Sudoku

[roger]

Hi,

I wondered whether there is any other way to solve these puzzles apart from just scanning and trial and error ?

Also is there any relevance to group theory or any other mathematics ?

Is there any practical way of solving it mentally ? because I have to write out the various possibilties on paper then try to figure it out..

[matt grime]

I'm not sure what you mean by trial and error. All sudoku's can be solved by deductive reasoning by eliminating options and seeing what is left.

There is some relation to group theory. They were originally called latin squares and are due to Gauss I believe.

[shmoe]

They are an example of a graph colouring problem with 81 vertices, an edge between two vertices if they are in the same 3x3 block, or the same column, or the same row, and 9 colours.

Note that while every sudoku is a latin square, not every latin square is a sudoku (latin squares just have the row and column restrictions). Also, "latin square" originated with Euler in the "36 officers problem" (essentially a 6x6 square).

[matt grime]

It seems you could interpret them as defining elements in S_9 that lie in certain subgroups with certain properites. Vague, but then sudoku's really aren't very interesting.

[shmoe]

What do you have in mind? Each row, column or 3x3 box could be considered as an element of S_9, but I can't think of any nice properties they'd have that aren't very artificial looking and not algebraic in nature. Ideas?

You may consider sudoku's uninteresting, but I do think they have value. Bare minimum they are practice at deductive reasoning and possibly an algorithmic approach to doing them. They could also be a springboard for introducing "more" interesting ideas, like graph colouring in general as well as ideas of isomorphisms (i.e permute the numbers and you get a different looking puzzle that's essentially the same).

[roger]

I'm not sure what you mean by trial and error. All sudoku's can be solved by deductive reasoning by eliminating options and seeing what is left.

There is some relation to group theory. They were originally called latin squares and are due to Gauss I believe.

But can sudoku be solved entirely mentally ? As I said earlier, I find that I have to write down the possibilities on paper and then use deductive reasoning.

And how does it relate to groups ? since there's no operation defined in sudoku, unlike in group theory.

[chets]

I'm not sure what you mean by trial and error. All sudoku's can be solved by deductive reasoning by eliminating options and seeing what is left.

There is some relation to group theory. They were originally called latin squares and are due to Gauss I believe.
i cant get that wats its relation with group theory...??

[masudr]

A Latin square or a Cayley table is a table which defines what a combination of 2 group elements would give. Of course there is closure since they only 1-9 are in the table, etc. etc.

[shmoe]

A Latin square or a Cayley table is a table which defines what a combination of 2 group elements would give. Of course there is closure since they only 1-9 are in the table, etc. etc.

Latin square's aren't all Cayley tables though, same with sudoku's. You can use an n by n Latin square to define a binary operation on a set of n elements, but you won't expect the group properties to hold except of course closure (it's a quasigroup).

But can sudoku be solved entirely mentally ? As I said earlier, I find that I have to write down the possibilities on paper and then use deductive reasoning.

Anything you can do on paper you can do in your head if your memory is good enough.

[masudr]

Latin square's aren't all Cayley tables though, same with sudoku's. You can use an n by n Latin square to define a binary operation on a set of n elements, but you won't expect the group properties to hold except of course closure (it's a quasigroup).

Sorry my error. Thank you for correcting me.