The New Sudoku Players' Forum

[dobrichev]

Below are 12 minimal puzzles with 38 symmetric givens.

Hidden Text: Show

Code: Select all

. . . . . . . . .
. . 1 . 2 3 5 . 4
. 2 3 4 . 5 1 6 .
. . 4 . 6 2 3 8 .
2 . 6 3 . 8 4 . 5
. 3 8 5 4 . 6 . .
. 4 7 2 . 6 8 5 .
6 . 2 8 5 . 7 . .
. . . . . . . . . #1 pi rotational (180 degrees)

. . . . . 1 . . .
. . . . 3 4 . 2 1
. . . 6 2 . 5 3 .
. . 2 . 7 3 . 6 .
. 4 6 1 8 . 3 7 .
7 3 . 4 . . 1 . 2
. . 5 . 4 7 . 1 6
. 6 4 2 1 . 7 5 .
. 7 . . . 6 2 . . #2 diagonal (main)

. . . . . 1 . . .
. . . . 3 4 . 2 1
. . . 6 2 . 5 3 .
. . 2 . 7 3 . 6 .
. 4 6 1 8 2 3 7 .
7 3 . 4 6 . 1 . .
. . 5 . 4 7 . 1 6
. 6 4 2 1 . 7 5 .
. 7 . . . . 2 . . #3 diagonal (main)

. . . . . 1 . . .
. . . . 3 4 . 2 1
. . . 6 2 . 5 3 .
. . 3 . 7 2 . 6 .
. 6 4 1 8 3 2 7 .
7 2 . 4 6 . 1 . .
. . 5 . 4 7 . 1 6
. 4 6 3 1 . 7 5 .
. 7 . . . . 3 . . #4 diagonal (main)

. . . . . . . 1 .
. . 2 . . 1 4 . 3
. 1 5 3 2 4 6 7 .
. 4 1 2 . 8 . 3 7
. . 8 . . . 1 . .
5 2 . 1 . 3 8 6 .
. 5 4 6 3 2 7 8 .
2 . 6 8 . . 3 . .
. 8 . . . . . . . #5 pi rotational (180 degrees)

. . 1 . . . . . .
. . 5 3 4 . . 1 2
7 4 . . 6 1 . 3 5
. 8 . . 1 . . . .
. 5 6 7 . 8 . 4 1
. . 7 . 3 6 . 5 8
. . . . . . . . 4
. 7 8 . 5 4 . 2 3
. 2 4 . 7 3 5 8 . #6 diagonal (main)

. . 1 . . . . . .
. . 5 3 4 . . 1 2
7 4 . . 6 1 . 5 3
. 8 . . 1 . . . .
. 5 6 7 . 8 . 4 1
. . 7 . 3 6 . 8 5
. . . . . . . . 4
. 7 4 . 5 3 . 2 8
. 2 8 . 7 4 5 3 . #7 diagonal (main)

. . . . . . 1 . .
. 3 4 . 2 1 5 . .
. 1 2 6 5 . . 3 4
. 7 . . . 8 4 5 .
4 5 . 2 . 6 . 8 3
. 8 3 5 . . . 1 .
3 4 . . 8 2 6 7 .
. . 8 7 6 . 3 4 .
. . 7 . . . . . . #8 pi rotational (180 degrees)

. 2 1 . . . . . .
4 . 5 2 3 . . 1 .
3 7 . . 1 6 . 4 2
. 4 . . 5 1 . . .
. 1 3 7 6 . . 5 4
. . 6 3 . . 1 7 .
. . . . . 3 . 2 5
. 3 2 . 7 5 4 8 .
. . 4 . 2 . 7 . . #9 diagonal (main)

. 1 2 . . . . . .
4 5 6 . 2 3 1 . .
3 7 8 . 6 1 4 . 2
. . . . . 7 2 . .
. 3 7 . 9 2 . 4 8
. 2 4 6 3 . . 7 .
. 4 1 3 . . 8 . .
. . . . 1 6 . . 4
. . 3 . 8 . . 1 . #10 diagonal (main)

