Monty Hall

[JohnW2]

OK John. I substituted an equation for the different use of radius in the equation from the paper you linked to, and got the result in the puzzle I linked to, so I'm happy they're the same.

You obviously did your algebra a bit more correctly than me, then - since, as you'll see from my recent post, I'm still trying

Kind Regards, John

 

[DetlefSchmitz]

Keep trying. I have had too many glasses to reproduce it here now! I feel a tiny bit smug though.

 

[JohnW2]

Keep trying. I have had too many glasses to reproduce it here now! I feel a tiny bit smug though.

Simples! Goodness knows what I did wrong first time (and I can't be bothered to look back to find out - just started again!) ...

Untitled

• JohnW2
• 17 Sep 2013
• 1

Now I need to open a bottle and see if I can catch up with you

Edit: image edited to correct 'r' to 'R' in starting equation, and 'h' to 'A' in final line
Cheers, John

 

[DetlefSchmitz]

Well done!

 

[viewer]

Wow, that took me back a year or two and made my brain hurt!
Can I suggest a different way of considering an answer?
The only way you can get a hole of exactly 10cm length through a sphere on an axis would be if it had negligible diameter. This was pointed out earlier. Any width (diameter) would have the result of having different, and smaller, lengths along the outside of the hole.
So, there are two basic answers. One is as above, i.e there is no difference you still have a 10cm diameter sphere. The other is that, in a real world with a real drill, you no longer have a sphere but a fat doughnut (also mentioned earlier). This takes you back to the first answer.
There were a few red herrings in some posts as assumptions were made about the environment, e.g. caves don't alter the volume of the earth spinning in space, but the solid volume is changed as could be seen if it was dunked in a huge bucket of water.

 

[ban-all-sheds]

Blimey - go out for the day and look what happens.

Apologies for any unintended imprecision - I did indeed mean the volume of the shape remaining, and even though I wasn't aware of the Scientific American article, Solution 2 is the one people are "expected" to find. It's not really weak, as this is a puzzle, not a question in a maths exam, so a bit of reasoning which says "the question would not be asked unless the diameter of the hole was irrelevant" is perfectly OK.

 

[JohnW2]

Can I suggest a different way of considering an answer? The only way you can get a hole of exactly 10cm length through a sphere on an axis would be if it had negligible diameter.

You are making the 'classic' error (for people looking at this puzzle) of assuming that the diameter of the sphere is 10cm (i.e. that the length of the hole is equal to the diameter of the sphere). You were not told that and, as you point out, if the sphere were 10cm in diameter, the only 10cm hole you could drill through it would be one of zero diameter. Hence, for any finite size of hole (10cm long), the diameter of the sphere has to be greter than 10cm.

It is perfectly possible to drill a 10cm long hole through a sphere with a diameter of, say, 40cm, albeit the hole would be of large diameter and there would not be much of the sphere left!

There were a few red herrings in some posts as assumptions were made about the environment, e.g. caves don't alter the volume of the earth spinning in space, but the solid volume is changed as could be seen if it was dunked in a huge bucket of water.

They weren't IMO really red herrings but, rather, an illustration that a lack of complete clarity in the question led to some ambiguities. This led to the fact that (by analogy) two people thought they were being asked about the volume of the earth ignoring the caves, whilst others (correctly) assumed that they were being asked about the volume of the material in the earth (i.e. with volume of caves etc. subtracted).

Kind Regards, John

 

[JohnW2]

Apologies for any unintended imprecision - I did indeed mean the volume of the shape remaining

Indeed. As you will have seen, that lack of precision led at least two people answering the wrong question (hence my very early request for 'increased clarity'). Ironically, one of them nevertheless got the right answer very quickly, but I presume that was due to a 'cancelling' second error (assuming the sphere was 10cm diameter).

... and even though I wasn't aware of the Scientific American article, Solution 2 is the one people are "expected" to find. It's not really weak, as this is a puzzle, not a question in a maths exam, so a bit of reasoning which says "the question would not be asked unless the diameter of the hole was irrelevant" is perfectly OK.

I'm not so sure. These 'puzzle questions' come in all sorts of forms, and with some of them the answer is "cannot be done with the information provided" (i.e. there is no unique solution possible with the information provided). That most often occurs when the puzzle is such that most people think that there is enough information to get a unique answer, but miss the 'catch'.

