy Stardust - was just a failed attempt at a random assumption. Would have been cool if I had been right though. (<- not a highly self-limiting statement.) ]

 

[Taliesin]

At this point switching will be a neutral gamble (+0.5X or -0.5X) since you don't know what you have. All you know is what to expect: 1.5X. That changes when you know one of the values in one enveloppe.

Evidently, but why? I haven't been able to think of a case where you've opened your envelope and subsequently don't want to switch: if it's odd, then you know you have the lesser value, so you switch; if it's even, it seems you always gain 1/4 of its value by switching. The other envelope always gives you more. So you don't even need to look in the one you're holding.

But then why isn't it to your advantage to switch again?

I guess the best way to look at the riddle is to find the difference between the two frames for your problem. In one framing, you've a 50/50 chance of picking the big envelope first, so there's no incentive to switch. In the other framing, the envelope you don't have always has an expected value of 1.25 times what you're holding, so you should always switch. Why the discrepancy?

 

[oagersnap]

Their server is responding as well as usual right now (i.e. not at all), so I can't link the thread or the article ATM, but IIRC the whole setup is impossible as it creates an infinite feedback loop between the wheel rotational speed and the conveyor belt speed. (For example, if the plane's engines have gotten enough thrust to normally propel the plane forward at 100 mph, the wheels would therefore be spinning opposite at a ground rate of 100 mph, but the conveyor belt would in turn have to go 'backwards' at 100 mph as well, so the wheels would actually have to be going 200 mph. Since the wheels would always have to be spinning twice as fast as any other component (or at the combined rate of speed of the conveyor belt and the airplane's theoretical speed), the whole setup is impossible.)

I don't understand what you (and that guy in the article that someone linked to earlier) are saying.
The plane (including the wheels) moves forward at 100 MPH, the conveyor belt moves backwards at 100 MPH, and 100 - 100 = 0, so the plane doesn't move relative to the air or the ground around it, and it moves 100 MPH relative to the conveyor belt.
Please tell me what, according to you, is wrong with this reasoning.

 

[Riffraff]

I don't understand what you (and that guy in the article that someone linked to earlier) are saying.
The plane (including the wheels) moves forward at 100 MPH, the conveyor belt moves backwards at 100 MPH, and 100 - 100 = 0, so the plane doesn't move relative to the air or the ground around it, and it moves 100 MPH relative to the conveyor belt.
Please tell me what, according to you, is wrong with this reasoning.

what happens is this:

the plane moves forward with 100 mph,

the conveyer belt moves backwards with 100 mph

The wheels of the plane spin at a rate as fast as they would be going 200 mph.

The conveyer belt doesn't influence the plane otherwise.

Relative to the air, the plane is moving at 100 mph.

 

[nonconformist]

what happens is this:

the plane moves forward with 100 mph,

the conveyer belt moves backwards with 100 mph

The wheels of the plane spin at a rate as fast as they would be going 200 mph.

The conveyer belt doesn't influence the plane otherwise.

Relative to the air, the plane is moving at 100 mph.

I don't get it; surely the plane is stationary relative to the ground?

 

[brennan]

I guess the best way to look at the riddle is to find the difference between the two frames for your problem. In one framing, you've a 50/50 chance of picking the big envelope first, so there's no incentive to switch. In the other framing, the envelope you don't have always has an expected value of 1.25 times what you're holding, so you should always switch. Why the discrepancy?

I think I get it:

You have an unknown amount X, and your choices appear to be to lose half that amount or gain that amount when you swap, a net gain of 1.25.

However you are not actually in that situation: If you can pick two envelopes with X or 2X in them then having made a choice of one (X or 2X) you are now choosing to either go from X to 2X (a gain of +1X), or go from 2X to X (a gain of -1X). IOW there is no point in swapping.

 

[oagersnap]

Evidently, but why? I haven't been able to think of a case where you've opened your envelope and subsequently don't want to switch: if it's odd, then you know you have the lesser value, so you switch; if it's even, it seems you always gain 1/4 of its value by switching. The other envelope always gives you more. So you don't even need to look in the one you're holding.

But then why isn't it to your advantage to switch again?

I guess the best way to look at the riddle is to find the difference between the two frames for your problem. In one framing, you've a 50/50 chance of picking the big envelope first, so there's no incentive to switch. In the other framing, the envelope you don't have always has an expected value of 1.25 times what you're holding, so you should always switch. Why the discrepancy?

