Circles

uzurpatorul · 2009-09-05 19:55:25

The following is a variant of a problem that appeared on a college entrance examination. In the figure below, the radius of circle A is 2 units, the radius of circle B is 3 units. Starting from the position shown in the figure, circle A rolls around circle B. At the end of how many revolutions of circle A will the center of circle A first reach its starting point?

Reference:
http://www.projecteureka.org/problem/question/561

bobbym · 2009-09-06 09:36:43

HI uzurpatorul;

phrontister · 2009-09-06 19:50:59

That's a very good brainteaser, uzurpatorul!

I thought it was easy and entered my answer into the projecteureka site link...which said I was wrong!

I played around with it some more but couldn't come up with anything different, and so I cheated by peeking at Bobby's answer - which was different from mine (as I knew it would be). Hmm...

So I did this (including some more cheating):-

Last edited by phrontister (2009-09-07 01:19:32)

bobbym · 2009-09-07 00:32:06

Hi phrontister;

Thats a really good, I also rolled circles around in a geometry program. Came up with 2 answers 2.5 or 1872653478596874152638476536475110010132, I chose the smaller one.

phrontister · 2009-09-07 14:41:54

Hi Bobby,

I decided against doing a practical coin test on that larger result found by your geometry program when I realised I'd have to stay up at least all night to do it...and I wanted to catch some of the US Open (live-streaming SopCast).

The threat of getting tenosynovitis from sending the smaller coin whizzing repeatedly around the larger one put me off too, as did having to find a calculator with a large enough mantissa to keep count...and I'd have to operate that with my other hand, unless I could devise some sort of sensor for it.

Keeping count mentally was, for me, quite out of the question (I'm not even mildly autistic)...once the number got too large to 'say' quickly the whizzing would slow down, eventually to less than a crawl, to the point where there wouldn't be enough time left for me to continue the test, as young as I might be right now.

Anyway, then I realised that 1872653478596874152638476536475110010132 was incorrect and switched on the tennis.

I wonder what your geometry program was looking for. I guess it thought that it found it, whatever it was, because it printed out the result. Or maybe all this spinning business was too much for its constitution...

For an integer solution I suppose the puzzle question could be extended with the addition of something like "and starting orientation" after "starting point".

Last edited by phrontister (2009-09-08 05:00:49)

bobbym · 2009-09-07 15:58:42

Hi phrontister;

I was just joking about that number. It took me a long time to get the answer even using the program, and now I can't duplicate it.

phrontister · 2009-09-07 17:05:20

Hi Bobby,

I was just joking about that number.

Yes...I'd spotted it and ran with it until my tongue started to hurt my cheek.

I thought it was very unlike you to post an incorrect answer.

bobbym · 2009-09-07 17:11:36

Hi phrontister;

Heck, I make plenty of mistakes, math is very demanding. Most of my posts are done in obscurity. I could say anything I want, like 4 is a prime and no one would ever see it. I tend to like it that way.

Last edited by bobbym (2009-09-07 17:15:08)

phrontister · 2009-09-07 17:22:31

Hi Bobby,

I was going to say - before I saw your edit - that I've joined quittyqat's team.

I'm just waiting now for someone to comment about my comment that 1872653478596874152638476536475110010132 is incorrect.

bobbym · 2009-09-08 10:23:14

Hi phrotister;

Did you know it factors into:

Isn't that amazing.

phrontister · 2009-09-08 11:31:14

Hi Bobby,

Yes...that is amazing. I'd have thought (without thinking) that there would be more unique factors than just six, even though a prime that large, doubled, has only 2 factors (obviously) and would look just as multi-factorizable.

I used a little 84KB freeware calculator with a 5011-digit accuracy to check your findings...which took it some time, and my cpu began to glow during the crunching process. But I wanted to reply to you before I died, which wouldn't have been possible if I'd tried to do the factorization longhand.

Did you look at the two questions I posed earlier? They may not look like real questions because I didn't use question-marks...but they are.

For an integer solution I suppose the puzzle question could be extended with the addition of something like "and starting orientation" after "starting point".

I'm just waiting now for someone to comment about my comment that 1872653478596874152638476536475110010132 is incorrect.

Last edited by phrontister (2009-09-08 11:31:57)

bobbym · 2009-09-08 14:01:54

Hi phrontister;

Yes, I am sorry. I didn't comment about the first question because I didn't want to give the puzzle poster any ideas to make it harder. I have a general solution on hand, which I am keeping secret from him.

