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

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 DENNIS GELLER Posts: 16 Registered: 12/6/04
Re: how many coins?
Posted: Apr 7, 1995 5:01 PM

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,
etc.

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.

===============================

Jim:

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?

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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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