Monty Hall

[JohnW2]

Mmmm well he did ask the volume of the sphere rather than the volume of the solid still remaining therefore the volume of the sphere is still the same.

Yes, so this is essentially a 'semantics' puzzle (some would call it a 'trick question') but, given that, I don't think BAS really was precise enough with his semantics - hence my original question.

He asked "...what is the volume of the original sphere remaining". If one is going to be semantically pedantic (which I guess one has to be, if the puzzle is essentially semantic!), there in NO 'sphere remaining' - there is a name (which I've forgotten!) for a sphere with a circular hole right through one axis (the first step in changing a sphere into a toroid), and it's not "sphere"! ... which makes one wonder what he actually meant (and, of course, most people will probably assume that he means 'the amount of material remaining in the solid sphere').

The "volume" of a 3D object is defined as the volume contained by its (continuous) surface - and that certainly does not remain unchanged if you drill a hole through a sphere. The "has not changed" answer is an answer to a different question - something like "what is the volume after drilling if one imagines that the entry/exit holes are not there and that the part of the ('containing') surface of the hole joining the poles does not exist". If I gave you a large washer (or even a doughnut) and asked you to determine its 'volume', would you really calculate the volume it would have had if the hole was not there?

Kind Regards, John

 

[DetlefSchmitz]

John, I've read your post twice and I'm still not sure if you get it or not, but I don't think you do. So look at this: http://www.puzzles.com/puzzleplayground/holeinthesphere/holeinthesphereprintplay.pdf and let me know if it makes it clearer or not...

or if I'm just not following you.

 

[JohnW2]

John, I've read your post twice and I'm still not sure if you get it or not, but I don't think you do. So look at this: http://www.puzzles.com/puzzleplayground/holeinthesphere/holeinthesphereprintplay.pdf and let me know if it makes it clearer or not... or if I'm just not following you.

Yes, I go 'get it' - but what you're 'not following' is that what I wrote was specifically a response to what ebee had written (and I quoted). ebee appeared to be saying that, because of the words used in the question, one was being asked what was the volume of the 'imaginery sphere' which remained if one imagined that the holes were not there. That is clearly wrong, and that was my point.

In fact, if I understood him correctly, a calculation based on ebee's view would actually give the wrong answer, and this highlights the thing that many people 'miss' about the problem. Many people (wrongly) assume that the length of the hole is equal to the diameter of the original sphere - whereas, in truth, it is obviously a bit less than the sphere's diameter. ebee's "the sphere hasn't changed" approach (if that is what he was saying) would lead the answer being the volume of the original sphere, rather than the (correct, and smaller) answer which is numerically equal to the volume of a sphere whose diameter was equal to the length of the hole (i.e.less that the diameter of the original sphere).

As for the true answer, I'm not sure that one can really avoid plodding through the geometrical calculations ('Solution 1' in your link) - even though the algebra is a bit tedious. The basis of John Campbell's 'easy' solution ('Solution 2') is not really very robust, because it relies on the assumption "The problem would not be given unless it has a unique solution.", which is a rather strange mathematical basis for anything!! As he goes on to say, if one does make that (IMO iffy) assumption, then it follows that "If it has a unique solution, the volume must be a constant which would hold even when the hole is reduced to zero radius" - which reduces the problem to a triviality - but I really don't think that it is in any way safe to have a 'mathematical proof' which is based on the assumption that the question wouldn't have been asked if it did not have a unique solution!! My maths teachers would certainly be turning in their graves if they knew I had based a 'proof' on such an assumption! There may be other 'simple' (and robust) proofs which avoid all the algebra, but I'm not aware of any.

Kind Regards, John

 

[DetlefSchmitz]

Ah! That's what you meant!

 

[EFLImpudence]

Ah! That's what you meant!

Wasn't it obvious?

 

[DetlefSchmitz]

Ah! That's what you meant!

Wasn't it obvious?

Not to me, anyway.
Starting the post as below threw me. I don't believe it was correct.

