[seqfan] Re: (no subject)
Rick Shepherd
rlshepherd2 at gmail.com
Sun Jun 13 22:07:29 CEST 2010
There are many ways these (9x9) solutions could be categorized.
For example, has anyone yet found a solution with palindromic strings in all
rows?
(This would be significant in checking that solution counts are accurate
considering symmetries.)
Analyzing the 5 solutions posted here and in sci.math so far, 5 palindromic
rows *and* columns is the record, split 3 and 2.
I've been running a(n inefficient PARI) program since yesterday evening
looking only for such but haven't found any yet.
Specifically, I'm exhaustively shuffling these nine specific row strings and
their independent bit reversals:
[1 0 1 0 1 0 1 0 1] (1)
[0 1 0 1 1 1 0 1 0] (3)
[1 1 0 1 0 1 0 1 1] (4)
[1 0 1 1 1 1 1 0 1] (5)
[1 0 0 0 0 0 0 0 1] (7)
[1 1 1 1 1 1 1 1 1] (9)
[1 0 0 1 1 1 0 0 1] (12)
[0 0 1 1 0 1 1 0 0] (16)
[0 0 0 1 1 1 0 0 0] (27)
1 and 27 must be represented by these two specific palindromic strings or
their bit reversals. The product 20, not included in the current search,
can also be represented by a palindromic string -- and alternate strings are
also possible for some of these products.
So, even if this run actually completes without finding any solutions (and
because I didn't expect it to run nearly this long to find just one and
stop, I didn't build in any progress markers), there are large numbers of
possibilities above I'm not even checking (yet).
If such a solution isn't possible, what is the maximum total number of rows
and columns possible with palindromic strings?
Thanks,
Rick
On Sat, Jun 12, 2010 at 2:16 PM, Rob Pratt <Rob.Pratt at sas.com> wrote:
> Yes, I used a computer. I've found over 1600 solutions so far...
>
> -----Original Message-----
> From: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
> On Behalf Of Leroy Quet
> Sent: Friday, June 11, 2010 3:04 PM
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] (no subject)
>
> Way to go, Rob! Did you use a computer or find this solution completely by
> hand?
>
> I wonder how many 18-distinct-products solutions there are. (There are at
> least 16, with rotations and reflections and flipping all the bits. But
> besides those solutions, I wonder how hard it is to come across an
> arrangement of 0's and 1's that work.)
>
> Thanks,
> Leroy Quet
>
>
> [ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]
>
>
> --- On Fri, 6/11/10, Rob Pratt <Rob.Pratt at sas.com> wrote:
>
> > From: Rob Pratt <Rob.Pratt at sas.com>
> > Subject: [seqfan] Re: Unique Products Regarding Binary Matrices
> > To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> > Date: Friday, June 11, 2010, 6:34 PM
> > 18:
> >
> > 0 0 0 0
> > 0 1 1 1
> > 1
> > 0 0 1 0
> > 0 0 0 0
> > 1
> > 0 0 0 0
> > 0 0 0 1
> > 1
> > 0 1 0 1
> > 0 1 1 1
> > 0
> > 0 1 0 1
> > 0 1 0 1
> > 0
> > 1 1 0 1
> > 0 1 1 1
> > 0
> > 0 0 0 1
> > 1 1 0 1
> > 0
> > 0 0 0 0
> > 1 0 1 0
> > 1
> > 0 0 0 0
> > 1 0 0 1
> > 0
> >
> > Note that for n = 9, there are exactly 18 possible products
> > (the ones you list below, and 1), so each one must be used.
> >
> > Rob
> >
> > -----Original Message-----
> > From: seqfan-bounces at list.seqfan.eu
> > [mailto:seqfan-bounces at list.seqfan.eu]
> > On Behalf Of Douglas McNeil
> > Sent: Friday, June 11, 2010 11:57 AM
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] Re: Unique Products Regarding Binary
> > Matrices
> >
> > > I too have been able to get 16 unique products. But
> > getting those last 2 products to differ is a *little*
> > tricky, perhaps, if possible.
> >
> > I can get 17 -- set([2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14,
> > 15, 16, 18,
> > 20, 24, 27]) -- but not 18 yet.
> >
> >
> > Doug
> >
> > --
> > Department of Earth Sciences
> > University of Hong Kong
> >
> >
> > _______________________________________________
> >
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> >
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