. 2 . 1 . . . . .
5 . 1 . . 4 . . 3
. 4 6 . 2 3 . 5 .
4 . . . 7 . . . .
. . 5 3 8 1 . 4 7
. 1 7 . 4 6 8 . 5
. . . . . 8 . 7 2
. . 2 . 1 . 3 8 4
. 7 . . 3 2 5 1 . #11 diagonal (main)

. . 2 . . . . 1 .
. 5 1 . 2 3 4 . .
6 3 . . 1 4 2 5 .
. . . . . 7 . . 1
. 6 3 . 8 1 7 4 .
. 1 7 4 3 . 8 . 5
. 7 6 . 4 8 . . .
3 . 5 . 7 . . . 4
. . . 3 . 5 . 7 . #12 diagonal (main)


[dobrichev]

Below are 90 minimal puzzles with 37 symmetric givens.

Hidden Text: Show

Code: Select all

. . . . . . . . .
. . . . 2 3 . . 1
. . 4 . . 5 . 2 3
. . . . 7 2 . 5 6
. 6 . 3 . . 7 . .
. 7 5 6 . 8 3 1 2
. . . . 3 7 . 6 9
. . 6 5 . 9 2 . 7
. 9 7 2 . 6 1 3 5 #1 diagonal (main)

. . . . . . . . .
. . 1 . . 2 . . 3
. 2 . . 4 3 . 5 1
. . . . 6 1 5 . 4
. . 4 3 . 7 . 1 6
. 1 6 2 5 . . 3 7
. . . 7 . . . 6 .
. . 7 . 2 6 3 . 5
. 6 2 4 3 5 . 7 8 #2 diagonal (main)

. . . . . . . . .
. . 2 3 . . . 1 .
. 4 5 6 . 1 . 3 2
. 7 6 . 8 . . 4 1
. . . 1 . 7 . . 8
. . 8 . 3 6 7 2 .
. . . . . 3 . 8 4
. 8 7 2 . 4 1 5 3
. . 4 8 1 . 2 7 . #3 diagonal (main)

. . . . . . . . .
. . 1 . . 2 . . 3
. 2 4 . 5 3 . 6 1
. . . . 7 1 6 . 5
. . 5 3 . 8 . 1 7
. 1 7 2 6 . . 3 8
. . . 8 . . . 7 .
. . 8 . 2 7 3 . 6
. 7 2 5 3 6 . 8 . #4 diagonal (main)

. . . . . . . . .
. . 2 . . 1 3 4 .
. 1 5 . 3 . 6 . 2
. . . . . 7 . . .
. . 7 . 4 8 2 3 6
. 2 . 3 6 5 8 7 .
. 7 8 . 1 6 . 2 3
. 4 . . 7 3 1 . 8
. . 1 . 8 . 7 6 . #5 diagonal (main)

. . . . . . . . .
. . 1 . 3 4 . . 2
. 3 . . 2 1 . 5 6
. . . . 7 3 . . .
. 1 7 2 4 5 3 6 .
. 4 3 1 6 8 . 2 7
. . . . 1 . . 7 5
. . 6 . 5 7 2 4 .
. 7 5 . . 2 6 . . #6 diagonal (main)

. . . . . . . . .
. . 1 . 3 4 . . 2
. 3 5 . 2 1 . 6 7
. . . . 8 3 . . .
. 1 8 2 4 6 3 7 .
. 4 3 1 7 . . 2 8
. . . . 1 . . 8 6
. . 7 . 6 8 2 4 .
. 8 6 . . 2 7 . . #7 diagonal (main)

. . . . . . . . .
. . 1 . . 2 . 3 4
. 3 5 1 4 . 6 . 2
. . 3 . 2 7 . 6 8
. . 7 8 . . 4 2 .
. 8 . 4 . . 7 . .
. . 6 . 7 4 . 8 1
. 1 . 6 8 . 2 . 7
. 7 8 2 . . 3 4 . #8 diagonal (main)

. . . . . . . . .
. . 1 . 3 4 . . 2
. 3 5 . 2 . 6 4 .
. . . . 8 2 . 7 1
. 1 7 4 5 3 8 2 .
. 8 . 7 1 . 4 . .
. . 6 . 4 8 . . 7
. . 8 2 7 . . 6 4
. 7 . 3 . . 2 8 . #9 diagonal (main)

