Number of plates that can be placed on the table so that they neither overlap each other nor the edge of the table?

Question

Table in my room is round in shape and its radius is 15 times the radius of our plates, which are also round in shape. Find the number of plates that can be placed on the table so that they neither overlap each other nor the edge of the table ?

MY Solution :- let the radius of plates be $r$ Then radius of table is $15 r$

Number of plates = $225πr^2/πr^2$ = $225$

I have doubt that is my solution is correct or not ?

Answer 1

You have correctly obtained an upper bound, but the best known lower bound seems to be $187$: http://hydra.nat.uni-magdeburg.de/packing/cci/d16.html

Answer 2

The area of $225$ plates equals the area of the table exactly. So we could only place the $225$ plates on the table if there were no gaps between them. This would be possible with square plates on a square table, but it is impossible to arrange round plates in a manner that does not leave holes.

The exact answer to this problem is open, and it is open even for much smaller problems; http://hydra.nat.uni-magdeburg.de/packing/cci/ summarizes the state of the art. See also Wikipedia's article on circle packing in a circle.

In particular, as shown below:

• We know a way to arrange $187$ plates on a table with radius slightly above $14.989$ times the radius of a plate. So we could also arrange those $187$ plates on a table with radius $15$ times the radius of a plate.
• The best arrangement of $188$ plates known requires a table with radius more than $15.028$ times the radius of a plate. So it cannot be implemented here; to put $188$ (or more) plates on your table would require a better solution than what is currently the state of the art.

(The images are also taken from http://hydra.nat.uni-magdeburg.de/packing/cci/)