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

Fred Lunnon fred.lunnon at gmail.com
Tue Oct 13 18:34:07 CEST 2020


  Looks good to me.    WFL

On 10/13/20, Leonardo Costa <leonardocostalesage at gmail.com> wrote:
> Part (A) can be settled by using the same argument as in the original 23x23
> problem. The idea is to paint the squares of the grid white and black.
> To be more precise, all squares in odd columns will be black and all even
> ones will be white. Observe that there are as many black and
> white squares in any 2x2 square you place and any 3x3 square you place will
> have 3 more white squares than black ones or vice versa.
>
> Now suppose we can cover a nxn square (n = 6k+1, 6k-1) with 2x2 and 3x3
> pieces. Then, the difference between the amount of black and white squares
> will be n, but it will also be a multiple of 3, because only 3x3 squares
> cover a different amount of black and white squares. But 3 doesn't divide
> n, contradiction.
>
> So if I haven't made any mistakes, I think this problem is settled. Good
> job everyone!
>
> Leonardo Costa
>
> On Tue, 13 Oct 2020 at 03:39, Fred Lunnon <fred.lunnon at gmail.com> wrote:
>
>>   Doh --- what's worse, JG had just beaten me to making the the exact
>> same remark myself!
>>
>> WFL
>>
>>
>>
>> On 10/13/20, Rob Pratt <robert.william.pratt at gmail.com> wrote:
>> > (B) is already settled by Jack Grahl’s argument.
>> >
>> >> On Oct 12, 2020, at 9:11 PM, Fred Lunnon <fred.lunnon at gmail.com> wrote:
>> >>
>> >>   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/
>> >>>>>>>>
>> >>>>>>>
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>> >>>>
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