It's been suggested that the minimum number of initially-fiiled cells to appear on a valid Su Doku puzzle is 18 if the puzzle has rotational symmetry of order two or 17 otherwise. On this page, I list the puzzles with the least number of initially-filled cells of those I've created, sorted according to the symmetric properties possessed by the puzzle. Where there are several candidates, I give the puzzle(s) with the simplest solution.
1.
5 . . | . 2 . | . . 3 . 7 . | . . . | . 9 . . . 6 | . . . | 4 . . -------+-------+------ . . . | 8 . 9 | . . . 3 . . | . . . | . . 1 . . . | 4 . 7 | . . . -------+-------+------ . . 4 | . . . | 6 . . . 9 . | . . . | . 7 . 2 . . | . 5 . | . . 9
2.
4 . . | . . . | . . 7 . 1 . | . . . | . 8 . . . 5 | . 9 . | 2 . . -------+-------+------ . . . | 3 . 8 | . . . . . 2 | . . . | 6 . . . . . | 1 . 7 | . . . -------+-------+------ . . 9 | . 5 . | 8 . . . 8 . | . . . | . 1 . 7 . . | . . . | . . 4
These very similar puzzles have 20 initially-filled cells. It's unlikely to be possible to beat this figure because it's impossible to create a puzzle with four lines of symmetry with 18 or 19 initially-filled cells. The puzzles have to be solved with Many-Valued Chains.
. . . | . . . | . . . 7 . . | . 3 . | . . 4 . 8 . | 9 . 4 | . 6 . -------+-------+------ . . . | . 4 . | . . . . 9 2 | . . . | 5 8 . . . . | . 7 . | . . . -------+-------+------ . 6 . | 5 . 8 | . 9 . 3 . . | . 9 . | . . 1 . . . | . . . | . . .
The puzzle has 20 initially-filled cells, just like the puzzles with four lines of symmetry. However, unlike those, this puzzle is straightforward to solve.