.

Since there are 5 "houses" we have 45 constraints (9R + 9C + 9B + 9P + 9W), and 81 + 45 = 126, so adapting Mladen's 128-bit

fsss2 solver is a simple task, and we have 2 spare bits so diagonal-mode (

SudokuPWX) also fits (just!).

I built my catalogs of ED grids for PW and PWX by finding all solutions from 1's templates, then canonicalising the results and removing duplicates.

I had thought that no transform other than rotation/reflection would preserve both W and P properties, but discovered that in fact the "W" transform that

blue identified (see

here) does this, so we have 16 transforms all up (for both PW and PWX).

My counts for the number of ED grids are

177,564 for PW, and just

2,922 for PWX.

(Hopefully

blue will do a quick "Burnside" check and confirm these counts).

Meanwhile, I have the HS engine running on both sets, and have found (not unexpectedly)

8-clue puzzles! (Yay!!)

An 8-clue SudokuPW:

- Code: Select all
` +-------+-------+-------+`

| . 5 . | . 6 . | . . . |

| 8 . . | . . . | 1 . . |

| . . . | 4 . . | . . . |

+-------+-------+-------+

| . . 9 | 3 . . | . . . |

| . . . | . . . | . 2 . |

| . . . | . . . | . . . |

+-------+-------+-------+

| . . . | . . . | . . . |

| . . . | . . . | . . . |

| . . . | . . . | . . . |

+-------+-------+-------+

.5..6....8.....1.....4.......93............2.....................................

There appear to be loads of 8-clue PWX puzzles. I've checked 360 grids so far (12%) and 160 of those grids have 8C puzzles, 4000 puzzles in total.