A really simple question.

Quote Mungo "I drilled a hole right through the middle of a sphere. I forget how big the sphere was; maybe . . . Ahhh; who cares? You've seen one sphere you've seen them all, right?.

The hole had a diameter of blaah blaah blaah whatever - sort of medium-ish wide - and the length of the hole was three feet or inches or metrical thingies or . . . Gookies perhaps. I was never very good at longitude!

The question is, what is the volume of what remains of the sphere?"

Quote Frank. "The volume is the same as it was before you did the drilling -- and the hole does not impact on the volume of a sphere."

Whilst I agree, with what you say Frank. The ?'problem' would appear to lie in the definition of the word ?'volume'. With synonyms such as, size, quantity, weight, amount and dimensions, I think you can come up with a number of answers, which would all be valid.

For example, if the object was square and was drilled, the size and dimensions would remain the same. However, the quantity and weight would reduce.
However, what if the object was a balloon, the volume would be drastically reduced. If, on the other hand, you bought a sphere of ice cream which weighed 1kilo and Mungo scooped out the middle so it now weighed .75. I am sure you would agree the volume had been reduced.

Although I have been accused of many things, fence sitting has not been one of them. So, IMHO for a volume to be valid it has to be enclosed. Otherwise it becomes a ?'void' the outside dimensions (volume) remain the same, but not it's mass.

A lot depends on the thought processes of the person setting the question, irrespective of how it appears in print. In this case, as the formula shows it is about more than the external dimensions. Says I whilst falling off the fence. Embarrassed

Set up your own problem Frank. :wink:

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Trial

"Irrespective of the wording" !!!!!

I would claim that ' The volume of WHAT REMAINS of the sphere" needs no further explanation. Within the context it can only mean 'after the volume removed by the drilling is discounted', surely?

It would indeed be interesting if Frank was to set up some puzzles. He has one of the keenest brains around here and, he himself has said, he has been doing puzzles for many years. He should know some good ones.

BTW; you called one of your puzzles 'The worlds oldest puzzle'. Surprisingly enough, it could be that one of the oldest puzzles is the old one of 'As I was going to St Ives'. There is an Egyptian papyrus that has the problem, "The wealthy man had seven barns. In each barn were seven cats. Each cat killed seven mice. Each mouse would have eaten seven sheafs of wheat. Seven sheafs of wheat make one cran of flour. How much flour did the wealthy man save?" Too similar in wording, IMHO, for it to be a coincidence. The one, methinks, must be a survival of, or was influenced by, the other.

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Mungo wrote:

Trial

"Irrespective of the wording" !!!!!

I would claim that ' The volume of WHAT REMAINS of the sphere" needs no further explanation. Within the context it can only mean 'after the volume removed by the drilling is discounted', surely?

It would indeed be interesting if Frank was to set up some puzzles. He has one of the keenest brains around here and, he himself has said, he has been doing puzzles for many years. He should know some good ones.

You are absolutely correct -- I really gotta contribute a puzzle or two -- and I promise I will do so right away...

...but...

...I gotta comment a bit on what you said here.

I am a puzzler of long standing -- more than I care to think about right now.

One thing you learn very early in puzzling is that you must not assume everything is exactly as it sounds. (My first easy, little puzzle will give an example of that.)

When you used the expression "the VOLUME of what remains in the sphere" -- one of the things that I had to consider was:

What remains in the sphere? Is it just the solid part?

My answer (and the answer any puzzler of long standing would give) was -- Hell no. In many, many, many puzzles -- the solution would involve understanding that the AIR that remains IS NOT A NOTHING. It is a SOMETHING -- not only in science, but even more frequently (of a sort) in puzzles. To discount the fact that the air in the sphere is a part of what remains in the sphere -- is the kind of mistake that often causes puzzles to seem unsolveable. (The "gry" puzzle comes to mind in this context!)

You seem to be well versed in math -- or at least the explanation you offered seems to indicate that you are well versed to a math moron like myself. If so, you should realize that the volume of a sphere is a mathematical abstract -- and that the sphere itself is a mathematical model independent even of the first layer of molecules that might compose any specific sphere.

It truly doesn't matter if anything is removed from the inside of the sphere -- the volume of the sphere remains the same as long as the radius of the sphere remains the same.

