How many can we place in this manner?
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?
We can consider a saucer as a circle of Radius R.
Draw a circle and 6 circles can be drawn on its circumference touching each other. If you join centers of the circles a hexagon will be created having edge of length 2R. Similarly another 12 circles can be drawn on the circumference of these six circles. A hexagon can be made by joining centers of these 12 circles having edge length = 3R.
So, concentric hexagons can be created having an A.P. of circles: 1, 6, 12,18, 24, 30, 36, 42. We need to stop here because the if we join any two opposite vertices of the largest hexagon, there will be 15 circles in line.
If there are even no. of circles per side of the outer most hexagon, an outsider can be placed centrally. and 2 more can be placed on circumference of this one. Like this, 3 outsiders can be there.
So in total, 1+6+12+18+24+30+36+42+ (3*6) = 187 saucers.
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