## Re: Sudoku - is there more than one?

Expand Messages
• ... Correction: The second and third of these are equivalent to each other under the permutation 2 3 plus reflection across the main diagonal, so there are
Message 1 of 9 , Feb 1, 2008
--- In mathforfun@yahoogroups.com, "video_ranger" <video_ranger@...>
wrote:
>
> I tried looking at the simpler problem of a sudoku square based on
> the number 2 instead of 3. In other words 16 numbers such that each
> column, row and 4x4 quadrant contain the digits 1,2,3,4. If there's
> more than one fundamental arrangement in this simple case it seems
> likely there are more in the regular (sudoku based on 3) case.
> Tentatively there appear to be three distinct arrangements unless
> some combination of those symmetry operations listed by Peter can
> turn one into another:
>
> 1243
> 3421
> 4312
> 2134
>
> 1243
> 3421
> 2314
> 4132
>
> 1234
> 3421
> 4312
> 2143
>

Correction: The second and third of these are equivalent to each
other under the permutation 2<-->3 plus reflection across the main
diagonal, so there are at most two (rather than three) distinct:

1243
3421
4312
2134

1243
3421
2314
4132

unless these two are equivalent also (in which case Peter's
conjecture that there's only one) would hold for these
simplified "sudoku"
• In a message dated 2/1/2008 6:47:51 PM Central Standard Time, ... Can determinants of these sudoku treated as matrices be used to determine if they are
Message 2 of 9 , Feb 2, 2008
In a message dated 2/1/2008 6:47:51 PM Central Standard Time,
video_ranger@... writes:

> there are at most two (rather than three) distinct:
>
> 1243
> 3421
> 4312
> 2134
>
> 1243
> 3421
> 2314
> 4132
>
> unless these two are equivalent also (in which case Peter's
> conjecture that there's only one) would hold for these
> simplified "sudoku"
>

Can determinants of these sudoku treated as matrices be used to determine if
they are possibly equivalent?

stevo </HTML>

[Non-text portions of this message have been removed]
• ... wrote: Interesting, and logical, approach - thanks. I did find that the two are equivalent. Swap bottom pair of rows with top pair of rows, then replace 2
Message 3 of 9 , Feb 2, 2008
--- In mathforfun@yahoogroups.com, "video_ranger" <video_ranger@...>
wrote:

Interesting, and logical, approach - thanks.

I did find that the two are equivalent.

Swap bottom pair of rows with top pair of rows, then replace 2 with
1; 3 with 2, 4 with 3 and 1 with 4 and you have the other example.

At least I now have a method to check for equivalence of some 1-9
SUDOKU.

I plan to do numeric replacement so that the top left corner of a
SUDOKU being tested is
123
456
789
then check if simple row/column swapping can prove equivalence.

I also realise that if all the examples of SUDOKU I find ARE
equivalent, it does not prove there are not more than one - just that
the "authors" always muck around with a single example.

Peter

>
> --- In mathforfun@yahoogroups.com, "video_ranger" <video_ranger@>
> wrote:
> >
> > I tried looking at the simpler problem of a sudoku square based
on
> > the number 2 instead of 3. In other words 16 numbers such that
each
> > column, row and 4x4 quadrant contain the digits 1,2,3,4. If
there's
> > more than one fundamental arrangement in this simple case it
seems
> > likely there are more in the regular (sudoku based on 3) case.
> > Tentatively there appear to be three distinct arrangements unless
> > some combination of those symmetry operations listed by Peter can
> > turn one into another:
> >
> > 1243
> > 3421
> > 4312
> > 2134
> >
> > 1243
> > 3421
> > 2314
> > 4132
> >
> > 1234
> > 3421
> > 4312
> > 2143
> >
>
> Correction: The second and third of these are equivalent to each
> other under the permutation 2<-->3 plus reflection across the main
> diagonal, so there are at most two (rather than three) distinct:
>
> 1243
> 3421
> 4312
> 2134
>
> 1243
> 3421
> 2314
> 4132
>
> unless these two are equivalent also (in which case Peter's
> conjecture that there's only one) would hold for these
> simplified "sudoku"
>
• ... Are you sure? Applying those to the second array seems to return to it although I might be confused: 1243 3421 2314 4132 -- 2314 4132 1243 3421 -- 1243
Message 4 of 9 , Feb 2, 2008
--- In mathforfun@yahoogroups.com, "Peter Otzen" <pmaxotzen@...>
wrote:
>
> --- In mathforfun@yahoogroups.com, "video_ranger" <video_ranger@>
> wrote:
>
> Interesting, and logical, approach - thanks.
>
> I did find that the two are equivalent.
>
> Swap bottom pair of rows with top pair of rows, then replace 2 with
> 1; 3 with 2, 4 with 3 and 1 with 4 and you have the other example.
>

