You are either not logged in or not registered with our community. Click here to register.
 
December 10, 2016, 08:59:08 AM

Welcome, Guest. Please login or register.
Did you miss your activation email?

Login with username, password and session length

Click here if you are having problems.
Default Wide Screen Beige Lilac Rainbow Black & Blue October Send us your theme!

Hark!  The Herald!
Holiday Issue 2016

Wiki Blogs Dicebot

Author Topic: Trouble with Circles. (Mathematical Proof)  (Read 614 times)

0 Members and 1 Guest are viewing this topic.

Offline ReanimateMagnusTopic starter

Trouble with Circles. (Mathematical Proof)
« on: July 31, 2012, 01:02:39 PM »
I thought it would be interesting to see if anyone could come up with this answer.

Assume you have thirteen equal circles each repeating a pattern of symetry touching each at one point. The first layer would have 1 circle. The second would have 6 and the third would also have 6. If you draw a line from the center of each circle to each other center. How many of those lines would only go through one other circles center?
« Last Edit: July 31, 2012, 01:04:37 PM by ReanimateMagnus »

Offline Oniya

  • StoreHouse of Useless Trivia
  • Oracle
  • Carnite
  • *
  • Join Date: Sep 2008
  • Location: Just bouncing through. Hi! City of Roses, Pennsylvania
  • Gender: Female
  • One bad Motokifuka. Also cute and FLUFFY!
  • My Role Play Preferences
  • View My Rolls
  • Referrals: 3
Re: Trouble with Circles. (Mathematical Proof)
« Reply #1 on: July 31, 2012, 02:21:00 PM »
Insufficient information.

Are you talking about circles (arranged with their centers on the same line) or spheres (arranged with their centers on the same plane)?  Also, the arrangement of the 6 in the second and third layers becomes very important if they are spheres.  Are they arranged in a ring-shape (which would have a hole large enough to fit another equally-sized sphere), or a triangle?  Is the third layer the same arrangement?  If so, is it simply raised up, or is it offset like in a stack of oranges?  Is the singleton centered under the second layer?

Offline Trieste

  • Faerie Queen; Her Imperial Lubemajesty; Willing Victim
  • Dame
  • Carnite
  • *
  • Join Date: Apr 2005
  • Location: In the middle of Happily Ever After with a dark Prince Charming.
  • Gender: Female
  • I am many things - dull is not one of them.
  • My Role Play Preferences
  • View My Rolls
  • Referrals: 4
Re: Trouble with Circles. (Mathematical Proof)
« Reply #2 on: July 31, 2012, 02:29:36 PM »
Needs a three-dimensional representation, seems like. Something like Jmol for spatial representations.

Offline ReanimateMagnusTopic starter

Re: Trouble with Circles. (Mathematical Proof)
« Reply #3 on: July 31, 2012, 03:08:22 PM »
Insufficient information.

Are you talking about circles (arranged with their centers on the same line) or spheres (arranged with their centers on the same plane)?  Also, the arrangement of the 6 in the second and third layers becomes very important if they are spheres.  Are they arranged in a ring-shape (which would have a hole large enough to fit another equally-sized sphere), or a triangle?  Is the third layer the same arrangement?  If so, is it simply raised up, or is it offset like in a stack of oranges?  Is the singleton centered under the second layer?

I am talking about circles of equal area. There is only one arangment you can possibly do with 6 equal circles around a center equal circle. I didn't say anything about spheres so I don't know where you got that from. Also I said they create a symetrical pattern. Oranges? Again I don't know how you got spheres from this.

Offline Oniya

  • StoreHouse of Useless Trivia
  • Oracle
  • Carnite
  • *
  • Join Date: Sep 2008
  • Location: Just bouncing through. Hi! City of Roses, Pennsylvania
  • Gender: Female
  • One bad Motokifuka. Also cute and FLUFFY!
  • My Role Play Preferences
  • View My Rolls
  • Referrals: 3
Re: Trouble with Circles. (Mathematical Proof)
« Reply #4 on: July 31, 2012, 06:51:00 PM »
It was your word 'layers' that confused the issue, as that normally indicates something flat (like a line or a plane).  I'm going to assume you meant something like 'rounds'.  Even so, there are two distinct symmetrical arrangements of circles that can be made with the conditions you described:  One like a large asterisk * with two circles on each branch:

..O....O.
....OO...
OOOOO
....OO...
..O....O.

and one like a six-pointed star:

....O...
OOOO
.OOO.
OOOO
....O...

Offline ReanimateMagnusTopic starter

Re: Trouble with Circles. (Mathematical Proof)
« Reply #5 on: July 31, 2012, 06:53:36 PM »
I see the confusion it would look more like the first one. When I hear layer I think of like an onion.

Offline Shjade

Re: Trouble with Circles. (Mathematical Proof)
« Reply #6 on: July 31, 2012, 06:57:14 PM »
Since the question doesn't restrict the lines to being straight or to passing through only the centers of circles, they would cross through as many or as few as you choose to intersect.

Offline Oniya

  • StoreHouse of Useless Trivia
  • Oracle
  • Carnite
  • *
  • Join Date: Sep 2008
  • Location: Just bouncing through. Hi! City of Roses, Pennsylvania
  • Gender: Female
  • One bad Motokifuka. Also cute and FLUFFY!
  • My Role Play Preferences
  • View My Rolls
  • Referrals: 3
Re: Trouble with Circles. (Mathematical Proof)
« Reply #7 on: July 31, 2012, 07:09:57 PM »
In mathematical exercises, 'straight line' is redundant.  All lines are defined as straight.  The fact that endpoints are given (from the center, to the center), further defines them as segments and not infinite lines.

Let's see.  Taking the upper left circle to start with, the line from that one to the four circles on the adjacent branches, as well as the one touching it would not intersect any other centers.  Likewise, the ones going to the 'roots' of the non-opposite branches would not intersect any centers.  The one to the center circle would go through one, as would the two that go to the tips of the non-opposite branches.  The one going to the 'root' of the opposite branch would go through two centers, and the one going to the tip of the opposite branch would go through three, so those don't count.

So, for each outer circle, there would be three lines that go through one other circle's center.

Each inner circle would have exactly one line that goes through one other circle's center (namely, the one opposite the center circle).

The center circle would have six lines (from the center to the tip of each branch) that go through one other circle's center.

So, 6+6+18 = 30 - but, each of those segments is actually counted twice (once from each end), so my final answer is 15.

Offline ReanimateMagnusTopic starter

Re: Trouble with Circles. (Mathematical Proof)
« Reply #8 on: July 31, 2012, 09:08:44 PM »
Here's the answer

Spoiler: Click to Show/Hide