bonanova Posted January 9, 2009 Report Share Posted January 9, 2009 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. 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. 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 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? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted January 9, 2009 Report Share Posted January 9, 2009 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. 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. 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 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? Quote Link to comment Share on other sites More sharing options...
0 Guest Posted January 9, 2009 Report Share Posted January 9, 2009 (edited) After three attempts i got 181 using the "pencil & stencil" methodThe 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. 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. 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 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 January 9, 2009 by Barron Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted January 9, 2009 Author Report Share Posted January 9, 2009 One calculation + guess saying no / One analog simulation saying yes, but barely.Both in ballpark. No definitive optimal answer - yet. Quote Link to comment Share on other sites More sharing options...
0 HoustonHokie Posted January 9, 2009 Report Share Posted January 9, 2009 I get 187 plates Quote Link to comment Share on other sites More sharing options...
0 Guest Posted January 9, 2009 Report Share Posted January 9, 2009 I get 187 plates Doesn't look like the big shape is actually a circle. Is that an image-reduction issue? Quote Link to comment Share on other sites More sharing options...
0 bonanova Posted January 9, 2009 Author Report Share Posted January 9, 2009 I get 187 plates Tipping the hat to HH! Nice job! The ratio of 15 exactly permits the 6 groups of 3 plates to be added to the outermost hexagon. Quote Link to comment Share on other sites More sharing options...
0 Guest Posted January 9, 2009 Report Share Posted January 9, 2009 I can fit...173 Quote Link to comment Share on other sites More sharing options...
0 unreality Posted January 10, 2009 Report Share Posted January 10, 2009 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 Quote Link to comment Share on other sites More sharing options...
0 Guest Posted January 10, 2009 Report Share Posted January 10, 2009 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... Quote Link to comment Share on other sites More sharing options...
0 unreality Posted January 10, 2009 Report Share Posted January 10, 2009 Good point - I stand corrected Quote Link to comment Share on other sites More sharing options...
Question
bonanova
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.
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.
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
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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