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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 14, 2010 4:08 pm

Joined: Thu Jun 03, 2010 2:25 pm
Location: Farmington, NM
PuzzleMaster6262 wrote:
Sjoerd wrote:
In my opinion, everything ends with g.

It's not just an opinion. The word "everything" does, IN FACT, end with g.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 3:54 am

Joined: Mon Mar 30, 2009 5:13 pm
PuzzleMaster6262 wrote:
Sjoerd wrote:
In my opinion, everything ends with g.
It's not just an opinion. The word "everything" does, IN FACT, end with g.

But here's the paradox: nothing ends with g.

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If you want something youâ€™ve never had, youâ€™ve got to do something youâ€™ve never done - Thomas Jefferson

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 9:13 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
Riddle: What word starts with an e, ends with an e, and has one letter in it?

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 9:37 am

Joined: Mon Mar 30, 2009 5:13 pm
GuiltyBystander wrote:
Riddle: What word starts with an e, ends with an e, and has one letter in it?
envelope

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 9:56 am

Joined: Sat Jul 11, 2009 1:09 pm
Location: My House
What's green, hangs on a wall and whistles?

Alex

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 11:27 am

Joined: Sat Jan 16, 2010 11:48 am
Location: In Front Of My Teraminx (saying WTF?)
APJ wrote:
What's green, hangs on a wall and whistles?

Alex

Me

- Greg

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 12:28 pm

Joined: Fri Feb 18, 2000 8:50 am
Location: chicago, IL area U.S.A
Here is my favorite riddle.

Why do firemen wear red suspenders?

-d

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 15, 2010 1:33 pm

Joined: Mon Mar 30, 2009 5:13 pm
darryl wrote:
Here is my favorite riddle.

Why do firemen wear red suspenders?

-d

To stop their trousers falling down.

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If you want something youâ€™ve never had, youâ€™ve got to do something youâ€™ve never done - Thomas Jefferson

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Sat Oct 16, 2010 1:53 am

Joined: Fri Jul 17, 2009 5:32 pm
Location: Tokyo
Houlis and Hearn are a pair o' docs.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 12:37 pm

Joined: Fri Feb 06, 2009 2:57 pm
Location: Pittsburgh
PuzzleMaster6262 wrote:
Sjoerd wrote:
Can you come up with a word, thing, or number that does not lead to 4?

Or what about $, â‚¬, ÂŁ, and ÂĄ? (the symbols, not what they represent ) What, all FOUR of them? _________________ 3x3x3 PB: 00:48.10 "Study gravitation, it's a field with a lot of potential." Top  Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 12:41 pm Joined: Sat Jan 16, 2010 11:48 am Location: In Front Of My Teraminx (saying WTF?) theVDude wrote: PuzzleMaster6262 wrote: Sjoerd wrote: Can you come up with a word, thing, or number that does not lead to 4? Or what about$, â‚¬, ÂŁ, and ÂĄ? (the symbols, not what they represent )

What, all FOUR of them?

four doesn't lead to four, it IS four!

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 12:54 pm

Joined: Sun Mar 11, 2007 3:11 am
Location: Oregon, USA
When in Rome...all roads lead away from Rome.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 1:00 pm

Joined: Sat Jan 16, 2010 11:48 am
Location: In Front Of My Teraminx (saying WTF?)
Rich people don't have it.
Poor people have it.
It's better than heaven.
It's worse than hell.

What is it?

- Greg

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 1:07 pm

Joined: Thu Sep 24, 2009 12:21 pm
Location: Chichester, England
RubixFreakGreg wrote:
Rich people don't have it.
Poor people have it.
It's better than heaven.
It's worse than hell.

What is it?

- Greg

Drugs?

Well, with me being athiest I'm going to ignore the Hell and Heaven part and just use the other two lines. Is it living on the street?

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 1:18 pm

Joined: Mon Mar 30, 2009 5:13 pm
RubixFreakGreg wrote:
Rich people don't have it.
Poor people have it.
It's better than heaven.
It's worse than hell.

What is it?

- Greg

Oskar's gear shift puzzle?