. . . . . . . . .
. . 1 . . 2 . 4 3
. 5 . . 6 1 7 2 .
. . . . 3 8 . . 7
. . 3 6 . 5 2 8 .
. 8 5 2 1 7 . 3 6
. . 7 . 8 . . . 2
. 4 8 . 2 6 . . 5
. 6 . 7 . 3 8 1 . #10 diagonal (main)

. . . . . . . . .
. . 1 . . 2 . 3 4
. 5 3 . 6 4 1 2 7
. . . . 4 8 . . .
. . 4 6 . 1 . 8 2
. 8 6 2 5 . . 4 .
. . 5 . . . . . 8
. 3 8 . 2 6 . 7 5
. 6 7 . 8 . 2 1 3 #11 diagonal (main)

. . . . . . . . .
. . 2 . 4 3 . 1 .
. 6 4 1 2 7 . 3 5
. . 8 . . 6 . . .
. 4 6 . 7 8 . 5 1
. 5 7 2 1 4 . . 8
. . . . . . . 8 3
. 8 5 . 3 . 1 . 6
. . 3 . 8 1 5 2 . #12 diagonal (main)

. . . . . . . . .
. . 1 . 2 3 . 4 5
. 2 4 . 6 . 7 1 3
. . . . . . . . 8
. 1 7 . 5 8 4 3 .
. 8 . . 3 . . 7 1
. . 6 . 4 . . 8 .
. 4 2 . 8 6 3 5 7
. 5 8 3 . 2 . 6 . #13 diagonal (main)

. . . . . . . . .
. . 1 . 4 5 . 2 3
. 2 4 1 . 3 5 . .
. . 2 . 5 7 . 6 .
. 4 . 8 . 2 3 . 7
. 8 7 3 1 . . 5 2
. . 8 . 7 . . 3 .
. 1 . 6 . 8 7 . 5
. 7 . . 3 1 . 8 . #14 diagonal (main)

. . . . . . . . .
. . 1 . 5 4 . 2 3
. 6 2 . 3 8 7 4 .
. . . . 4 3 . . .
. 4 8 5 . 1 . 3 6
. 5 3 8 6 . 4 1 .
. . 7 . . 5 . 6 .
. 2 5 . 8 6 1 7 .
. 8 . . 1 . . . 2 #15 diagonal (main)

. . . . . . . . .
. 2 3 5 . 4 . 1 .
. 5 6 . 1 3 2 . 4
. 3 . . 4 . . . 5
. . 7 8 . 5 1 . .
. 8 5 . 3 . 7 4 .
. . 2 . 7 1 . 5 8
. 7 . . . 8 3 6 1
. . 8 3 . . 4 7 . #16 diagonal (main)

. . . . . . . . .
. 3 2 . . 1 5 4 .
. 5 6 4 2 . . 1 3
. . 7 . . . 1 . .
. . 5 . . 7 . 8 4
. 8 . . 4 . . 5 7
. 2 . 8 . . . 7 1
. 7 8 . 1 2 4 3 5
. . 3 . 7 4 8 2 . #17 diagonal (main)

. . . . . . . . .
. 2 1 3 . 4 6 . 5
. 5 3 . 1 6 . 2 4
. 3 . . 7 5 . 8 .
. . 5 6 . 8 . . 7
. 8 7 1 4 3 . 5 .
. 7 . . . . . 6 .
. . 2 4 . 1 7 . 8
. 1 8 . 6 . . 4 . #18 diagonal (main)

. . . . . 1 . . .
. . 2 3 4 . . . .
. 1 5 2 6 . . 4 3
. 8 1 6 2 3 . 7 4
. 7 6 1 . 4 . . 8
2 . . 8 7 . 1 . .
. . . . . 2 . 8 7
. . 7 4 . . 3 5 .
. . 8 7 3 . 4 . . #19 diagonal (main)

