# [seqfan] Re: Unique Products Regarding Binary Matrices

Leroy Quet q1qq2qqq3qqqq at yahoo.com
Fri Jun 11 17:13:02 CEST 2010

Yes, a(9) of my sequence is >=16 because of rotations, reflections, and flipping of digits.

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 posted the 9-by-9 problem to sci.math and rec.puzzles (in the guise of black stones and white stones on a grid). I will report back here if anything develops there.

Thanks,
Leroy Quet

[ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )]) ] ) ]

--- On Fri, 6/11/10, Rick Shepherd <rlshepherd2 at gmail.com> wrote:

> From: Rick Shepherd <rlshepherd2 at gmail.com>
> Subject: [seqfan] Re: Unique Products Regarding Binary Matrices
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Date: Friday, June 11, 2010, 2:43 PM
> Hi Leroy,
>
> I also highly suspect that solutions exist:  It only
> took me a few tries by
> hand to find a 9-by-9 with 16 unique products from the 18
> possible {1, 2, 3,
> 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24,
> 27}.  If I have time
> later, I'll give this more thought.
>
> I also agree with Hugo's statement:  Exchanging all 0s
> and 1s in any given
> solution would also result in a solution (and of course
> there would be
> symmetries of the square by rotating and flipping).
> Different sequences
> could be generated by including or ignoring these.
>
> If solutions do exist and they're not too numerous, I could
> imagine this
> being a good somewhat Sudoku-like puzzle (with some entries
> given upfront).
>
> Rick
>
> On Thu, Jun 10, 2010 at 11:16 AM, Leroy Quet <q1qq2qqq3qqqq at yahoo.com>wrote:
>
> > I'm actually most interested in the 9-by-9 case.
> >
> > I am wondering if it would make a good puzzle, trying
> to come up with a
> > solution by hand.
> >
> > But I don't even know if it is possible, although I
> highly suspect that it
> > is.
> >
> > Thanks,
> > Leroy Quet
> >
> >
> >
> > [ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ] )])
> ] ) ]
> >
> >
> > --- On Thu, 6/10/10, Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
> wrote:
> >
> > > From: Leroy Quet <q1qq2qqq3qqqq at yahoo.com>
> > > Subject: [seqfan]  Unique Products Regarding
> Binary Matrices
> > > To: seqfan at seqfan.eu
> > > Date: Thursday, June 10, 2010, 11:24 AM
> >  > Say we have an n-by-n binary matrix
> > > (all elements either 0 or 1).
> > >
> > > For a given row of the matrix, take the lengths
> of the runs
> > > of 0's and 1's and multiply these lengths.
> > > (By "run", it is meant a string of consecutive
> elements in
> > > the row (or column) all of the same value b,
> bounded by the
> > > value 1-b or by the edge of the row (or
> column).)
> > >
> > > Do this for all rows and all columns to get 2n
> products.
> > >
> > > Let a(n) = the number of such n-by-n binary
> matrices such
> > > that the 2n products are all unique.
> > >
> > > I know that a(n) = 0 for n <= 8, since
> A034891(n) <
> > > 2n for n <= 8.
> > >
> > > Is {a(n)} in the EIS already? It would seem a bad
> idea to
> > > compute this sequence via brute-force search of
> all the
> > > 2^(n^2) matrices for a given n, since the number
> of
> > > potential matrices grows so quickly as n grows.
> > >
> > > Thanks,
> > > Leroy Quet
> > >
> > >
> > > [ ( [ ([( [ ( ([[o0Oo0Ooo0Oo(0)oO0ooO0oO0o]]) ) ]
> )]) ] )
> > > ]
> > >
> > >
> > >
> > >
> > >
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