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Symmetry

Symmetry in Sudoku usually refers to the pattern of givens in the initial grid. Some Sudoku Variations, like Killer and Jigsaw, can have symmetry in the additional constraints.

There are several types of symmetry. The type depends on the operation that results in the same pattern. Most commonly used are rotation and reflection. These operations can also be combined to increase the level of symmetry. However, there is a drawback. The higher the level of symmetry, the greater the chance that there will be redundant givens in the grid. It is not easy to find Sudokus which are fully symmetrical and locally minimal at the same time.

Symmetry Classes

180 degrees rotational

A popular type, because it very easy to create, has small orbits and the whole pattern looks very balanced.

Orbits: 40x2, 1x1

90 degrees rotational

Less popular, because the patterns often look alike.

Orbits: 20x4, 1x1

Reflection on the horizontal axis

On its own, this type is rarely used.

Orbits: 36x2, 9x1

Reflection on the vertical axis

This type of symmetry is similar to horizontal reflection, but has a higher aestethic value.

Orbits: 36x2, 9x1

Reflection on both the horizontal and vertical axes

A very nice combination, but has relatively high redundancy. It is also 180 degrees rotational.

Orbits: 16x4, 8x2, 1x1

Reflection on the main (/) diagonal

Some people like it, others hardly recognize it as symmetry.

Orbits: 36x2, 9x1

Reflection on the anti- (\) diagonal

Same as the main diagonal

Orbits: 36x2, 9x1

Reflection on both diagonals

Can produce nice patterns, but you pay for it in increased redundancy.

Orbits: 16x4, 8x2, 1x1

Full dihedral

Maximum symmetry, includes all other classes. Due to the high orbit size, only a limited number of patterns are possible.

Orbits: 6x8, 8x4, 1x1

Asymmetrical

No symmetry.

Alternative Symmetry Classes

Fractal Symmetry

The pattern of boxes containing givens is repeated in each of those boxes.

Orbits: 9x9

Chute Symmetry

The 3 towers or floors contain the same or a reflected pattern.

Box Symmetry

Each box contains a pattern which is isomorph to the other boxes.

Orbits: 9x9

10-1 Symmetry

Can only be used for 180 degrees rotational symmetry. Each given and its symmetrical counterpart add up to 10. The cell in the center always contains digit 5.

Notes on Symmetry

A symmetrical and an asymmetrical puzzle can be mathematically equivalent. The symmetry has no effect on the difficulty, other than the redundancy. The scrambled puzzle has the same redundancy.

This page was last modified 11:32, 3 November 2007.  

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