[seqfan] Re: Squares packings

Rob Pratt robert.william.pratt at gmail.com
Sun Oct 11 20:05:20 CEST 2020

```Via integer linear programming, I get instead
1 0 0 0 4 0 3 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1

Here's an optimal solution for n = 17, with the only 1x1 appearing in cell
(12,6):
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1 2 2 3 3 38 38 38 4 4 5 5 6 6 7 7 8 8
2 2 2 3 3 38 38 38 4 4 5 5 6 6 7 7 8 8
3 9 9 10 10 38 38 38 11 11 12 12 13 13 14 14 15 15
4 9 9 10 10 39 39 39 11 11 12 12 13 13 14 14 15 15
5 16 16 17 17 39 39 39 18 18 19 19 40 40 40 41 41 41
6 16 16 17 17 39 39 39 18 18 19 19 40 40 40 41 41 41
7 42 42 42 20 20 21 21 22 22 23 23 40 40 40 41 41 41
8 42 42 42 20 20 21 21 22 22 23 23 24 24 25 25 26 26
9 42 42 42 27 27 43 43 43 44 44 44 24 24 25 25 26 26
10 45 45 45 27 27 43 43 43 44 44 44 28 28 29 29 30 30
11 45 45 45 31 31 43 43 43 44 44 44 28 28 29 29 30 30
12 45 45 45 31 31 1 46 46 46 47 47 47 32 32 48 48 48
13 33 33 34 34 35 35 46 46 46 47 47 47 32 32 48 48 48
14 33 33 34 34 35 35 46 46 46 47 47 47 36 36 48 48 48
15 49 49 49 50 50 50 51 51 51 52 52 52 36 36 53 53 53
16 49 49 49 50 50 50 51 51 51 52 52 52 37 37 53 53 53
17 49 49 49 50 50 50 51 51 51 52 52 52 37 37 53 53 53

Clearly, even 2n can be partitioned into 2x2 only, and 3n can be
partitioned into 3x3 only.  Otherwise, it appears that the minimum is 1 for
n >= 11.

On Sun, Oct 11, 2020 at 3:11 AM <michel.marcus at free.fr> wrote:

>
> Hello Seqfans,
>
>
> "Images du CNRS" French site 4th problem for September 2020 is:
> .
> What is the minimum number of 1X1 pieces to fill a 23X23 square with 1X1,
> 2X2, and 3X3 pieces.
> Problem is at http://images.math.cnrs.fr/Septembre-2020-4e-defi.html.
> Solution is at http://images.math.cnrs.fr/Octobre-2020-1er-defi.html
> (click in Solution du 4e défi de septembre)
>
>
> I wondered what we get for other squares and painstakingly obtained for
> n=1 up to n=23:
> 1 0 0 0 4 0 3 0 0 0 1 0 1 0 0 0 4 0 4 0 0 0 1
>
>
> Do you see some mistakes? Is it possible to extend it?
>
>
> Thanks. Best.
> MM
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```