Yes, so this is essentially a 'semantics' puzzle (some would call it a 'trick question')

 

[EFLImpudence]

I would say it is a trick question - that's what these things are, with irrelevant details to confuse - not actually semantics.

Isn't it just based on the fact that the volume of a sphere is the same whatever is inside?

Dare I say some overthinking is being employed?

 

[DetlefSchmitz]

I would say it is a trick question - that's what these things are, with irrelevant details to confuse - not actually semantics.

Isn't it just based on the fact that the volume of a sphere is the same whatever is inside?

Dare I say some overthinking is being employed?

Seriously?

Did you look at the link I posted? The question relates to what is left of the sphere after you've drilled a hole through it and removed that volume. It sounds like you are thinking the way I thought John was.

Kind regards, Detlef

 

[JohnW2]

Wasn't it obvious?

Not to me, anyway. Starting the post as below threw me. I don't believe it was correct.

Yes, so this is essentially a 'semantics' puzzle (some would call it a 'trick question')

You're right. Apologies. The way I started my response to ebee was very badly written. I probably should have written something like:

Yes, so you feel that this is essentially a 'semantics' puzzle (some would call it a 'trick question')? If so ...

Kind Regards, John

 

[JohnW2]

Isn't it just based on the fact that the volume of a sphere is the same whatever is inside?

It almost sounds as if you are making the same mistake that I think ebee was making (and the mistake that Detlef probablythought I was making).

If so, I wonder how you got the right answer so quickly. Maybe two errors which 'cancelled'? - firstly what you appear to be saying above and, secondly, assuming that the diameter of the sphere was the same as the length of the hole?

... or am I misunderstanding you?

Kind Regards, John

 

[EFLImpudence]

It was BaS asking the question, don't forget, so you must look at what it actually says.

If he had wanted the volume of the drilled cylinder deducted from the sphere I am sure he would have asked for the volume of the remaining shape and would have had to include the diameter of the drilling.
Then it would be just Mathematics.

Is the volume of the Earth less because of caves?

He'll come along and tell me I'm wrong now.

 

[DetlefSchmitz]

To be honest, I'm still trying to find a link which explains how to derive the volume of a spherical cap from first principles. I suppose it's a double integration but I haven't done that for some time.

 

[DetlefSchmitz]

He'll come along and tell me I'm wrong now.

I suspect so, because this was a question which appeared in Scientific American (I'm guessing in the 60s without looking it up). The follow-up was John Campbell's witty alternative approach to finding the answer.

I doubt that Bas was both aware of this and then asking a different question.

The question has been around since at least 1932.

 

[EFLImpudence]

Isn't it just based on the fact that the volume of a sphere is the same whatever is inside?

It almost sounds as if you are making the same mistake that I think ebee was making (and the mistake that Detlef probablythought I was making).

Which was what, exactly? I'm getting confused.

If so, I wonder how you got the right answer so quickly. Maybe two errors which 'cancelled'? - firstly what you appear to be saying above and, secondly, assuming that the diameter of the sphere was the same as the length of the hole?

Has he actually said it is the right answer?

I just assumed the volume of a sphere is the same regardless of the inside being solid or empty or part of each.

I suppose pedantically you could dispute the length of the drilling, hence the diameter, by measuring from the point of contact of the drill to first point of egress or the length of the sides of the empty cylinder after drilling but it is just a puzzle question.

 

[JohnW2]

It was BaS asking the question, don't forget, so you must look at what it actually says. If he had wanted the volume of the drilled cylinder deducted from the sphere I am sure he would have asked for the volume of the remaining shape and would have had to include the diameter of the drilling.

That's why I sympathised with the view that it might be down to semantic imprecision on the part of BAS. Unless he was talking about a totally different problem from the one we are all discussing (per Detlef's link), he did want you to deduct the volume of the drilled 'cylinder' (actually a cylinder with domed ends) from the volume of the original sphere.

As I said, since you got the right answer despite apparently misunderstanding what was being asked, I can but presume that you made the two errors I mentioned, which cancelled

Kind Regards, John