In the example where you know your envelope has 20 $ in it, you may either lose 10 $ or gain 20 $.
But in the riddle, the scenario is different. You don't know what you have, so for example if you know the envelope you chose either has 20 $ or 10 $ in it, you could either gain 10 $ or lose 10 $.
So the example you gave is meant to confuse people, because there are actually 3 different possibilities: 10 $, 20 $ and 40 $.
In other words, the choice you have in the riddle is not between X, 2X or ½X, but between X and 2X. However, you don't know which one you have, so there is no reason to switch.

 

[brennan]

I don't get it; surely the plane is stationary relative to the ground?

No: If you turn on your jet engine or your propellor, then your plane is exerting a thrust force relative to the air it is in. So, unless the thrust is so low that it is negated by the friction at the wheels, your plane will move with respect to the air, independant of what is happening to the ground underneath it. Relative to a stationary observer the plane will move in one direction and the conveyor will move in the other (so the plane is moving twice as fast wrt the conveyor as the observer).

 

[nonconformist]

No: If you turn on your jet engine or your propellor, then your plane is exerting a thrust force relative to the air it is in. So, unless the thrust is so low that it is negated by the friction at the wheels, your plane will move with respect to the air, independant of what is happening to the ground underneath it. Relative to a stationary observer the plane will move in one direction and the conveyor will move in the other (so the plane is moving twice as fast wrt the conveyor as the observer).

So this is an infinitely lengthed conveyor belt?

 

[brennan]

Runway length would suffice.

 

[Riffraff]

Think of this situation: you are rolling a bowling ball towards a conveyer belt going at the same speed in the reverse direction.

Wouldn't you find it unexpected that the instant the bowling ball rolls on the belt, it would stop immedeatly?

 

[Eran of Arcadia]

yeah, but the wheels aren't what makes the plane move. They spin more or less freely relative the motion of the plane, which is dependent on the engines.

 

[brennan]

In the example where you know your envelope has 20 $ in it, you may either lose 10 $ or gain 20 $.
But in the riddle, the scenario is different. You don't know what you have, so for example if you know the envelope you chose either has 20 $ or 10 $ in it, you could either gain 10 $ or lose 10 $.
So the example you gave is meant to confuse people, because there are actually 3 different possibilities: 10 $, 20 $ and 40 $.
In other words, the choice you have in the riddle is not between X, 2X or &#189;X, but between X and 2X. However, you don't know which one you have, so there is no reason to switch.

This way just occurred to me: You have picked an envelope containing either $10 or $20. You pick the 20$ envelope, the other envelope now contains $10 but you just don't know whether it contains $10 or $40. However; there is is not actually a possibility of you ending up with $40, hence swapping will not mean you gain anything.

 

[nonconformist]

Runway length would suffice.

Ah, I get it, it moves along the belt.

I was imagining a scenarion on a treadmill sized belt, meaning it had to remain stationary.

 

[brennan]

Yeah, it's a sneaky little problem.

 

[ainwood]

A plane is standing on a runway that can move (some sort of band conveyer). The plane moves in one direction, while the conveyer moves in the opposite direction. This conveyer has a control system that tracks the plane speed and tunes the speed of the conveyer to be exactly the same (but in the opposite direction). Can the plane take off?

No.

The plane needs air moving across its wings at a suitably high velocity.

 

[nonconformist]

Yeah, it's a sneaky little problem.

It's the wording that's the problem

 

[ainwood]

No.

The plane needs air moving across its wings at a suitably high velocity.

Actually, yes it can take off.

The plane needs to generate velocity relative to the air - as the air moving across the wings creates lift.

If the wheels were responsible for generating this velocity (eg. like a car), then the conveyor moving in the opposite direction to the wheels would negate this; the plane would not move relative to the air, so it could not take off.

But: Its not the wheels that generate the velocity. The velocity is instead generated by the engines, which pull against the air to create thrust - it doesn't matter what the conveyor does - its just spinning the wheels.

 

[Erik Mesoy]

Censored plane problem and its censored wording.

I ask: Where would the plane get upwards acceleration from?

 

[brennan]

Ignore anything anyone ever tells you about air pressure. When air passes over an aircraft wing the shape of the wing forces the air to leave the trailing edge with a downward component. Newton's first Law means that the reaction pushes the plane up.

In this problem the wording implies that you have a stationary plane, when in reality you'd get a moving plane, regardless of the conveyor.