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Trouble with Circles. (Mathematical Proof)

Started by ReanimateMagnus, July 31, 2012, 01:02:39 PM

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ReanimateMagnus

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?

Oniya

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?
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Trieste

Needs a three-dimensional representation, seems like. Something like Jmol for spatial representations.

ReanimateMagnus

Quote from: Oniya 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?

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.

Oniya

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...
"Language was invented for one reason, boys - to woo women.~*~*~Don't think it's all been done before
And in that endeavor, laziness will not do." ~*~*~*~*~*~*~*~*~*~*~Don't think we're never gonna win this war
Robin Williams-Dead Poets Society ~*~*~*~*~*~*~*~*~*~*~*~*~*~Don't think your world's gonna fall apart
I do have a cause, though.  It's obscenity.  I'm for it.  - Tom Lehrer~*~All you need is your beautiful heart
O/O's Updated 5/11/21 - A/A's - Current Status! - Writing a novel - all draws for Fool of Fire up!
Requests updated March 17

ReanimateMagnus

I see the confusion it would look more like the first one. When I hear layer I think of like an onion.

Shjade

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.
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Oniya

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.
"Language was invented for one reason, boys - to woo women.~*~*~Don't think it's all been done before
And in that endeavor, laziness will not do." ~*~*~*~*~*~*~*~*~*~*~Don't think we're never gonna win this war
Robin Williams-Dead Poets Society ~*~*~*~*~*~*~*~*~*~*~*~*~*~Don't think your world's gonna fall apart
I do have a cause, though.  It's obscenity.  I'm for it.  - Tom Lehrer~*~All you need is your beautiful heart
O/O's Updated 5/11/21 - A/A's - Current Status! - Writing a novel - all draws for Fool of Fire up!
Requests updated March 17

ReanimateMagnus

Here's the answer

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