. . . 1 . . . . .
. 2 . . 3 4 . . .
. . 5 . 2 6 . 3 4
6 . . . 7 8 . . 1
. 7 2 3 . 1 . 8 .
. 8 1 4 6 2 3 7 .
. . . . . 7 . 1 3
. . 7 . 4 3 6 . 8
. . 8 6 . . 7 4 . #20 diagonal (main)

. . . 1 . . . . .
. 2 . . 3 4 . . .
. . 5 . 2 6 . 3 4
6 . . . 7 8 . . 1
. 7 2 3 . . . 8 6
. 8 1 4 . 2 3 7 .
. . . . . 7 . 1 3
. . 7 . 4 3 6 . 8
. . 8 6 1 . 7 4 . #21 diagonal (main)

. 1 . . . . . . .
3 . 4 . 2 1 . . .
. 2 . 5 4 . . 3 1
. . 6 . . 2 . 5 .
. 4 2 . 7 5 . 6 3
. 3 . 4 6 . . . 7
. . . . . . . 1 6
. . 1 6 5 . 3 7 2
. . 3 . 1 7 5 4 8 #22 diagonal (main)

. 1 . . . . . . .
3 . 4 . 2 1 . . .
. 2 . 5 4 . . 3 1
. . 6 . . 4 . 5 .
. 4 2 . 7 5 . 6 3
. 3 . 2 6 . . . 7
. . . . . . . 1 6
. . 1 6 5 . 3 7 4
. . 3 . 1 7 5 2 8 #23 diagonal (main)

. . 1 . . . . . .
. 3 . . 2 . . 1 .
5 . 6 . 1 3 . 2 4
. . . . 8 7 . . .
. 7 5 4 3 1 . . 2
. . 3 2 5 . . 4 7
. . . . . . . 7 8
. 5 7 . . 8 2 9 1
. . 8 . 7 2 4 5 3 #24 diagonal (main)

. . . . . . . . 1
. . 1 . . 2 . 3 4
. 2 4 . 1 3 5 . 6
. . . . . 1 . . .
. . 2 . 7 8 1 . 9
. 1 9 2 6 4 8 . 3
. . 5 . 2 6 . 9 8
. 9 . . . . 3 . .
2 4 8 . 3 9 6 . . #25 diagonal (main)

. . . . . . 1 . .
. . 2 . . 1 . 4 3
. 1 4 . 3 5 . 6 2
. . . . . 7 . . 6
. . 7 . . 8 2 3 .
. 2 5 3 6 . . . 7
2 . . . 1 . . 7 8
. 4 8 . 7 . 3 . 1
. 7 1 8 . 3 6 2 . #26 diagonal (main)

. . . . . . . . 1
. . 1 . 2 3 . 4 .
. 2 5 . 4 . 6 . .
. . . . 7 4 . 1 8
. 1 8 3 5 2 7 . 4
. 7 . 8 1 . 3 . .
. . 6 . 3 7 . 8 .
. 8 . 2 . . 4 . 7
2 . . 4 8 . . 3 6 #27 diagonal (main)

. 1 . . . . . . .
4 . . . . 1 . 3 2
. . . . 2 5 4 1 .
. . . . . 2 . 4 6
. . 6 . 7 . 3 . .
. 4 3 6 . 8 7 2 .
. . 1 . 5 7 . 6 4
. 5 4 1 . 6 2 7 3
. 6 . 2 . . 1 5 . #28 diagonal (main)

. . . . . 1 . . .
. . 1 . 4 . . 2 3
. 5 2 . 3 7 . 4 6
. . . . 8 3 . . .
. 8 7 4 . 5 . 3 .
5 . 3 7 1 . . 8 4
. . . . . . . . 8
. 2 8 . 7 4 . 6 5
. 7 6 . . 8 4 1 2 #29 diagonal (main)

. . . . . 1 . . .
. . 1 . 4 . . 2 3
. 5 2 . 3 . 1 . 4
. . . . . 7 3 . 6
. 7 6 . 8 3 . 4 5
5 . . 4 6 . . . .
. . 5 6 . . . 3 7
. 2 . . 7 . 6 . 1
. 6 7 3 1 . 4 5 2 #30 diagonal (main)

