[seqfan] Re: Squares packings

Jack Grahl jack.grahl at gmail.com
Mon Oct 12 00:39:57 CEST 2020


If we have a solution with 1 small square for n=11 and n=13, doesn't that
imply a solution with 1 small square for all larger numbers 6k+1 and 6k-1
(ie all for which the solution is greater than zero)?

Simply form an L-shape with width 6, to extend a solution for 6k+1 to a
solution for 6(k+1)+1. The L shape is made up of a 6x6 square, and two
6x(6k+1) strips. Since any number >1 can be formed by a sum of 2's and 3's,
we can always make the strips. Same for 6k-1 of course.

Jack Grahl

On Sun, 11 Oct 2020, 20:25 Rob Pratt, <robert.william.pratt at gmail.com>
wrote:

> 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/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>


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