5
$\begingroup$

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.

$\endgroup$
3

1 Answer 1

14
$\begingroup$

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

$\endgroup$
7
  • $\begingroup$ 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
    Jan 29, 2020 at 14:33
  • $\begingroup$ I added your suggested modification, but I guess the 4 is bigger than it looks. $\endgroup$
    – RobPratt
    Jan 29, 2020 at 19:32
  • $\begingroup$ Sad. Thanks. Well done on your configuration! $\endgroup$
    – user44143
    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. $\endgroup$ 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. $\endgroup$ Feb 1, 2020 at 22:11

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.