Kind Regards, John

 

[JohnW2]

From a mathematical/philosphical point-of-view, the really interesting question about the 'hole through a sphere puzzle' is whether there is some conceptual/physical reason/rationalisation as to why the answer is what it is.

We know, from the maths, that, provided only that the diameter of the hole is less than the diameter of the original sphere, the volume of material left after drilling will always be numerically equal to what would be the volume of a sphere whose diameter was equal to the final length of the drilled hole - regardless of the diameters of sphere and hole (with above proviso).

Is this purely a 'mathematical co-incidence' (seems unlikely), or is there some way of rationalising why this is how the answer works out? No such rationalisation has yet come to my mind.

Kind Regards, John

 

[JohnW2]

I may be moving towards (but not yet quite reaching!) an answer to the conceptual question I posed in my last post ...

... there is another way of looking at the (correct) result. If one starts with a solid sphere of diameter D and drills a hole through the centre which ends up being of length L, then the volume of material removed would be just enough to create a hollow sphere of external diameter D (i.e. the same as the original sphere) and wall thickness D-L.

I can't help but feel that somewhere therein may lay the 'conceptual explanation'

Kind Regards, John

 

[ebee]

So,
in the example of a six inch hole the answer is 36 pi

does that mean that if the hole is three inches long then the answer is 9 pi

(both in cubic inches)

Therefore Ban`s 10cm hole the answer would be 100 pi ?

(answer in cubic centimetres)

 

[JohnW2]

So, in the example of a six inch hole the answer is 36 pi

Yes. If the length of the hole is 6", the answer is the volume of a sphere of diameter 6" (i.e. radius 3"). That volume is (4/3)πr³ = (4/3) x π x 3 x 3 x 3 = 36π

does that mean that if the hole is three inches long then the answer is 9 pi

No - 4.5π. If the length of hole is three inches, the answer would be the volume of a sphere with diameter 3" (i.e. radius 1.5"). As above, that would be 4.5π ... (4/3) x π x 1.5 x 1.5 x 1.5

Therefore Ban`s 10cm hole the answer would be 100 pi ?

No. As above, the answer is (4/3) x π x 5 x 5 x 5 = 166.67π, which is about 523.6 cu cm.

Kind Regards, John

 

[ebee]

Right got it.
Thanks.

Phew, it took some thinking about.

It is amazing how it works out that the meat left remaining from our sphere equals that of a sphere whose diameter would be the length of the hole.

 

[viewer]

Can I suggest a different way of considering an answer? The only way you can get a hole of exactly 10cm length through a sphere on an axis would be if it had negligible diameter.

JohnW2 wrote, "You are making the 'classic' error (for people looking at this puzzle) of assuming that the diameter of the sphere is 10cm (i.e. that the length of the hole is equal to the diameter of the sphere). You were not told that and, as you point out, if the sphere were 10cm in diameter, the only 10cm hole you could drill through it would be one of zero diameter. Hence, for any finite size of hole (10cm long), the diameter of the sphere has to be greater than 10cm."

True, but my assumption was not that of the diameter of the sphere, but that a finite answer was required. A 10cm sphere is the only scenario which gives a finite answer to the puzzle. The given is that the hole is 10cm and through the sphere. If the hole is zero diameter then it must be a 10cm sphere. If the sphere was larger than 10cm a zero diameter hole wouldn't go through

The reason it is the only finite solution is in your second point to which I (and others) referred: "

JohnW2 wrote "It is perfectly possible to drill a 10cm long hole through a sphere with a diameter of, say, 40cm, albeit the hole would be of large diameter and there would not be much of the sphere left!"

Clearly true (I called it a fat doughnut), but as this is not specified in the puzzle, then the answer cannot be specified except to say that as the sphere and drill increase in diameter the cord must always be 10cm and thus there are infinite answers.

So, I suppose there are two correct answers:
This is a 10cm sphere with a zero diameter hole
or
There are an infinite number of (theoretical) answers if the drill has diameter.

My assumption was that only the finite answer was acceptable and in this I failed.

 

[Jackrae]

The other schoolboy howler was "what date is Christmas day in Australia"
25th June came up quite often