# Packing circles with radii 1, 2, 3, ..., n in a rectangle

For each positive integer n, let $$a_n$$ be the area of the smallest rectangle whose area is a whole number, and inside which it is possible to pack all n circles of radii 1, 2, 3, ..., n respectively (with no overlaps).

Is it possible to determine $$a_n$$ precisely?

For example $$a_{12}$$ is at most 2466 (https://puzzling.stackexchange.com/questions/92949/my-mothers-dish-collection), and can perhaps be proved to be precisely that.

• It's certainly not precisely that. One of the comments links a configuration in a square of area ~2518.16. There has been a lot of work into packing circles into squares, with very few bounds actually proven optimal. Jan 25 '20 at 16:03
• @Wojowu You are right. Have edited accordingly. Jan 25 '20 at 16:22
• On the same website (last updated in 2015) are the best known results up to N = 72. Jan 26 '20 at 4:14

Here's a better solution for $$n=12$$, with area approximately 2496:

Even better, with area approximately 2463:

Here's @MattF's suggestion, but it's worse in both dimensions:

@GerhardPaseman, if I consider only circles 6 through 12, this is the best solution I have found:

• If you replace 4 by 3, move 12 up and to the left, and then put 4 tangent to 12 and 7, can you cut off some area on the right? Jan 29 '20 at 14:33
• I added your suggested modification, but I guess the 4 is bigger than it looks. Jan 29 '20 at 19:32
• Sad. Thanks. Well done on your configuration! Jan 29 '20 at 19:33
• Suppose you ignore plates one through five temporarily. How small a rectangle do you get for the other seven? (and does it involve a four grouping of 9 12 10 11 with 8 6 7 in a column?) Gerhard "Looking For A Dish Pattern" Paseman, 2020.02.01. Feb 1 '20 at 17:47
• Thanks. I was expecting something slightly different, which would lead to an easy insertion of the remaining disks. However, I am not seeing how to improve the arrangement you provided. Gerhard "Now Looking For Place Settings" Paseman, 2020.02.01. Feb 1 '20 at 22:11