Are you sure? Applying those to the second array seems to return to
it although I might be confused:

1243
3421
2314
4132
-->
2314
4132
1243
3421
-->
1243
3421
4132
2314

> At least I now have a method to check for equivalence of some 1-9
> SUDOKU.
>
> I plan to do numeric replacement so that the top left corner of a
> SUDOKU being tested is
> 123
> 456
> 789
> then check if simple row/column swapping can prove equivalence.
>
> I also realise that if all the examples of SUDOKU I find ARE
> equivalent, it does not prove there are not more than one - just
that
> the "authors" always muck around with a single example.
>
> Peter
>
• ... determine if ... Maybe but it would be surprising. There s an overlap with the types of transformations that preserve determinants (i.e. row and column
Message 5 of 9 , Feb 2, 2008
>
> In a message dated 2/1/2008 6:47:51 PM Central Standard Time,
> video_ranger@... writes:
>
>
> > there are at most two (rather than three) distinct:
> >
> > 1243
> > 3421
> > 4312
> > 2134
> >
> > 1243
> > 3421
> > 2314
> > 4132
> >
> > unless these two are equivalent also (in which case Peter's
> > conjecture that there's only one) would hold for these
> > simplified "sudoku"
> >
>
> Can determinants of these sudoku treated as matrices be used to
determine if
> they are possibly equivalent?
>
> stevo </HTML>
>

Maybe but it would be surprising. There's an overlap with the types
of transformations that preserve determinants (i.e. row and column
permutations) but they're not exactly the same.

>
> [Non-text portions of this message have been removed]
>
• ... Then again, I might be the one who is confused, as you numbers below look convincing. I will check what I did - I may have described it poorly; or made a
Message 6 of 9 , Feb 2, 2008
--- In mathforfun@yahoogroups.com, "video_ranger" <video_ranger@...>
wrote:
>
> Are you sure? Applying those to the second array seems to return to
> it although I might be confused:

Then again, I might be the one who is confused, as you numbers below
look convincing. I will check what I did - I may have described it
poorly; or made a mistake [it was midnight here at the time]

Peter

>
> 1243
> 3421
> 2314
> 4132
> -->
> 2314
> 4132
> 1243
> 3421
> -->
> 1243
> 3421
> 4132
> 2314
>
> > At least I now have a method to check for equivalence of some 1-9
> > SUDOKU.
> >
> > I plan to do numeric replacement so that the top left corner of a
> > SUDOKU being tested is
> > 123
> > 456
> > 789
> > then check if simple row/column swapping can prove equivalence.
> >
> > I also realise that if all the examples of SUDOKU I find ARE
> > equivalent, it does not prove there are not more than one - just
> that
> > the "authors" always muck around with a single example.
> >
> > Peter
> >
>
• ... I notice in the above two arrays; the diagonals have two different numbers and the diagonals in the second have three different numbers. That is, the
Message 7 of 9 , Feb 3, 2008
--- In mathforfun@yahoogroups.com, "video_ranger" <video_ranger@...>
wrote:
>
> Correction: The second and third of these are equivalent to each
> other under the permutation 2<-->3 plus reflection across the main
> diagonal, so there are at most two (rather than three) distinct:
>
> 1243
> 3421
> 4312
> 2134
>
> 1243
> 3421
> 2314
> 4132
>
I notice in the above two arrays; the diagonals have two different
numbers and the diagonals in the second have three different numbers.
That is, the diagonals in the first are 1414, two different numbers,
and 3232, two different numbers. The diagonals in the second are
1412, three different numbers, and 3234 three different numbers.

Regardless what transformation I perform on the second there are
always three different numbers in each diagonal. On the first I can
get two different numbers or four different numbers in the diagonals
but never three different numbers.

If this is always true then these two solutions to 4 x 4 Sudoku must
be unique under all "legal" transformations.

John
> unless these two are equivalent also (in which case Peter's
> conjecture that there's only one) would hold for these
> simplified "sudoku"
>
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