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Topic: how many coins?
Replies: 5   Last Post: Apr 8, 1995 3:01 PM

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Posts: 16
Registered: 12/6/04
Re: how many coins?
Posted: Apr 7, 1995 5:01 PM
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I can't draw with dots as well as Jim, but suppose we take his configuration.
Alas, I cleverly deleted the problem. If I don't have it right, I think the
same construction can be used to explore the correct parameters.)
I'm taking that to be :
Coins with a diameter of one cm
A band 2 cm high and very long (1000 cm?)
Two coins on top of each other, flush left, followed by a coin whose
center is 1 cm high, wedged next to the other two, followed by two
coins on top of each other, as close as possible to the third,

Consider just the first three coins; Let A and B be the ones on top of each
other (A on top) and C be the third. Draw the equilater triangle connecting
their centers. Now drop a vertical line from the center of C to its
circumference (point x). Point x, together with the centers of B and C form a
right triangle, with the right angle at x. The hypotenuse of the triangle is
1 cm; it is a 30-60-90 triangle with the 30 degree angle at the center of B.
So, the line from the center of B to x is sin 60 = .8660.

Now consider the rectangle formed by the upper, lower and left sides of the
band, and the line from x through the center of C, extended to the top of the
band. This rectangle contains 2.5 circles. It uses up 0.5 + .866 = 1.366 of
the length of the band. The next 1.366 of the band also contains 2.5 circles,
and so on.

Therefore, this packing method yields a density of 2.5 circles/1.366 linear cm,
or 1.830 ci/cm. By contrast, the obvious packing, 2 above 2, next to 2 above 2,
etc, has 2 circles every cm. This would suggest that Jim's construction does
not pack coins with sufficient density to meet the constraints.



Can you prove this?

The area of the rectangle is 2000 cm^2, and the circle is pi/4 cm^2. So,
if the coing was "soft", we can fit 2000 / (pi/4) ~ 2500. The question
is, off setting the coins will really reduce the waste.

By the way, the name Graham sounds familiar. Does anyone have a reference?

* Tad Watanabe e-mail *
* Dept. of Mathematics watanabe-t@toe.towson.edu *
* Towson State University watanabe-t@towsonvx.bitnet *
* Towson, MD 21204 tad@midget.towson.edu *
* (410) 830 - 3585 (410) 830 - 4149 FAX *

On Fri, 7 Apr 1995, Jim Osborn wrote:

> There are three rows of coins instead of two
> .. ..
> . . . .
> . . .. . .
> .. . . ..
> .. . . ..
> . . .. . .
> . . . .
> .. ..
> There are more coins packed into the middle row then are lost by the
> missing pairs of coins.

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