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If you want something youâ€™ve never had, youâ€™ve got to do something youâ€™ve never done - Thomas Jefferson

Last edited by KelvinS on Tue Oct 19, 2010 2:13 pm, edited 1 time in total.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 1:19 pm

Joined: Fri Feb 20, 2009 6:38 pm
RubixFreakGreg wrote:
Rich people don't have it.
Poor people have it.
It's better than heaven.
It's worse than hell.

What is it?

- Greg

Why it's nothing, of course, nothing at all

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doctor who wrote:
I don't think I can make her pose without heavy sedation. The rendering doesn't have to be perfect, it just can't look like Oskar in drag.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 1:25 pm

Joined: Fri Feb 06, 2009 2:57 pm
Location: Pittsburgh
Nothing is better than heaven, right? And $.10 is better than nothing, right? So$.10 is better than heaven!

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 2:19 pm

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
A quick number puzzle (with only one number!).

I have a sphere. I drill a hole 10 cm long right through it which passes through the centre. What is the volume of the material which remains?

HINT: This is not a trick question, or a play on words.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Oct 19, 2010 2:27 pm

Joined: Mon Mar 30, 2009 5:13 pm
Gus wrote:
A quick number puzzle (with only one number!).

I have a sphere. I drill a hole 10 cm long right through it which passes through the centre. What is the volume of the material which remains?

HINT: This is not a trick question, or a play on words.

Depends on the radius (r) of the hole, no?

Anyway the answer is ~ (4/3*3.14*5^3) - (3.14*r^2*10)

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 12:25 am

Joined: Wed May 13, 2009 4:58 pm
Location: Vancouver, Washington
theVDude wrote:
Nothing is better than heaven, right? And $.10 is better than nothing, right? So$.10 is better than heaven!
That reminds me of:
Code:
G = L    God is love
L = B    Love is Blind
R = B    Ray Charles is Blind
By the transitive property
R = G    Ray Charles is God

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 8:07 am

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
You do not need to know the radius of the hole, so your answer is wrong

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 8:12 am

Joined: Mon Mar 30, 2009 5:13 pm
Gus wrote:
You do not need to know the radius of the hole, so your answer is wrong

Well, OK, then the answer is simply the volume of the original sphere = 4/3*3.14*5^3 = 523.6 cm3, on the basis that the material which is drilled out still "remains".

But if you ask me that IS a trick question, not just a simple "number puzzle" as you say, on the basis that the wording is intentionally misleading. So to say it is "not a trick question or play on words" is doubly misleading and (I think) quite dishonest.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 1:22 pm

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
I'm sorry, but nowhere in the question do I state that the diameter of the "original" sphere is 10cm. But my point is that you do not need to know either the radius of the hole OR the diameter of the original sphere.

And I reiterate, the wording of the question is not dishonest.

So, to make things clear:

The hole is 10cm long.
The radius of the hole is not required to solve the question.
The diameter of the sphere is not required to solve the question.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 2:20 pm

Joined: Wed Apr 01, 2009 2:59 pm
Gus wrote:
I'm sorry, but nowhere in the question do I state that the diameter of the "original" sphere is 10cm. But my point is that you do not need to know either the radius of the hole OR the diameter of the original sphere.

And I reiterate, the wording of the question is not dishonest.

So, to make things clear:

The hole is 10cm long.
The radius of the hole is not required to solve the question.
The diameter of the sphere is not required to solve the question.

It's not V_sphere - V_cylinder, since the cylinder also has rounded caps on both ends...

Picture a cross-section so the cylinder drilled out is a vertical rectangle inside a circle. Let h be half the height of the hole (5), r the radius of the hole, and R the radius of the sphere. By Pythagoras, r^2+h^2=r^2+25=R^2. If you put the center of this picture as (0,0,0) in R^3, a suitable double integral should give us the volume. In particular, the surface of the sphere is given by z^2=R^2-(x^2+y^2), or f(x,y)=\sqrt(R^2-x^2-y^2). What's the volume of the hole? Well, it should be the double integral over the region x^2+y^2 < 1 of 2f(x,y) (the times 2 is because that's only the top half of the hole). Putting this in polar coordinates, V_hole=2 \iint_{s=0}^r \sqrt{R^2-s^2} s ds d\theta, which is (4\pi/3)(R^3) - (4\pi/3)( R^2 - r^2)^(3/2). (Thanks, Mathematica...)