. . . . . 1 . . .
. . 1 . 4 . . 2 3
. 5 2 . 3 . 1 . 4
. . . . . 7 4 . 6
. 6 7 . 8 4 . 3 5
5 . . 3 6 . . . .
. . 5 6 . . . 4 7
. 2 . . 7 . 6 . 1
. 7 6 4 1 . 3 5 2 #31 diagonal (main)

. . . . . 1 . . .
. . 1 . 4 . . 2 3
. 5 2 . 3 . 4 . 1
. . . . . 7 6 . 4
. 6 7 . 8 4 . 3 5
5 . . 3 6 . . . .
. . 6 4 . . . 5 7
. 2 . . 7 . 1 . 6
. 7 5 6 1 . 3 4 2 #32 diagonal (main)

. . . . . . . . 1
. . 1 . 2 3 . 4 5
. 2 5 . 1 . 3 6 .
. . . . 7 1 . . .
. 1 2 3 5 8 . 7 6
. 7 . 2 9 . . 1 3
. . 7 . . . . . .
. 4 6 . 3 2 . 5 7
2 5 . . 6 7 . 3 . #33 diagonal (main)

. 1 . . . . . . .
4 5 . . . 1 . 3 2
. . . . 2 6 4 1 .
. . . . . 2 . 4 7
. . 7 . 8 . 3 . .
. 4 3 7 . . 8 2 .
. . 1 . 6 8 . 7 4
. 6 4 1 . 7 2 8 3
. 7 . 2 . . 1 6 . #34 diagonal (main)

. 1 . . . . . . .
4 . 5 . . 1 3 2 .
. 6 . 2 4 . 1 5 .
. . 7 . 1 6 . . 2
. . 1 4 . 7 6 8 .
. 4 . 5 2 . 7 1 .
. 3 4 . 5 2 8 7 .
. 7 6 . 8 4 2 . .
. . . 7 . . . . . #35 diagonal (main)

. 1 . . . . . . .
5 . 4 2 3 . . 1 .
. 6 . 1 . . 2 5 4
. 7 5 3 2 . . 4 .
. 3 . 7 . . . 2 5
. . . . . 8 7 . .
. . 7 . . 2 . 6 1
. 5 1 6 7 . 4 8 2
. . 6 . 1 . 5 7 . #36 diagonal (main)

. 1 . . . . . . .
4 6 5 3 2 . . . 1
. 3 . . 5 1 . 2 4
. 5 . 8 . . . . 7
. 7 3 . . 5 . 4 8
. . 4 . 3 . . 6 .
. . . . . . . 1 2
. . 7 . 1 6 4 8 3
. 4 1 2 8 . 7 5 . #37 diagonal (main)

. . 1 . . . . . .
. . 4 . 2 . . 3 .
2 6 . . 3 . 4 5 .
. . . . 7 4 . 1 5
. 1 7 3 5 2 8 6 .
. . . 6 1 . 3 7 .
. . 6 . 8 7 . . 3
. 7 5 2 4 3 . . 6
. . . 5 . . 7 4 . #38 diagonal (main)

. . . . 1 . . . .
. . 3 . . 4 . 2 .
. 5 4 2 . . . 3 1
. . 6 4 . 7 . 1 .
7 . . . 8 6 . . 2
. 4 . 1 2 . . 6 7
. . . . . . . 7 6
. 6 5 7 . 2 1 8 3
. . 7 . 6 1 2 5 4 #39 diagonal (main)

. . . 1 . . . . .
. . 3 . 5 4 . . 2
. 1 5 . 2 3 . 4 .
3 . . . 7 6 . . 1
. 5 7 2 . 1 . . 6
. 6 1 4 3 5 . 2 7
. . . . . . . 7 4
. . 6 . . 7 2 8 .
. 7 . 3 4 2 6 . . #40 diagonal (main)

. 1 . . . . . . .
4 . 5 . 3 . . . 2
. 3 6 5 2 . . 4 .
. . 3 . 4 2 . 5 7
. 5 7 1 8 3 4 2 .
. . . 7 5 . 8 . .
. . . . 1 8 . 7 5
. . 1 3 7 . 2 8 .
. 7 . 2 . . 3 . . #41 diagonal (main)