I'm just waiting now for someone to comment about my comment that 1872653478596874152638476536475110010132 is incorrect.

I kept silent and am keeping silent about this one for a different reason.

phrontister · 2009-09-09 11:16:57

bobbym wrote:

Did you know it factors into:
Isn't that amazing.

What is amazing about that, Bobby?

bobbym · 2009-09-09 11:20:06

Hi phrontister;

Thats the point, it's very uninteresting, but that makes it interesting.

phrontister · 2009-09-09 11:55:55

Oh...I see. And there I was looking for something D&M! The degree of my amazement is waning rapidly.

Last edited by phrontister (2009-09-09 22:34:31)

bobbym · 2009-09-09 12:54:22

Hi phrontister;

Look at it like this: Take a finite list of numbers that are uninteresting. Now order them by how uninteresting they are. Look at the top one, it is the most uninteresting. But the fact that it is the most uninteresting gives it a certain distinction, a certain degree of interest. So it is not uninteresting, it is interesting. You must remove it from the list. Now you look at the list again the new top of the list is the most uninteresting but that makes it distinctive and interesting. You remove and repeat until the list is empty. Proof that they are all interesting.

Last edited by bobbym (2009-09-09 12:55:31)

mathsyperson · 2009-09-09 22:34:37

That assumes that the amount of uninteresting numbers is finite.

phrontister · 2009-09-09 22:39:39

Hi Bobby,

I tried that, but it didn't work for me.

I got together a finite list of uninteresting numbers, as you said to do, but failed to achieve a listing based on their uninterestingness as they were all equally uninteresting.

I'm sure that my lack of sleep from having stayed up half the night to watch the US Open tennis had nothing to do with that perception.

Finding that equally-boring group of numbers was probably just a fluke, but I wonder if (a) it is unique; or (b) there are a finite number of others; or (c) there are infinitely more such equally-boring groups.

I even considered listing them Every Which Way But Loose (starring Clint Eastwood) but dispensed with Every Which Way because of any preconceived notions I may have had that the number at the top (or bottom, or elsewhere) or left (or right, or elsewhere) - or any other Which Way - was of most interest, and so I went for Loose, listing them circularly in random order in an ever-spinning motion.

And, wouldn't you know it, after watching a few rounds of that the numbers were even more equally-uninteresting than before!

Last edited by phrontister (2009-09-10 01:52:24)

phrontister · 2009-09-09 22:56:55

Hi mathsyperson,

mathsyperson wrote:

That assumes that the amount of uninteresting numbers is finite.

I think that the number of uninteresting numbers in existence (or even not yet in existence) may, or may not (possibly depending on, inter alia, the observer and their state of mind), be either finite or infinite, and the finite list for Bobby's exercise is chosen from them.

Last edited by phrontister (2009-09-10 03:59:37)

phrontister · 2009-09-10 03:57:00

Hi Bobby,

bobbym wrote:

I could say anything I want, like 4 is a prime and no one would ever see it.

These six people certainly would've missed it! Scary!

Four is a prime number

Last edited by phrontister (2009-09-10 04:02:13)

bobbym · 2009-09-10 04:39:22

Hi phrontister;

Great vid, I don't think that contestant was picked because of her mathematical knowledge.

Math Is Fun Forum

#1 2009-09-05 19:55:25

Circles

#2 2009-09-06 09:36:43

Re: Circles

#3 2009-09-06 19:50:59

Re: Circles

#4 2009-09-07 00:32:06

Re: Circles

#5 2009-09-07 14:41:54

Re: Circles

#6 2009-09-07 15:58:42

Re: Circles

#7 2009-09-07 17:05:20

Re: Circles

#8 2009-09-07 17:11:36

Re: Circles

#9 2009-09-07 17:22:31

Re: Circles

#10 2009-09-08 10:23:14

Re: Circles

#11 2009-09-08 11:31:14

Re: Circles

#12 2009-09-08 14:01:54

Re: Circles

#13 2009-09-09 11:16:57

Re: Circles

#14 2009-09-09 11:20:06

Re: Circles

#15 2009-09-09 11:55:55

Re: Circles

#16 2009-09-09 12:54:22

Re: Circles

#17 2009-09-09 22:34:37

Re: Circles

#18 2009-09-09 22:39:39

Re: Circles

#19 2009-09-09 22:56:55

Re: Circles

#20 2009-09-10 03:57:00

Re: Circles

#21 2009-09-10 04:39:22

Re: Circles

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