But R^2-r^2=h^2=25, so really V_hole=(4\pi/3)(R^3)-(4\pi/3)(25^1.5)=4R^3\pi/3-500\pi/3.

But V_sphere=4R^3\pi/3 exactly, so V_sphere-V_cyl is precisely 500\pi/3, which is independent of r and R. Hooray! This is an awesome problem. (Done a really easy way, if you're explicitly told it doesn't depend on the radii, suppose the hole as radius 0, that should give the same answer. Well, the hole as volume 0, and the sphere has radius 5 cm, so the 'remaining' volume is just the volume of the sphere, 4*5^3\pi/3=500\pi/3 cm^3...) You can also avoid calculus by using the volume formula for a spherical cap, but I don't remember that off hand.

Also, how do you make LaTeX show up on this forum?

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 4:54 pm

Joined: Tue Aug 11, 2009 2:44 pm
This is one of those problems that you can solve without any heavy lifting, on the assumption that sufficient information is given.

First, note that the sphere radius is only 5cm if the hole is infinitesimally narrow. If the diameter of the hole is larger, then the 10cm-long cylinder which forms the hole does not extend the entire diameter of the sphere -- the end caps have been removed.

So. The implication is that whatever the diameter of the sphere, and hole, the answer will be the same. This can make sense once you realize that those two numbers are not independent. E.g. if the diameter of the sphere is 10cm, then the diameter of the hole must be 0. If the sphere is larger, there will still be a unique hole diameter such that the length of the hole is 10cm. Based on the problem statement, we can assume that it just happens to work out that volume remaining when the hole material is removed is constant.

We could no doubt work that fact out with a bit of clever geometric reasoning, but we don't have to. Just assume the hole is diameter 0, and the sphere is diameter 10. Then the volume of remaining material equals the entire sphere, which is

4/3 * pi * 5^3 cubic centimeters.

Edit - just read the above. So yeah, you can do it the explicit way if you feel like thinking that much.

You're in good company, Iranon. There's a famous story about John von Neumann where he also didn't take the easy way out.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 5:45 pm

Joined: Mon Mar 30, 2009 5:13 pm
Thanks Iranon for the full mathematical analysis and Bhearn for the clear and intuitive explanation. All of this is very logical, but the ultimate solution is that the "hole" does not really exist. This completely contradicts the wording of the riddle, which states that the hole is "drilled" through the sphere. I mean what drill has zero width? Basically there is no drill or hole and so I still find this riddle very misleading and completely pointless.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 7:11 pm

Joined: Tue Aug 11, 2009 2:44 pm
What do you mean the hole doesn't really exist? I got the answer by assuming it had 0 diameter, which was logically sound. But even if you want to say "that's not a hole", the point is that *any* diameter hole will give you the same answer.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 7:34 pm

Joined: Mon Mar 30, 2009 5:13 pm
bhearn wrote:
What do you mean the hole doesn't really exist? I got the answer by assuming it had 0 diameter, which was logically sound. But even if you want to say "that's not a hole", the point is that *any* diameter hole will give you the same answer.

Only if you assume the length of the hole is measured from perimeter to perimeter (rather than from one side to the other), but then you don't drill all the way through the sphere as stated by the riddle: the edge caps remain so you shouldn't remove this volume. Just try to draw any diagram (cross section of the drilled sphere) with dimensions indicated, that fulfil all the requirements *as stated* by the riddle. My point is that either you have to give the hole zero width, in which case it doesn't really exist, or the hole won't really go all the way through the sphere (including the end caps), or it won't really be 10cm long.

