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skeptical scientist

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« Reply #90 on: November 07, 2006, 07:32:20 PM »
Oooh, it looks like we're giving Erasmus brainteasers, so I have one! This deals with infinite numbers, so I predict that I will pose the question, Erasmus will correctly answer it, and then five people will jump up to tell us both that not only is Erasmus wrong, but the question itself is meaningless gibberish.

Ok, here goes:
An infinitely long tube filled with orange pingpong balls numbered 1,2,3,4,... is pouring balls (in order) at an ever-increasing rate: at first they are going at 1 ball per minute, but after every 10 balls pour, the rate doubles. The balls are being poured into a bucket, over which an evil gremlin is standing chucking out balls, starting with the lowest numbered ball in the bucket. At first he chucks out one ball in 10 minutes, but then waits half as much time after each ball before chucking out the next, so he chucks out ball #1 at the 10 minute mark, ball #2 at the 15 minute mark, ball #3 at 17:30, etc.

Question: how many balls are in the bucket after 10 minutes? After 15 minutes? After 17 and a half minutes? After 19:59? After 20 minutes?

p.s. If you have any other mathematical brainteasers, throw 'em at me!
-David
E pur si muove!

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« Reply #91 on: November 07, 2006, 07:34:53 PM »
Prove that any reciprocal lattice vector K is an integral multiple of the shortest parallel reciprocal lattice vector K0.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

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BOGWarrior89

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« Reply #92 on: November 07, 2006, 07:36:16 PM »
Quote from: "skeptical_scientist"
Oooh, it looks like we're giving Erasmus brainteasers, so I have one! This deals with infinite numbers, so I predict that I will pose the question, Erasmus will correctly answer it, and then five people will jump up to tell us both that not only is Erasmus wrong, but the question itself is meaningless gibberish.

Ok, here goes:
An infinitely long tube filled with orange pingpong balls numbered 1,2,3,4,... is pouring balls (in order) at an ever-increasing rate: at first they are going at 1 ball per minute, but after every 10 balls pour, the rate doubles. The balls are being poured into a bucket, over which an evil gremlin is standing chucking out balls, starting with the lowest numbered ball in the bucket. At first he chucks out one ball in 10 minutes, but then waits half as much time after each ball before chucking out the next, so he chucks out ball #1 at the 10 minute mark, ball #2 at the 15 minute mark, ball #3 at 17:30, etc.

Question: how many balls are in the bucket after 10 minutes? After 15 minutes? After 17 and a half minutes? After 19:59? After 20 minutes?

p.s. If you have any other mathematical brainteasers, throw 'em at me!


MEANINGLESS GIBBERISH!

Sorry, I seem to have jumped the gun.

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« Reply #93 on: November 07, 2006, 08:04:33 PM »
Quote from: "fathomak"
Prove that any reciprocal lattice vector K is an integral multiple of the shortest parallel reciprocal lattice vector K0.

From my googling, it seems that this has more to do with crystallography than with mathematics, so I'll try to put it into mathematical terms:

Claim: If L is a lattice in R^n, and v is a lattice vector, then v is an integral multiple of the shortest parallel lattice vector v'.
Proof: Suppose L is a lattice in R^n and v, v' are in L, with v' parallel to v. Then v is rv' for some real number r. If r is irrational, we can find an arbitrarily good approximation to 0 of the form n-mr for some natural numbers n and m, and so nv-mv' can be made arbitrarily close to 0, but is of course in the lattice. This contradicts the fact that lattices are discrete. Hence r is of the form p/q for some integers p and q, relatively prime. Since p and q are relatively prime, but the chinese remainder theorem, we can find integers m and n so that mp+nq=1. Then nv+mv'=nv+m(p/q)v=1/qv, so (1/q)v is in the lattice, and hence there is a lattice vector (1/q)v of which both v and v' are integer multiples. If v' is a shortest lattice vector parallel to v, v and v' are both integer multiples of some (1/q)v, so we must have (1/q)v=v', or else (1/q)v would be shorter than v'. Hence v is an integer multiple of v'.

In any case, by "mathematical brainteaser" I meant a problem which could be stated in simple terms but with an interesting and creative solution. This would seem to be a problem in complicated terms but with a fairly straightforward solution, assuming I am correctly guessing at the underlying mathematics.
-David
E pur si muove!