. 1 . . . . . . .
4 . 5 . 2 . . 3 .
. 3 6 . . 4 5 2 .
. . . . 7 5 . 1 8
. 7 . 2 . . 3 . .
. . 1 3 . 8 2 7 .
. . 3 . 5 7 . 4 2
. 5 7 4 . 2 1 8 3
. . . 8 . . 7 5 . #42 diagonal (main)

. 1 . . . . . . .
5 . 4 3 . . 2 . .
. 3 2 . 1 5 6 . 4
. 4 . . . 1 7 . .
. . 5 . . 7 1 . 6
. . 1 5 6 . . 4 8
. 2 7 6 5 . 8 . 3
. . . . . 3 . . 7
. . 3 . 7 8 4 6 2 #43 diagonal (main)

. . . . . 1 . . .
. . 2 4 5 . . 3 .
1 4 . 2 3 . 5 . 6
2 . 9 . 1 4 . 7 5
. 7 . . 9 . . 1 .
4 1 . 5 7 . 9 . 3
8 . 4 . 2 3 . 5 7
. 2 . . 4 5 3 . .
. . . 7 . . . . . #44 pi rotational (180 degrees)

. . . . . 1 . . .
. . 2 4 5 . . 3 .
1 4 . 2 3 . 5 . 6
2 1 . 3 7 . 9 . 5
. 7 . . 9 . . 1 .
4 . 9 . 1 2 . 7 3
8 . 4 . 2 3 . 5 7
. 2 . . 4 5 3 . .
. . . 7 . . . . . #45 pi rotational (180 degrees)

. . 1 . . . . . .
. . 4 . . 2 3 1 .
2 3 . . 1 5 . 4 6
. . . . . 1 . . 7
. . 2 . 6 7 1 5 .
. 1 7 2 5 8 . 3 .
. 4 . . 2 . . 7 3
. 2 3 . 7 4 5 6 .
. . 6 5 . . 4 . . #46 diagonal (main)

. . 1 . . . . . .
. . 4 1 . 3 . 2 .
6 7 . 4 . 2 1 . 5
. 6 7 3 . 4 . 5 .
. . . . . . . . 7
. 3 5 7 . 8 6 . 4
. . 6 . . 1 . 7 2
. 5 . 2 . . 4 . 1
. . 2 . 4 7 5 6 3 #47 diagonal (main)

. . . . . 1 . . .
. 3 1 4 . . . 2 .
. 2 5 . 3 6 . 4 1
. 7 . . . . . . .
. . 3 . 6 4 . 7 2
2 . 6 . 7 5 . 3 4
. . . . . . . 1 7
. 1 7 . 4 3 2 . 8
. . 2 . 1 7 4 9 3 #48 diagonal (main)

. . 1 . . . . . .
. 4 5 1 3 . . 2 .
3 2 . . 4 . . 5 1
. 3 . . 6 5 . 1 .
. 1 4 7 8 3 . 6 5
. . . 2 1 . 7 . .
. . . . . 6 . . 2
. 5 2 3 7 . . 8 6
. . 3 . 2 . 5 7 . #49 diagonal (main)

. . . . . . . . 1
. . 2 . 1 3 . 4 5
. 1 5 . 2 . 3 6 .
. . . . 7 1 . . .
. 2 1 3 5 8 . 7 6
. 7 . 2 9 . . 1 3
. . 7 . . . . . .
. 4 6 . 3 2 . 5 7
2 5 . . 6 7 . 3 . #50 diagonal (main)

. . . . . . . . 1
. . 2 . 1 3 5 4 .
. 1 3 4 . . 2 6 .
. . 7 . . . . 8 .
. 2 . . . 4 6 . .
. 3 . . 7 8 4 . 2
. 6 1 . 5 7 8 2 4
. 7 5 8 . . 1 . 6
2 . . . . 1 7 5 . #51 diagonal (main)

. . . . . . . . 1
. . 2 . 1 3 4 5 .
. 1 4 . 6 . 3 2 .
. . . . 7 . . 3 .
. 2 6 3 . 5 . 8 .
. 7 . . 8 . 5 . 2
. 4 7 . . 8 . 1 5
. 8 1 7 5 . 2 . 3
2 . . . . 1 8 7 . #52 diagonal (main)