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Last edited by KelvinS on Wed Oct 20, 2010 7:38 pm, edited 1 time in total.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 7:35 pm

Joined: Mon Jul 21, 2008 4:52 am
Location: Brighton, UK
Kelvin Stott wrote:
Thanks Iranon for the full mathematical analysis and Bhearn for the clear and intuitive explanation. All of this is very logical, but the ultimate solution is that the "hole" does not really exist. This completely contradicts the wording of the riddle, which states that the hole is "drilled" through the sphere. I mean what drill has zero width? Basically there is no drill or hole and so I still find this riddle very misleading and completely pointless.
The hole does exist with larger spheres though.
bhearn wrote:
If the sphere is larger, there will still be a unique hole diameter such that the length of the hole is 10cm. Based on the problem statement, we can assume that it just happens to work out that volume remaining when the hole material is removed is constant.
So if the sphere is the size of the earth, the hole would be almost the diameter of the earth, but still a cylinder 10 cm long, and the remaining part of the earth would be a skinny girdle around the equator totalling 500/3 * pi in volume.

To me, the wording of the question is rather dubious (though I am sincerely impressed with Iranon and bhearn for solving it):

The assumption that when the length of a hole is stated, "of course" only the cylindrical part of the hole is meant. I do understand that if this is stated clearly, such as "... the cylindrical part of which is 10 cm long...", then the clever part would be revealed and the riddle aspect disappears. (In general, who says that only the cylindrical part of a hole counts towards its length? If a circular drill bit is imagined penetrating the sphere, perimeter only, then this is a natural jump to make. However if a solid circular drill bit is envisioned, the whole circular cross-section doing the drilling, then the sphere is penetrated before the 10 cm starts and the assumption seems mighty arbitrary to me.)

And of course there is casually throwing away the end caps beyond the 10 cm cylindrical hole, which has to be done to make the problem work.

"I would like to have my cake and eat it."

"I would like to drill a 10 cm length hole which actually removes 1,000,000 km length of material from a vast body in space. Because, you see, the 10 cm only really begins and ends for the duration of the cylindrical part."

It is an interesting question though, despite the problem with the unstated assumptions.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 20, 2010 7:43 pm

Joined: Mon Mar 30, 2009 5:13 pm
Exactly Juian, this is my issue with the riddle: there is no solution which really satisfies all stated criteria (a 10cm hole drilled all the way through the sphere, where the radii of the hole and the sphere don't matter).

Anyway, despite the misleading wording I admire Iranon and Bhearn for their independent analyses of the problem.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 5:35 am

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
Well done bhearn, your analysis is completely correct. As you worked out, a 10cm "hole" can be drilled through a 10cm sphere only if the hole is infinitesimally small. As the sphere gets larger, the radius of the hole must increase to keep the hole length 10cm. Here is a picture of a 10cm hole drilled through some transparent plastic spheres of various radii:

As you can see, as the sphere size increases, the ring of material left gets thinner and thinner, but maintains a volume of 4/3 * pi * 5^3 cm3.

Kelvin, the question as stated did not say the radius of the hole or sphere do not matter, just that you did not need to know them to solve the problem. All of these spheres have a hole in them which you can measure as 10cm long. The hole actually exists. You only have to use the zero size hole in order to simply solve the problem.

I like this sort of problem, because you go from "impossible" to "complex math" to "obvious" in order to solve it.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 5:47 am

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
Julian, in the real world, if you take a real sphere and a real drill and drill a real hole you have to lose the endcaps (how else could you drill the hole in the sphere without removing this material?), and you are left with a hole of circular cross-section and a measureable length. Even if you "drilled" a square hole or an octaganal hole the analysis by bhearn is still correct.

I don't see this as "having my cake and eating it"

Of course, the point of this problem is to give as little information as possible while the problem is still solvable. Again, well done bhearn and Iranon.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 6:35 am

Joined: Mon Mar 30, 2009 5:13 pm
Gus wrote:
Kelvin, the question as stated did not say the radius of the hole or sphere do not matter, just that you did not need to know them to solve the problem. All of these spheres have a hole in them which you can measure as 10cm long. The hole actually exists. You only have to use the zero size hole in order to simply solve the problem.
I guess I'm just not happy with the way you define and measure the "length" of the hole drilled, from perimeter to perimeter rather then between opposite extremities of the material which is actually removed (i.e. the diameter of the original sphere).

