Sudoku and Symmetry

Most sudoku puzzles that you see - although not all - contain symmetry.

Sepcifically, they tend to contain rotational symmetry such that if the puzzle is rotated 180 degrees it looks the same.

This is visually quite pleasing, and infact some publishers require that their sudokus display symmetry or else they do not count it as a valid sudoku: for instance Nikoli in Japan apparently stipulates a sudoku must have symmetry and 32 givens or less to count as a valid sudoku or else it is rejected.

There are however many other types of symmetry that could be introduced: for instance different degrees of rotational symmetry, reflectional symmetry, combinations of all, and these have interesting names for non-mathematicians such as "full dihedral symmetry" which would leave most people none the wiser as to what exactly that meant! There are various websites that list many types of sudoku symmetry with examples of each.

So does symmetry help with solving sudoku? The jury is out, with most people based on reading various sudoku blogs thinking symmetry is just for effect, and that if anything it might make puzzles easier, often by introducing a given (starting number) that is not required to solve but was required for the symmetry to hold.

As soon as you add one number to a sudoku puzzle with the most common form of symmetry then the symmetry is broken (well, unless you enter the middle digit in a 9x9 grid!).

However there are occasions on which symmetry can help out, and particularly if you have the same digit in two cells that share symmetry you may be able to repeat a logical rule you used on one to make progress with your solve on its partner cell.

If you are creating your own sudoku, then you should definitely spend the extra few minutes to introduce symmetry to your puzzles. Most people who play sudoku are used to seeing symmetrical grids too, just like with crosswords, and therefore something that is non-symmetric can strike the player as odd. Many publishers stipulate that they want symmetrical sudoku grids.

And of course the most common type of symmetry is trivial for a programmer to introduce to your sudoku puzzles.

If you are someone who hand writes sudoku puzzles, do you find creating a puzzle with symmetry any easier or harder than creating one without symmetry? Do you have any comments in general about sudoku and symmetry? If so please do post below in the comments section!
Date written: 01 Dec 2010

Category: sudoku | Keywords: symmetry

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