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« Reply #94 on: November 07, 2006, 08:06:08 PM »
Prove that a group of order 56 must have a nontrivial proper normal subgroup.


...Yes, I'm stealing these from textbooks.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

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Erasmus

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« Reply #95 on: November 07, 2006, 08:06:51 PM »
Quote from: "skeptical_scientist"
Question: how many balls are in the bucket after 10 minutes?


Assuming it takes no time for the pipe to spew a ball or for our evil gremlin to chuck one, and that both of these operations are completed before any point after 10 minutes has elapsed, I feel the best answer to this question is: nine balls.

Quote
After 15 minutes?


Ditto assumptions: eighteen balls.

Quote
After 17 and a half minutes?


Ditto, twenty-seven balls.

Quote
After 19:59?


Bite me.

Quote
After 20 minutes?


Zero balls.  Consider the state of ball 10n, numbering the balls from zero instead of one: it was placed in the bucket at time 10 + 5 + 2.5 + ... + 10/2^n minutes = 10 * (1 + 1/2 + ... + 1/2^n) minutes < 20 minutes.  Obviously balls 10(n-1)+1 through 10n-1 made it in before this time.  In other words, every ball has gotten into the bucket before the 20 minute mark.

However, by similar reasoning any ball you choose has also been removed by the 20 minute mark, though it's important to note that the gremlin is always removing balls slower than they are being added before the 20 minute mark has been reached, so he never tried to remove a ball that has not yet been added.  Just in case you were concerned.

Thus, there are no balls remaining in the bucket.
Why did the chicken cross the Möbius strip?

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Erasmus

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« Reply #96 on: November 07, 2006, 08:11:42 PM »
Quote from: "skeptical_scientist"
p.s. If you have any other mathematical brainteasers, throw 'em at me!


Hmm!  Stop me if you've heard this one: you are Little Red Riding hood and you are off to grandma's house.  Aside from cookies or whatever, it has been requested that you bring water from the river, a perfectly linear object a known distance that your house and grandma's house are both on the same side of, on the perfect Euclidean plane that is the countryside.

You want to take the shortest path to the river and thence to grandma's house... but your houses are not the same distance from the river!  What point on the river should you go to to get the water?  (Answers not using calculus are preferred).
Why did the chicken cross the Möbius strip?

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« Reply #97 on: November 07, 2006, 08:19:46 PM »
I've done something similar regarding the path of light moving through different mediums, though that was solved with calculus.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

?

Erasmus

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« Reply #98 on: November 07, 2006, 08:21:10 PM »
Quote from: "fathomak"
I've done something similar regarding the path of light moving through different mediums, though that was solved with calculus.


Well if you really want to use calculus that's fine.... really it's a hint that you don't need to.
Why did the chicken cross the Möbius strip?

0.9... = 1
« Reply #99 on: November 07, 2006, 08:21:33 PM »
what is the one thing that begins going, but ends coming?

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GeoGuy

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« Reply #100 on: November 07, 2006, 08:22:18 PM »
Quote from: "woopedazz"
what is the one thing that begins going, but ends coming?


"G"

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« Reply #101 on: November 07, 2006, 08:22:36 PM »
Perhaps I'll come back to it, but now I'm trying to focus on some physics homework, which is why I announced my resignation that seems to have had no effect on my posting thus far.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

?

Erasmus

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« Reply #102 on: November 07, 2006, 08:25:42 PM »
Quote from: "fathomak"
Perhaps I'll come back to it, but now I'm trying to focus on some physics homework, which is why I announced my resignation that seems to have had no effect on my posting thus far.


Oooh ooh what's the homework?
Why did the chicken cross the Möbius strip?

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« Reply #103 on: November 07, 2006, 08:27:23 PM »
Currently working on solid state physics.  I can't do a lot of notation for the problem I'm currently working on, so instead I'll divert my attention to paint so I can draw all of the notation and put it into a picture.  Give me some time and I'll post it if you like.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

0.9... = 1
« Reply #104 on: November 07, 2006, 08:27:47 PM »
Quote from: "Erasmus"
you are Little Red Riding hood and you are off to grandma's house. Aside from cookies or whatever, it has been requested that you bring water from the river...What point on the river should you go to to get the water?