I agree it's a very nice mathematical problem, but the wording should be clarified in terms of what you mean by the length of the hole that is drilled (or at least indicate there are two possible ways to interpret this). Otherwise the "quick number problem" is tainted by a misleading (or at best ambiguous) statement. It's a shame because if I had understood your definition (or at least anticipated that some lateral thinking was required to interpret the wording) I would have enjoyed thinking about the maths as you had intended. On the basis that this was supposed to be a "quick number puzzle" and "not a trick question or play on words", my brain was prepared for lateral thinking in the maths, but not in interpretation of the wording. Oh well.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 8:17 am

Joined: Mon Jul 21, 2008 4:52 am
Location: Brighton, UK
Gus wrote:
Julian, in the real world, if you take a real sphere and a real drill and drill a real hole you have to lose the endcaps (how else could you drill the hole in the sphere without removing this material?)
We could use a drillbit with a cross-section that is just the perimeter of a circle (but importantly with some thickness to it), seal both ends of the cylinder perpendicularly to the cylinder's length when the cylinder is complete, then drain off the volume through a slim pipe (if not a gas or liquid, then crush the volume up into tiny grains first with an additional super-powerful, expandable mini-drill), then seal and unseal everything up afterwards as necessary so the volume has been drained and the caps are still there. (If we assume a real sphere, drill, and hole, then it seems reasonable to assume that there is thickness/room for such a draining pipe.) But that's a digression. The key issue follows...

Gus wrote:
and you are left with a hole of circular cross-section and a measureable length. Even if you "drilled" a square hole or an octaganal hole the analysis by bhearn is still correct.
I do understand the analysis. There will always be people who see beyond a (potential) wording issue to arrive at the intent of the questioner. There are two main options in response to this kind of feedback. One is, "At least two people understood me fine, so that confirms my wording is fine, the problem must be with Julian and Kelvin, and I won't change a thing." The other is, "Hmmm, maybe the wording of my question could be improved, let's see how..." I think all of us are agreed that it's a fascinating and stimulating/surprising mathematical result. I think that in essence it's a great problem, and I hope you are willing to tweak the wording. (How about "cylindrical hole" instead of "hole"? Is that enough of a indication without blowing the lateral thinking aspect? Just a suggestion.)

Gus wrote:
I don't see this as "having my cake and eating it"
No, but that isn't what I suggested. My following sentence showed I was referring to the huge disparity between the "real", total length of the hole and the mere cylindrical part of the hole, as the "having cake and eating it". "I will drill a hole longer than the diameter of a googol of suns side by side, but I will call it just 10 cm long because the cylindrical part of the hole is 10 cm long." I think that kind of sophistry is "too far", "too misleading".

Gus wrote:
Of course, the point of this problem is to give as little information as possible while the problem is still solvable. Again, well done bhearn and Iranon.
Yes, but there is always a threshold beyond which people will feel that commonsense meanings/concepts/definitions are being violated too far and they will say, "Hey! No fair!" I had a Captain Picard moment with this puzzle.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 9:14 am

Joined: Sun Mar 15, 2009 12:00 am
Location: Jarrow, England
OK gentlemen/ladies, I suppose that I will have to accept that some of you (you know who you are! ) will argue about how to measure a hole. I concede that if you used a drill exactly 10cm long (and of the required radius) then it would not be able to drill right through the sphere due to the thickness of the end-caps.

However, if I gave you a sphere with a hole through the middle, and asked you to measure the length of the hole, I'm sure that you would get 10cm.

Stating some "mathematical" problems like this will always lead to different interpretations of the words used - English is pretty flexible. It was not my intention to deceive, it's just that I stated the problem exactly as I first read it. And I still think that it is a pretty clever problem.

Anywhoo, if my intention was to pose a question which made people think and generated some debate, then I am happy. Congrats to bhearn and Irano for their solutions, and thanks to Julian and Kelvin and everyone else for taking the time to make comments.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 9:21 am

Joined: Fri Nov 04, 2005 12:31 am
Location: Greece, Australia, Thailand, India, Singapore.
ALL TWISTY PUZZLE FORUM PEOPLE ARE DECEIVERS!!!
(sorry ladies and gentlemen, I just couldn't help myself!)