Wording is everything in a puzzle -- as the "gry" puzzle illustrates.

Perhaps the wording should have been -- what is the volume of the solid material left inside the sphere -- in which case, I would have to ask if your explanation has to be revised. (Don't give a complete explanation -- I honestly don't understand the math well enough to appreciate it. But I would be interested in a YES or NO.)

Given that -- and despite the fact that both you and Try disagree with me -- to a true puzzler, the answer to the puzzle as you proposed it IS THE ANSWER I GAVE. Rather than being the math puzzle you conceive it to be -- it simply would be one of those trick puzzle, with the answer being "There is no change."

Several small puzzles to follow!!

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I am a huge fan of Raymond Smullyan -- probably the worlds foremost puzzle master.

I will borrow a few from him -- although the few I borrow are so easy, I suspect he borrowed them from antiquity. (The one he devises himself are OUTSTANDING!)

Here's a soft ball:

There are two camels facing in opposite directions. One facing due east; the other due west. How can they see each other without walking, turning around, or even moving their heads?

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Quote by Frank. "Given that -- and despite the fact that both you and Try disagree with me" Sad

Here's me thinking I was agreeing with you. Still, it shows you can never trust a fence-sitting fibber. Embarrassed

The camels are in fact facing each other? Very Happy

What, in mathematical terms would explain the answer?
(I am sorry I have ?'lost' trace of the first post)
"What would you get if you crossed the Atlantic in the Titanic?"

"There is an Egyptian papyrus" Is there anything ?'truly' original?

How many were going to St. Ives?
Just one - the narrator. All of the others were travelling the other way.
How many were going the other way? 1 man + 7 wives + 49 sacks + 343 cats + 2401 kits = 2801 :wink:

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This one is cute:

A man comes to a jeweler with six chains -- each of which has five links.

He wants the six chains to be joined into one large, circular chain and inquires as to cost.

The jeweler replies: "Every link I cut open and close will cost $20."

So...how much will the job cost?

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I can't do any better than $100. I get the feeling that I have missed something.

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The missing link!

You are so lucky, it cost me $120. :wink:

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$100 is correct -- and ya didn't miss anything. I was an easy one -- although it is easy to come up with $120.

Try, did you find the missing link?

A bunch of kids notice some cookies cooling on a windowsill -- and one of them decides to steal them. But as he starts to do so, his conscience gets the better of him, and he decides to take only half of them, which he stuffs in his pocket. But as he is leaving, he takes one more so he can start eating right away.

Than another kid decides to help himself from the remainder -- and he takes half of what is left plus another one. And after than, a third kid takes half of what is left plus one -- and a fourth takes half of what is left plus one.

When a fifth kid comes along to snag some cookies, he finds they are all gone.

So how many cookies were there to start with -- assuming nobody but the first four kids took any of 'em.

(No trickery in the wording. Straightforward math puzzle!)

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Put me down for 62 Confused

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Try

I hope you are not a school teacher, you can't count kids too well! :-)

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Sorry, Try, wanna try again?

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Now you know where the name came from. Sad

However 62 works for me.

62 - 30 - 14 - 6 - 2 - 0 Cool

If you dont believe me, which three digit number, when multipied by 4, is equal to 9. Question

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If there were 62 cookies -- and the first person took half of them plus 1 -- there would be 31 left -- not 30!

If you start with 30 cookies, and the first person takes half plus one -- he takes 16 -- leaving 14. The second person takes half plus one or 8 cookies -- leaving 6. The third takes half plus one or 4 -- leaving 2. The fourth takes half plus one -- leaving none for the fifth.

Mungo PM'ed me with a formula for the solution -- which he can share with you.

My solution to the problem (which is identical to his formula, but less sophisticated) started from the end result.

If the fifth kid found none -- the fourth had to find 2 cookies. That is the only number where if he took half of what was there plus one would leave zero.

If 2 cookies were there for the fourth kid, the third kid had to find 6 cookies. That is the only number where if he took half of what was there plus one would leave 2.

If 6 cookies were there for the third kid, the second kid had to find 14 cookies. That is the only number where if he took half of what was there plus one would leave 6.

If 14 cookies were there for the second kid, the first kid had to find 30 cookies. That is the only number where if he took half of what was there plus one would leave 14.

So 30 is the answer.

Right?

