How To Create Symmetrical Sudoku Puzzles

Two of the most popular puzzles in the world have something in common: and that commonality is symmetry.

We're talking about the great staple of the word puzzle world: the crossword, and the logic puzzle that has taken its place alongside the crossword in most newspapers around the world: sudoku.

And they both tend to exhibit the same rotational symmetry, such that rotating the grid 180 degrees results in the same pattern of givens, or of black/white squares in the case of the crossword grid.

Now, symmetry increases the aesthetic appeal: whether the crossword in its modern form or the sudoku would have been as popular without this is hard to know and much less so to prove - but it is at the very least interesting that both puzzles exhibit this pleasing symmetry.

Of course, grids don't have to exhibit symmetry, but in the vast majority of cases they do, and the newspaper rules for crosswords (and sometimes sudoku) specify that they do. For some, a well-formed grid of either type has to exhibit symmetry, and the majority of sudoku you see printed - and virtually all crosswords - do display it.

When you are creating a sudoku puzzle, how do you ensure that this symmetry is in the grid? The process is quite simple: if you put a number in the second square of the puzzle (square 2) then you must also place a number in the second square from the end of the puzzle grid (square 80, in a 9x9 puzzle). By always placing in pairs then you can be sure to create a symmetrical grid.

If you are filling a sudoku grid with a computer program, then you can program symmetry in the same way. The most popular method to get standard rotational symmetry into a grid is simply to fill the grid, then remove the given digits in their symmetric pairs, as described above. If both digits can be removed to create a valid puzzle, remove them - if not, then put them back in. By repeating this process for each pair in the grid, you are sure to create a valid sudoku that exhibits this symmetry.

Of course, there are other symmetries that you can try instead - reflectional symmetry (also called mirror symmetry, for obvious reasons), for instance; or full rotational symmetry (4-way) or indeed both rotation and reflection.

Symmetry is broken as soon as you place your first digit into the grid in most cases, of course, however by then the symmetry has done its job and drawn you into the puzzle.

Do you appreciate the symmetry of different sudoku puzzles? Do you have a favourite pattern or feel that it helps you solve a puzzle more quickly, or that it has no impact on solving above the initial aesthetic appeal? Feel free to comment below.
Date written: 13 Aug 2015

Category: sudoku | Keywords: Sudoku | symmetry | puzzle creation

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