. . 1 . . . . . .
. 5 6 4 3 . . 2 .
2 8 . . 5 1 . 3 7
. 4 . . 6 . . 7 .
. 7 5 8 4 . . 6 2
. . 2 . . 5 . . 8
. . . . . . . 1 3
. 1 7 3 8 . 2 . 6
. . 3 . 1 6 7 8 . #53 diagonal (main)

. 1 . . . . . . .
6 . 5 4 . 3 2 . .
. 4 2 . 1 6 3 . 5
. 5 . . . 1 . . .
. . 6 . . 7 1 . 3
. 7 1 6 3 . . 5 8
. 2 7 . 6 . 8 . 4
. . . . . 4 . . 7
. . 4 . 7 8 5 3 2 #54 diagonal (main)

. . 1 . . . . . .
. . 5 3 . 4 . 1 2
7 2 . . 1 6 . 5 3
. 8 . 6 . . . . 1
. . 7 . 4 3 . 8 .
. 4 6 . 8 . 3 2 .
. . . . . 8 . . 5
. 7 2 . 3 5 . 6 8
. 5 8 7 . . 2 3 . #55 diagonal (main)

. . 1 . . . . . .
. . 5 . 1 2 3 4 .
2 4 . . 5 6 . 1 7
. . . . . 1 . . 8
. 2 4 . 3 8 . 6 1
. 1 8 2 6 7 4 . .
. 3 . . . 5 . 8 .
. 5 2 . 8 . 6 . .
. . 7 6 2 . . . 3 #56 diagonal (main)

. . 1 . . . . . .
. . 5 3 . 4 2 . 1
4 2 8 . 7 1 3 . 6
. 6 . . 1 . 8 . .
. . 7 4 . . 1 . 2
. 1 4 . . 8 . 6 .
. 5 6 8 4 . 9 . 3
. . . . . 3 . . 5
. 4 3 . 5 . 6 2 . #57 diagonal (main)

. . 1 . . . . . .
. . 5 . 3 4 2 . 1
4 2 . . 6 1 . 3 5
. . . . 1 . . . .
. 6 3 4 . 5 1 . 7
. 1 4 . 2 7 . 8 3
. 5 . . 4 . 7 . .
. . 6 . . 8 . . 2
. 4 2 . 7 6 . 5 8 #58 diagonal (main)

. . 1 . . . . . .
. . 5 . 3 4 1 2 .
4 2 7 . 6 1 . 3 5
. . . . . 3 . . .
. 6 3 . 8 5 7 1 .
. 1 4 6 2 . . 8 3
. 4 . . 7 . . 5 .
. 5 6 . 4 8 2 . .
. . 2 . . 6 . . 8 #59 diagonal (main)

. . . 1 . . . . .
. 4 3 . 5 . . 1 2
. 1 6 . 4 2 . 5 3
3 . . . 7 . . 2 .
. 5 4 2 6 . . 3 7
. . 7 . . . 5 . .
. . . . . 5 . 7 1
. 3 5 7 1 . 2 . 8
. 7 1 . 2 . 3 9 . #60 diagonal (main)

. . 1 . . . . . .
. 5 6 2 . 1 3 4 .
2 4 . 3 . 6 . 5 1
. 1 8 . . . 7 3 .
. . . . . 8 . . 4
. 2 4 . 3 7 . 1 8
. 8 . 7 . . . 6 .
. 6 5 8 . 2 4 7 .
. . 2 . 6 3 . . . #61 diagonal (main)

. . 1 . . . . . .
. 5 6 . 1 2 . 4 3
2 4 . 3 . 7 1 . 5
. . 7 . . 3 . . 1
. 2 . . 8 1 . 3 .
. 1 3 7 2 . 4 8 .
. . 2 . . 6 . . 4
. 6 . . 7 8 . . .
. 7 5 2 . . 6 . 8 #62 diagonal (main)

