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Plates on a circular table


bonanova
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The waiter wondered whether the plates stacked in his cupboard,

180 in all, would fit on the circular table without overlapping -

each other or the edge of the table. That is, someone under the

table would not see any part of a plate hanging over the edge. :huh:

He began by centering a line of plates along a table diameter,

and found that 15 plates exactly fit the criteria he had in mind.

That is, the diameter of the table was exactly 15 times the

diameter of the plates. B))

A few more plates began to fit a pattern, and the waiter decided

to answer his question with pencil and paper, rather than to

move all the plates; and perhaps have to wash them all :o

Forgot to mention: you are the waiter. ;)

What did you find?

If 180 is not the exact number that will fit, what is that number?

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I found that

in fact quicker to lay them all out and see than hurt my head with math!

But back to the puzzle, I will say that 180 plates will *not* fit on the table.

Assume:

plate diameter = 2 units

Then:

table diameter = 30 units

plate area = pi square units

table area = 225 pi square units

Hexagonal packing of circles is the most efficient at about 90.1% space utilization, so with an optimal container shape you could get 202 plates in.

Square packing has an efficiency of 78%, so with an optimal container shape you could get 175.5 plates in.

The circular 'container' is far from optimal, so I'd guess you'd end up at around 75% efficiency, or

168 plates.

The waiter wondered whether the plates stacked in his cupboard,

180 in all, would fit on the circular table without overlapping -

each other or the edge of the table. That is, someone under the

table would not see any part of a plate hanging over the edge. :huh:

He began by centering a line of plates along a table diameter,

and found that 15 plates exactly fit the criteria he had in mind.

That is, the diameter of the table was exactly 15 times the

diameter of the plates. B))

A few more plates began to fit a pattern, and the waiter decided

to answer his question with pencil and paper, rather than to

move all the plates; and perhaps have to wash them all :o

Forgot to mention: you are the waiter. ;)

What did you find?

If 180 is not the exact number that will fit, what is that number?

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After three attempts i got 181 using the "pencil & stencil" method

The waiter wondered whether the plates stacked in his cupboard,

180 in all, would fit on the circular table without overlapping -

each other or the edge of the table. That is, someone under the

table would not see any part of a plate hanging over the edge. :huh:

He began by centering a line of plates along a table diameter,

and found that 15 plates exactly fit the criteria he had in mind.

That is, the diameter of the table was exactly 15 times the

diameter of the plates. B))

A few more plates began to fit a pattern, and the waiter decided

to answer his question with pencil and paper, rather than to

move all the plates; and perhaps have to wash them all :o

Forgot to mention: you are the waiter. ;)

What did you find?

If 180 is not the exact number that will fit, what is that number?

Edited by Barron
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plate radius = 1 u

table radius = 15 u

area of table: 225pi u2

area of single plate: pi u2

So if the circles fit perfectly (which they don't of course), 225 could fit.

So we just need to find a fitting algorithm that's 180/225 = 80% efficient or more (ie, 80% or more of the circle filled)

I'm in a hurry, so I don't have time to figure it out myself, but Wolfram says:

http://mathworld.wolfram.com/CirclePacking.html

efficiency of hexagonal packing = (1/6)*pi*sqrt(3) ~= .9069

90.7% is more than efficient enough to fill the area

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plate radius = 1 u

table radius = 15 u

area of table: 225pi u2

area of single plate: pi u2

So if the circles fit perfectly (which they don't of course), 225 could fit.

So we just need to find a fitting algorithm that's 180/225 = 80% efficient or more (ie, 80% or more of the circle filled)

I'm in a hurry, so I don't have time to figure it out myself, but Wolfram says:

http://mathworld.wolfram.com/CirclePacking.html

efficiency of hexagonal packing = (1/6)*pi*sqrt(3) ~= .9069

90.7% is more than efficient enough to fill the area

In theory, but in practice, the maximum efficiency is for an infinite plane. In this case, the optimum packing only occurs if the ratio of table diameter to plate diameter is a multiple of sqrt(3), and in that case it would still be less than pi/(2*sqrt(3)). You can guess that it is enough, but until you actually try it out, you will never know...

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