# [Maths-Education] RE: Group Theory and Soduko

romlins at rc.unesp.br romlins at rc.unesp.br
Mon Dec 1 22:13:22 GMT 2008

```Dear Paul,

I am not sure I quite understood your message, but the theme is
interesting and I would be grateful if you can help me to understand what
you are thinking of.

Something that came to my mind. If we call "Sudoko" a completed 9x9 table
that satisfies the rules of the game, then any permutation of the 9
digits, applied to all the entries of a given Sudoko will produce a new
(correct) Sudoko. The question is: given a Sudoko is there another Sudoko
that does not result from a permutation applied to it?

all the best
Romulo

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> Dear Colleagues
>
> My Group theory is a bit rusty, but I was looking at a completed Soduko
> which is a 9X9 grid and it is obviously a permutation - each line is a
> rearrangemnt of previous with no elements staying in same place - so my
> thought is - it should (or could?) be a permutation group of order 9. Now
> I only recall 2 finite groups of order 9 - C9 (cyclic group of order nine)
> and C3XC3 (cartesian product of 2 cyclic groups of order 3)
>
> But playing with transformations/rearrangements of the Soduko square did
> not get me anywhere.
>
> The number of nine element permutations is 9! so the 9 shown in an S grid
> is a tiny part of all possible perms - do these form a group? If so, what
> operation transforms them into each other?
>
> The answer may differ for different completed Sodukos as they constitute
> different selections of 9 permutations
>
> To be a group, one of them has to be a unit transformation, and each needs
> an inverse
>
> Actually I'm starting to have doubts as to where an aribtrary Soduko grid
> does form a group
>
> me!
>
> More generally, how do you determine a group from a finite group table
> (without the labels)
>
>
> Best wishes
>
> Paul
> __________________
> Paul Ernest
> Emeritus Professor
> University of Exeter
> SELL, St Lukes, Heavitree Road
> Exeter  EX1 2LU, UK
>
> Visiting Professor, HiST-ALT, Norway
> Visiting Professor, UiO, Norway
>
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```