Sudoku scanning and trial and error

In summary, the conversation discusses alternative methods for solving sudoku puzzles, such as using deductive reasoning and group theory, and the potential for sudoku puzzles to introduce more complex mathematical concepts. The relation between sudoku and group theory is explored, with the suggestion that sudoku can be seen as a quasigroup. The possibility of solving sudoku mentally is also mentioned.
  • #1
roger
318
0
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..
 
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  • #2
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.
 
  • #3
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).
 
  • #4
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.
 
  • #5
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).
 
  • #6
matt grime said:
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.
 
  • #7
sudoku

matt grime said:
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 can't get that wats its relation with group theory...??
 
  • #8
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.
 
  • #9
masudr said:
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).

roger said:
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.
 
  • #10
shmoe said:
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.
 

FAQ: Sudoku scanning and trial and error

What is Sudoku scanning and trial and error?

Sudoku scanning and trial and error is a solving technique used in solving Sudoku puzzles. It involves systematically scanning each row, column, and square of the puzzle to identify possible number candidates, and then using trial and error to test and eliminate these candidates until only one solution remains.

When should Sudoku scanning and trial and error be used?

Sudoku scanning and trial and error should be used when there are no immediate obvious solutions to the puzzle and other techniques, such as elimination and cross-hatching, have been exhausted. It is also useful for more challenging Sudoku puzzles that require a higher level of logical thinking.

How does Sudoku scanning and trial and error work?

Sudoku scanning and trial and error works by scanning each row, column, and square of the puzzle to identify possible number candidates. Once these candidates are identified, they are tested through trial and error by placing them in the puzzle and seeing if they lead to a solution. If they do not, they are eliminated and the process is repeated until only one solution remains.

Are there any tips for using Sudoku scanning and trial and error?

Yes, some tips for using Sudoku scanning and trial and error include starting with the easier numbers (1-4) and working your way up to the more challenging ones, using a pencil to make notes and keep track of potential candidates, and rechecking the puzzle after each trial and error to ensure no mistakes were made.

Is Sudoku scanning and trial and error the most effective way to solve Sudoku puzzles?

No, Sudoku scanning and trial and error is not always the most effective way to solve Sudoku puzzles. It should only be used as a last resort when other techniques have been exhausted. Other techniques, such as elimination and cross-hatching, can often lead to a quicker and more efficient solution.

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