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Frank Apisa wrote:

If there were 62 cookies -- and the first person took half of them plus 1 -- there would be 31 left -- not 30!

62
31
-1
=30

OK! so I am soft hearted and all 5 kids got some cookies. :wink:

Not wanting to let sleeping dogs lie, I hope this clears up any misunderstanding.

Cutting a Cylinder out of a Sphere
Subject: Remaining Volume of a cut Sphere

I have a geometry problem:

A cylindrical hole has been drilled directly through the centre of a
sphere. The length of the cylinder is 6 inches. What is the volume
remaining in the sphere?

Thank you.
Tryagain

From: Doctor Rob
Subject: Re: Remaining Volume of a cut Sphere

There does not seem to be enough data to solve this problem, yet it
does have a solution. In order for this to be true, the solution must
be independent of the radius of the cylindrical hole. That means that
we can assume that the cylindrical hole has radius zero, and compute
the correct answer. Then the diameter of the sphere is 6 inches, and
the volume of the sphere will give you the answer.

There is a more direct approach using the following diagram, with the
cylindrical hole bored horizontally with axis PQ through the centre O
of the sphere:

_..-----.._
.+'-----------`+.
,' |\ 6 | `.
,' | \ | `.
/ | \R |r \
/ r| \ | \
. | \ | .
| R-3 | 3 \ 3 | R-3 |
P+-------+------+------+-------+Q
| | O | |
. | | '
\ r| |r /
\ | | /
`. | | ,'
`. | 6 | ,'
`+._---------_.+'
''-----''

O is the centre of the sphere, R its radius, and r the radius of the
cylindrical hole. Then by the Pythagorean Theorem, r^2 = R^2 - 9.

for the formulas used below. The volume of the sphere is 4*Pi*R^3/3.
The two missing spherical caps have volume (Pi/6)*(3*r^2+[R-3]^2)*(R-
3), and the cylinder has volume Pi*r^2*6. The remaining volume is then

V = 4*Pi*R^3/3 - (Pi/3)*(3*r^2+[R-3]^2)*(R-3) - Pi*r^2*6,
= 4*Pi*R^3/3 - (Pi/3)*(3*[R^2-9]+[R-3]^2)*(R-3) - Pi*(R^2-9)*6,
= (Pi/3)*(4*R^3 - [R-3]^2*[4*R+6] - 18*[R^2-9]),

Which, simplifies to the correct answer independent of R or r.

Fairy nuff.

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Nice for Dr Rob. However, when he says, "it does have a solution. In order for this to be true, the solution must be independent of the radius of the cylindrical hole".

I would point out that the blunt assertion of "it does have a solution" does not constitute proof that it has a solution. He goes on to say "in order for this to be true . . . must . . . " But he has not yet shown that it is true. So we have a solution that is correct given the assumption that it is true.

Oh dear oh dear oh dear!

How one wonders would Dr Rob deal with the question, "What is the sum of anything at all plus two? It does have a solution" (As indeed it does. It has many of them) Would he reply, "In order for this to be true the solution must be independant of the value of the first number, therefore we can assume that the first number has a value of zero and compute". Without any evidence, one statement is as valid as the other.

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Try

Ooops.

My mistake!

But I did say that I stunk at math, right?

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I have just re-read the previous posts. My we are a fractious bunch - and so unlike good old Mungo who merely asserts the truth. :-)

A neat explanation of why no further information is needed in that sphere puzzle, Try.

Frank. I accept that you never, if you are a puzzler, accept words at their face value. Please, then, interpret the wording of my puzzle as being extra crafty in that it - knowing your hesitation in accepting things at face value - stated the problem in straightforward English, knowing you would not believe it. The best bluff of all is not to bluff when a bluff is expected.

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Quote. Oh dear, oh dear, oh dear!
"and so unlike good old Mungo who merely asserts the truth."

"I would point out that the blunt assertion of " merely asserts the truth " does not constitute proof that it is the truth. In order for it to be true . . . good old Mungo has not yet shown that it is true. So we have a solution that is correct given the assumption that it is true."
"Please, then, interpret the wording of my puzzle as being extra crafty"

After careful consideration Mungo the verdict is one of Guilty. Razz

Frank, not as bad as me! :wink:

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Try

"You smart-alecky big village feller you!" [Stan Freberg] :-)

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A really simple question.

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