Mirrored from **Sudopedia**, the Free Sudoku Reference Guide

# 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.