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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.

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

Even better, with area approximately 2463: enter image description here

Here's @MattF's suggestion, but it's worse in both dimensions: enter image description here

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

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  • $\begingroup$ 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? $\endgroup$
    – user44143
    Commented Jan 29, 2020 at 14:33
  • $\begingroup$ I added your suggested modification, but I guess the 4 is bigger than it looks. $\endgroup$
    – RobPratt
    Commented Jan 29, 2020 at 19:32
  • $\begingroup$ Sad. Thanks. Well done on your configuration! $\endgroup$
    – user44143
    Commented Jan 29, 2020 at 19:33
  • $\begingroup$ 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. $\endgroup$
    – Gerhard Paseman
    Commented Feb 1, 2020 at 17:47
  • $\begingroup$ 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. $\endgroup$
    – Gerhard Paseman
    Commented Feb 1, 2020 at 22:11

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