Walk in a straight line to grandma's house and tell her to make lazy-ass uncle Joe get off of the couch and go get her a pail of water!  I'm just a little "girl" that already has a long trip ahead of "her"--not to mention the dangers of wolves along the way that I'm risking.  Uncle Joe can then walk to the river in a perpindicular (sp.) line to the river and back to minimize his travelling distance.
ooyakasha!

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« Reply #105 on: November 07, 2006, 08:31:48 PM »
Now that I think about it, that problem isn't similar to the one with light in different mediums.  In this case, the shortest path would be the one where the point she reaches on the river forms a right triangle with her house and her grandmother's house.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

?

Erasmus

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« Reply #106 on: November 07, 2006, 08:38:43 PM »
Quote from: "fathomak"
Now that I think about it, that problem isn't similar to the one with light in different mediums.  In this case, the shortest path would be the one where the point she reaches on the river forms a right triangle with her house and her grandmother's house.


Definintely not... imagine the case when some line perpendicular to the river exactly or very nearly intersects both houses.  On the other extreme, imagine both houses very close the river and yet very far apart.
Why did the chicken cross the Möbius strip?

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skeptical scientist

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« Reply #107 on: November 07, 2006, 08:39:06 PM »
Quote from: "fathomak"
Prove that a group of order 56 must have a nontrivial proper normal subgroup.


...Yes, I'm stealing these from textbooks.

I can tell.

Let G be a group of order 56. By sylow's theorem, there is at least one subgroub of order 7; in fact, there are n of them, and n must divide 8=56/7. Furthermore, n=1 mod 7, so n=8 or n=1.

If n=1, then there is a unique sylow 7-subgroup H, and since all conjugates of H are sylow 7-subgroups, all conjugates of H must be H, and hence H is normal.

If n=8, then G has 6*8=48 elements of order 7, and hence 7 elements whose orders are neither 1 nor 7. Also by sylow's theorem, G has a subgroup of order 8, which must contain these elements, and so G has a unique sylow 2-subgroup H. By the same argument, all conjugates of H are H, so H is normal.

Alright, I'm not answering any more boring questions. Good math brainteasers require little knowledge but do require ingenuity. The ones you are giving require little ingenuity but very specific knowledge about certain areas of math.

P.S. a gold star to Erasmus, who not only solved the problem correctly, but gave an explanation which should help illuminate the unusual properties of infinite processes and sets.
-David
E pur si muove!

0.9... = 1
« Reply #108 on: November 07, 2006, 08:41:04 PM »
I can be driven, yet have no wheels or feet. I can be sliced yet remain complete. What am i? (This one's easy)

I am an eight letter word, yet i only have one inside me (Also easy)

now another one:

I am famous in China, but infamous in Berlin. Add a monarch and the excercise may begin.

What bodypart would Adam and Eve NOT have, that you and i do?

k, how did u do?

0.9... = 1
« Reply #109 on: November 07, 2006, 08:43:22 PM »
Quote from: "Erasmus"
Quote from: "fathomak"
Now that I think about it, that problem isn't similar to the one with light in different mediums.  In this case, the shortest path would be the one where the point she reaches on the river forms a right triangle with her house and her grandmother's house.


Definintely not... imagine the case when some line perpendicular to the river exactly or very nearly intersects both houses.  On the other extreme, imagine both houses very close the river and yet very far apart.


Ah, good point.
 captain is sailing through the arctic. The first mate runs up and says to him, "captain, there is an iceberg dead ahead. What should we do?" The captain looks at the iceberg, then glances at his map and says, "there's no iceberg here! Keep going!"

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BOGWarrior89

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« Reply #110 on: November 07, 2006, 08:43:57 PM »
Quote from: "woopedazz"
I can be driven, yet have no wheels or feet. I can be sliced yet remain complete. What am i? (This one's easy)

I am an eight letter word, yet i only have one inside me (Also easy)

now another one:

I am famous in China, but infamous in Berlin. Add a monarch and the excercise may begin.

What bodypart would Adam and Eve NOT have, that you and i do?

k, how did u do?


You were supposed to come up with MATHEMATICAL brainteasers.

