The Round Table
We have a circular dining table made of marble which had come down to us as a family heirloom. We also have some beautiful bone-china saucers that I recently brought from Japan.
Diameter of our table top is fifteen times the diameter of our saucers which are also circular. We would like to place the saucers on the table so that they neither overlap each other nor the edge of the table. How many can we place in this manner?
The answer is 187.
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1 solution
We can build concentric hexagons containing 1,6,12,18,24,30,36 and 42 circles. When R/r becomes sufficiently large there, will be room for extra circles.
if there is an even number of circles per side in last hexagon , an outsider can be placed centrally, if
R/r ≥ (1 + \sqrt { 3/2 } ) / (1 - \sqrt { 3/2 } ) i.e. if R/r ≥ 13.9
Two more "outsiders" can be put each side of this one,if
[ { (R + r) }^{ 2 }({ \sqrt { 3/2 } }^{ 2 } + { (2r) }^{ 2 } ] + r ≤ R
i.e if 0 ≤ { R }^{ 2 }/r^{ 2 } - 14 R/r - 15
ie if 0 ≤ (R/r+1)(R/r - 1)
i.e if R/r ≥ 15
There , in the given example three outsiders can be accommodated.
And the number of saucers that can be placed on the table are:
1 + 6 + 12 + 18 + 24 + 30 + 36 + 42 + (3 x 6) = 187