Pantazis

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 9:27 am

Joined: Mon Mar 30, 2009 5:13 pm
Gus wrote:
Anywhoo, if my intention was to pose a question which made people think and generated some debate, then I am happy. Congrats to bhearn and Irano for their solutions, and thanks to Julian and Kelvin and everyone else for taking the time to make comments.

No problem, and thanks for sharing this puzzle, and also for taking my (initially quite harsh) feedback so well.

PS. I grew up in Tynemouth, Whitley Bay and Seaton Deleval (just down the road from you), so I assume our misunderstanding was not down to any language/cultural differences.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 10:38 am

Joined: Fri Feb 06, 2009 2:57 pm
Location: Pittsburgh
kastellorizo wrote:
ALL TWISTY PUZZLE FORUM PEOPLE ARE DECEIVERS!!!
(sorry ladies and gentlemen, I just couldn't help myself!)

Pantazis

Except me.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 10:55 am

Joined: Tue Aug 11, 2009 2:44 pm
Kelvin Stott wrote:
I agree it's a very nice mathematical problem, but the wording should be clarified in terms of what you mean by the length of the hole that is drilled (or at least indicate there are two possible ways to interpret this). Otherwise the "quick number problem" is tainted by a misleading (or at best ambiguous) statement.

I don't agree... I don't think the wording could be improved one bit. Here's the original wording:

Gus wrote:
I have a sphere. I drill a hole 10 cm long right through it which passes through the centre. What is the volume of the material which remains?

Anything that attempted to "clarify" this would just make it less concise and appealing. There's no tricky interpretation required here. You drill an ordinary hole with an ordinary drill bit. And as Gus said, if you had such a sphere with a hole in it in your hands and were asked to measure the hole, no doubt you would measure it correctly. To the extent that one has to decide on an interpretation at all, the intended interpretation clearly makes the most sense (and makes the problem meaningful).

Gus wrote:
I like this sort of problem, because you go from "impossible" to "complex math" to "obvious" in order to solve it.

Yes, exactly. Very nice!

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 11:01 am

Joined: Mon Mar 30, 2009 5:13 pm
Bob, I'm glad you could understand the intended meaning/construct. I didn't.

Anyway that's enough from me on this, I think.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 12:18 pm

Joined: Sat Jul 11, 2009 1:09 pm
Location: My House
Gus wrote:
I have a sphere. I drill a hole 10 cm long right through it which passes through the centre. What is the volume of the material which remains?

If the hole is 10 cm long, you may end up with this (which has caused confusion):

Attachment:

10cm.jpg [ 124.48 KiB | Viewed 3309 times ]

If the hole goes right through, you get the intended (by the riddle) shape. I think that (although bhearn may disagree) the riddle could and should be reworded. I'd suggest something along the lines of:

"A hole is drilled all the way through a sphere, leaving a ring/torus with a height of 10cm; what is the volume of the material left?"

This may or may not make it too easy, but it certainly makes it a lot less ambiguous, making it a more interesting riddle (confusing or ambiguous wording ruins riddles imo).

*Btw, I know that the Earth isn't a perfect sphere (it was just an easy example), but you get what I mean.*

Alex

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 12:35 pm

Joined: Mon Mar 30, 2009 5:13 pm
APJ wrote:
"A [cylindrical] hole is drilled all the way through a sphere, leaving a ring/torus with a height of 10cm; what is the volume of the material left?"

Perfect!

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 12:37 pm

Joined: Fri Feb 06, 2009 2:57 pm
Location: Pittsburgh
Kelvin Stott wrote:
APJ wrote:
"A [cylindrical] hole is drilled all the way through a sphere, leaving a ring/torus with a height of 10cm; what is the volume of the material left?"

Perfect!

What else do you get when you drill a hole straight through a sphere though?

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 12:40 pm

Joined: Mon Mar 30, 2009 5:13 pm
theVDude wrote:
Kelvin Stott wrote:
APJ wrote:
"A [cylindrical] hole is drilled all the way through a sphere, leaving a ring/torus with a height of 10cm; what is the volume of the material left?"