0.9... = 1
« Reply #111 on: November 07, 2006, 08:44:30 PM »
Quote from: "BOGWarrior89"
Quote from: "woopedazz"
I can be driven, yet have no wheels or feet. I can be sliced yet remain complete. What am i? (This one's easy)

I am an eight letter word, yet i only have one inside me (Also easy)

now another one:

I am famous in China, but infamous in Berlin. Add a monarch and the excercise may begin.

What bodypart would Adam and Eve NOT have, that you and i do?

k, how did u do?


You were supposed to come up with MATHEMATICAL brainteasers.


just try them, u should all do well

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BOGWarrior89

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« Reply #112 on: November 07, 2006, 08:46:33 PM »
Quote from: "fathomak"
Quote
There is no such thing as 100% certainty.


Are you 100% certain?


Yes.

Quote from: "My profile"
Past the point of no contradictions.

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skeptical scientist

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« Reply #113 on: November 07, 2006, 08:51:46 PM »
Quote from: "Erasmus"
Quote from: "skeptical_scientist"
p.s. If you have any other mathematical brainteasers, throw 'em at me!


Hmm!  Stop me if you've heard this one: you are Little Red Riding hood and you are off to grandma's house.  Aside from cookies or whatever, it has been requested that you bring water from the river, a perfectly linear object a known distance that your house and grandma's house are both on the same side of, on the perfect Euclidean plane that is the countryside.

You want to take the shortest path to the river and thence to grandma's house... but your houses are not the same distance from the river!  What point on the river should you go to to get the water?  (Answers not using calculus are preferred).


Much better brainteaser! And no, I hadn't seen it before. Of course this is a simple plug-n-chug problem using calculus, but without calculus requires a more interesting solution:

Suppose path c is an optimal solution. Let c' be the portion of c before intersecting the river for the first time, and c" be the portion of c after that point. Now imagine c" reflected across the river. Now c' together with c" reflected takes you to the point which is the location of grandma's house, reflected across the river, and does so in the most efficient way possible, since otherwise we could reflect back to get a more efficient way of getting to grandma's house via the river. But any path to the reflection of grandma's house must cross the river, so the optimal solution is a straight line. Hence, the best point on the river to get water is the point where the lines to your house and grandma's house make the same angle to the river.
-David
E pur si muove!

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« Reply #114 on: November 07, 2006, 08:52:45 PM »
im moving my brain teasers to another topic...in this thread still, just my own little topic  :D

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skeptical scientist

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« Reply #115 on: November 07, 2006, 08:55:27 PM »
I can be driven, yet have no wheels or feet. I can be sliced yet remain complete. What am i? (This one's easy)
--

I am an eight letter word, yet i only have one inside me (Also easy)
--envelope

now another one:

I am famous in China, but infamous in Berlin. Add a monarch and the excercise may begin.
--

What bodypart would Adam and Eve NOT have, that you and i do?
--seen this one.
-David
E pur si muove!

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Erasmus

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« Reply #116 on: November 07, 2006, 08:58:13 PM »
Quote from: "skeptical_scientist"
Hence, the best point on the river to get water is ...


Well done... here is another one which has just been sent to me: http://xkcd.com/blue_eyes.html (btw, the comic on that website is quite good as well).
Why did the chicken cross the Möbius strip?

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BOGWarrior89

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« Reply #117 on: November 07, 2006, 09:02:46 PM »
Quote from: "Erasmus"
Quote from: "skeptical_scientist"
Hence, the best point on the river to get water is ...


Well done... here is another one which has just been sent to me: http://xkcd.com/blue_eyes.html (btw, the comic on that website is quite good as well).


You fool!  I have two comics in my signature, and you never mentioned a thing about them!  GAH!

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Erasmus

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« Reply #118 on: November 07, 2006, 09:04:24 PM »
Quote from: "BOGWarrior89"
You fool!  I have two comics in my signature, and you never mentioned a thing about them!  GAH!


If I had mentioned them earlier, presumably you would have posted exactly the same response...
Why did the chicken cross the Möbius strip?

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BOGWarrior89

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« Reply #119 on: November 07, 2006, 09:06:45 PM »
Quote from: "Erasmus"
Quote from: "BOGWarrior89"
You fool!  I have two comics in my signature, and you never mentioned a thing about them!  GAH!


If I had mentioned them earlier, presumably you would have posted exactly the same response...


You can't prove that.