# [seqfan] Re: [math-fun] Re: Squares packings

Fred Lunnon fred.lunnon at gmail.com
Tue Oct 13 03:11:34 CEST 2020

```  I don't know --- I was hoping that we have some seasoned packer of squares
on hand who can tell us what might be known.

In the meantime, just to set up a formal aunt Sally:
Conjecture: For natural  n mod 6 in {1, 5} :
(A) no packing of a  n x n  box by  1 x 1,  2 x 2,  3 x 3  tiles
exists with no  1 x 1  tile;
(B) there exists some packing with only a single  1 x 1  tile.

WFL

On 10/13/20, rcs at xmission.com <rcs at xmission.com> wrote:
> Is it established that large 6k+-1 squares require any 1x1s?  --Rich
>
> -----
> Quoting Fred Lunnon <fred.lunnon at gmail.com>:
>> << Sounds good for the upper bound.  I have also confirmed the lower bound
>> up
>> through n = 100. >>
>>
>>   Unclear: does your program establish that  _NO_ solution exists without
>> some 1x1 tile for edge length  n mod 6 in {1, 5}  &  n < 100 ?
>>
>>   If so, this suggests an interesting conjecture ...    WFL
>>
>>
>>
>> On 10/12/20, Rob Pratt <robert.william.pratt at gmail.com> wrote:
>>> Sounds good for the upper bound.  I have also confirmed the lower bound
>>> up
>>> through n = 100.
>>>
>>> On Mon, Oct 12, 2020 at 3:06 AM Jack Grahl <jack.grahl at gmail.com> wrote:
>>>
>>>> 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/
>>>> >
>>>>
>>>> --
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>>
>>>
>>> --
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>>>
>>
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```