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Re: Resolved: Even vs. Odd Latin Square Problem

From Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups comp.theory
Subject Re: Resolved: Even vs. Odd Latin Square Problem
Date 2022-06-27 02:15 +0100
Organization A noiseless patient Spider
Message-ID <877d52uct3.fsf@bsb.me.uk> (permalink)
References (12 earlier) <18c8fe7f-c245-4fee-b9de-334e723e6965n@googlegroups.com> <878rpjv8t7.fsf@bsb.me.uk> <7ad01d7a-0352-4efe-b81d-bf6c66bb9e08n@googlegroups.com> <87fsjrt8u5.fsf@bsb.me.uk> <151e87ad-992b-4eaf-bbfa-6a4b6d75759bn@googlegroups.com>

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"B.H." <xlt.pjw@gmail.com> writes:

> On Sunday, June 26, 2022 at 5:26:29 PM UTC-4, Ben Bacarisse wrote:
>> "B.H." <xlt...@gmail.com> writes: 
>> 
>> > On Sunday, June 26, 2022 at 9:44:07 AM UTC-4, Ben Bacarisse wrote: 
>> 
>> >> You were wrong about having resolved the open problem you posted about. 
>> >> It's still open. You don't appear to have changed you mind about that. 
>> > 
>> > I am not wrong at all, it is surprising that you claim to think that 
>> > that easy proof, which I will not publish, does not exist.
>> There are counterexamples. There can be no proof that (in your words)
>> "it is actually the case that for every positive integer m, the number 
>> of even Latin squares of order m and the number of odd Latin squares 
>> of order m are *the same*"
>> because there is (to take just one counterexample) no odd 1x1 Latin 
>> square. When I asked about this you said
>> "The odd Latin square of order 1 is simply the square with one cell 
>> containing -1"
>> but that is an even Latin square. 
>
> You finally stated your objection.

There are two: that the Latin square you gave is even, and that there
are counterexamples (included in the page you linked to) to your claimed
resolution.

> "A latin square is even if the product of the signs of all of the row
> and column permutations is 1 and is odd otherwise. "

Yes, so your example is an even Latin square, not an odd one.  Unless
there is (up to isomorphism) exactly one odd Latin square the simple
case of n=1 is a counterexample to your claimed resolution.  The page
you linked to included 7 other counterexamples -- basically every known
case is a counterexample.

> If you treat the same square as the selected row and column, then the
> "base case" would have to be changed to a 2 x 2 Latin square, where it
> would work.
>
> I am not sharing the rest of the details of the proof with you.

There can't be a proof of your claim because there are counterexamples.
They were even listed in the page you linked to.

-- 
Ben.

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Thread

Resolved:  Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-22 10:40 -0700
  Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-22 10:42 -0700
    Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-22 10:50 -0700
  Re: Resolved:  Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-22 19:25 +0100
    Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-23 07:03 -0700
      Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-23 23:52 +0100
        Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-23 16:07 -0700
          Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-24 02:28 +0100
            Re: Resolved: Even vs. Odd Latin Square Problem Malcolm McLean <malcolm.arthur.mclean@gmail.com> - 2022-06-24 01:00 -0700
              Re: Resolved: Even vs. Odd Latin Square Problem Paul N <gw7rib@aol.com> - 2022-06-24 04:50 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-24 08:38 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-24 21:38 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-24 14:04 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Jeff Barnett <jbb@notatt.com> - 2022-06-24 23:06 -0600
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-25 07:22 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Jeff Barnett <jbb@notatt.com> - 2022-06-25 11:25 -0600
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-25 14:06 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Jeff Barnett <jbb@notatt.com> - 2022-06-25 16:40 -0600
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-25 17:21 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-26 01:02 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-25 17:26 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-26 14:44 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-26 07:47 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-26 22:26 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-26 17:57 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-27 02:15 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-26 18:27 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-27 02:54 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-27 08:51 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-27 21:56 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-27 15:56 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-26 08:03 -0700
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-26 21:49 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-24 21:17 +0100
                Re: Resolved: Even vs. Odd Latin Square Problem Paul N <gw7rib@aol.com> - 2022-06-25 04:44 -0700
              Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-24 21:14 +0100
            Re: Resolved: Even vs. Odd Latin Square Problem "B.H." <xlt.pjw@gmail.com> - 2022-06-24 08:35 -0700
              Re: Resolved: Even vs. Odd Latin Square Problem Ben Bacarisse <ben.usenet@bsb.me.uk> - 2022-06-24 21:30 +0100

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