Perfect!

What else do you get when you drill a hole straight through a sphere though?

The difference is this makes it clear that 10cm is the height of the remaining ring, NOT the length of the material that is drilled away. After all, what does the "length of a hole" mean anyway?

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Thu Oct 21, 2010 1:08 pm

Joined: Mon Jul 21, 2008 4:52 am
Location: Brighton, UK
bhearn wrote:
You drill an ordinary hole with an ordinary drill bit.
But an ordinary drill bit is solid, isn't it? With a solid drill bit, when you drill a hole in straight line through the center of a sphere, the length of the hole is always the diameter of the sphere. In common sense terms, the hole begins where the burrowing begins, and ends where the burrowing ends. Unless you have a clever/freaky hollow drill bit, you smash into the sphere at its nearest point and continue burrowing for the length of the diameter of the sphere.

bhearn wrote:
And as Gus said, if you had such a sphere with a hole in it in your hands and were asked to measure the hole, no doubt you would measure it correctly.
That's a very interesting point, and I would say the key point in fact. The drilling operation has removed the evidence of how long the drilling operation was, and we would probably have no reason to view the remaining hollow solid as a sphere or a snubbed sphere or whatever. "Drill a hole through the center of a sphere so that the remaining part of the sphere is an object that has a 10 cm long hole in it, even if you drilled further through the sphere than that." That's what we really have to do. Either we use a completely hollow drill bit, or we ignore the length of the drilling operation and look at the appearance of the result. I partially withdraw my former objection, with thanks and humility.

I think this problem is ideally suited to an expert at manipulating 3D objects and visualizing their intersections in a hollow kind of way, CAD style, and the appearance of what is left after cuts have been made -- you know, the kind of whizz who would come up with a face turning icosahedron.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Fri Oct 22, 2010 6:18 am

Joined: Wed Jan 21, 2009 12:58 pm
Hello,

I really enjoyed reading the discussion about the drill-a-hole-through-a-sphere problem. I must admit that I would have liked to have a different wording right from the beginning... I don't know the original wording of the problem but it seems that it goes back to the 17th century, it is actually known as the Napkin ring problem. A little search proves that there are several ways of wording this problem:

Keith Devlin simply talks about "calculating the volume that remains when a circular cylinder is removed from the center of a sphere"

Everything2 says "[...] The hole is six inches long -- that is, from one point on one edge of the hole to the closest point on the other edge is a distance of six inches [...]"

Ask Marylin said "[...] boring through a sphere and leaving a 6-inch cylindrical hole through the center"

2000clicks says "Suppose a circular hole was drilled through the center of a sphere. When the length of the hole was measured along its wall, it was found to be six inches long [...]"

So, yes, I think the wording could be changed.

My 2Â˘ worth,

Skarabajo.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Sat Oct 23, 2010 6:26 am

Joined: Fri Feb 06, 2009 2:57 pm
Location: Pittsburgh
Skarabajo wrote:
"Suppose a circular hole was drilled through the center of a sphere. When the length of the hole was measured along its wall, it was found to be six inches long [...]"
That one sounds best IMO.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Mon Oct 25, 2010 5:41 pm

Joined: Thu Oct 16, 2008 9:46 pm
Location: Littleton CO
Wouldn't it be 0? Because it's a hole...

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Wed Oct 27, 2010 10:23 am

Joined: Fri Feb 06, 2009 2:57 pm
Location: Pittsburgh
Gus wrote:
A quick number puzzle (with only one number!).

I have a sphere. I drill a hole 10 cm long right through it which passes through the centre. What is the volume of the material which remains?

HINT: This is not a trick question, or a play on words.

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 Post subject: Re: Enigmas, Riddles, and Paradoxes... (post yours here!)Posted: Tue Nov 09, 2010 4:16 pm

Joined: Thu Jun 03, 2010 2:25 pm
Location: Farmington, NM
Is it possible to cut an obtuse triangle (a triangle with one obtuse angle) into smaller triangles, all of them acute? If such a dissection can be done, what is the smallest number of acute triangles into which any obtuse triangle can be dissected?

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