. . . . . . . 2 1
. . . . . 3 . 4 .
. . 4 . 5 2 3 . 6
. . . . 2 7 . . .
. . 2 5 8 1 . 7 3
. 7 5 3 9 4 . 1 2
. . 7 . . . . . 9
5 4 . . 3 9 . . 7
9 . 6 . 7 5 1 3 . #63 diagonal (main)

. . . . . . . 2 1
. . . . . 3 4 . .
. . 4 . 1 5 . . 6
. . . . 7 . 8 . 3
. . 7 1 . 8 2 6 .
. 8 2 . 3 6 7 1 .
. 4 . 3 5 1 . 7 .
5 . . . 6 7 1 4 8
7 . 6 8 . . . 3 . #64 diagonal (main)

. 1 2 . . . . . .
4 . 5 . 3 1 . . .
6 3 7 5 2 . . 4 1
. . 3 . . . . 6 .
. 5 6 . 7 3 . 2 4
. 4 . . 5 . . . 7
. . . . . . . 1 2
. . 1 2 6 . 4 . 3
. . 4 . 1 7 6 5 8 #65 diagonal (main)

. 1 2 . . . . . .
4 5 6 . 3 1 . . .
7 3 8 6 2 . . 4 1
. . 3 . . . . 7 .
. 6 7 . 8 3 . 2 4
. 4 . . 6 . . . 8
. . . . . . . 1 2
. . 1 2 7 . 4 . 3
. . 4 . 1 8 7 6 . #66 diagonal (main)

. 1 2 . . . . . .
4 5 . . . 1 . 3 .
7 . 8 . 2 6 4 1 .
. . . . . 2 . 4 7
. . 7 . 9 . 3 . .
. 4 3 7 . . 9 2 .
. . 1 . 6 9 . 7 4
. 6 4 1 . 7 2 . 3
. . . 2 . . 1 6 . #67 diagonal (main)

. 1 2 . . . . . .
4 . 5 1 . . . 2 3
3 6 7 2 . . 1 5 .
. 4 3 . . . 8 1 .
. . . . . 1 . . 6
. . . . 4 8 . 3 5
. . 4 8 . . . 6 2
. 3 6 4 . 2 5 8 .
. 2 . . 5 6 3 . . #68 diagonal (main)

. 2 1 . . . . . .
4 . 5 . 2 3 1 . .
3 7 . . 4 . . 2 6
. . . . 1 2 . . .
. 4 2 3 6 5 7 1 .
. 1 . 4 7 8 . 5 2
. 3 . . 5 . . . .
. . 4 . 3 7 . 6 5
. . 6 . . 4 . 7 . #69 diagonal (main)

. 2 1 . . . . . .
4 . 5 . 2 3 1 . .
3 8 7 . 4 . . 2 6
. . . . 1 2 . . .
. 4 2 3 6 5 8 1 .
. 1 . 4 8 . . 5 2
. 3 . . 5 . . . .
. . 4 . 3 8 . 6 5
. . 6 . . 4 . 8 . #70 diagonal (main)

. 2 1 . . . . . .
4 6 5 . 1 . . 2 3
3 7 . 5 . 2 1 . 4
. . 7 . . . 4 . .
. 3 . . . 5 . . 8
. . 4 . 7 8 . 3 6
. . 3 2 . . 6 . 1
. 4 . . . 1 . . 2
. 1 2 . 8 6 3 4 . #71 diagonal (main)

. 1 2 . . . . . .
5 . 6 . 4 1 . . 3
4 3 7 . 2 . . 6 1
. . . . 1 2 . . .
. 2 4 5 . 3 . 1 8
. 5 . 4 6 8 7 . 2
. . . . . 7 . . 6
. . 3 . 5 . . 8 .
. 6 5 . 8 4 3 . . #72 diagonal (main)

. . . . . 1 . . 2
. . . . 3 . 4 1 .
. . 2 4 5 . 6 . 3
. . 4 . 1 5 3 . 7
. 5 3 7 2 . 1 . 6
7 . . 3 . . . . .
. 4 8 5 7 . . 6 1
. 7 . . . . 8 . .
2 . 5 1 8 . 7 . 4 #73 diagonal (main)

. . 1 . 2 . . . .
. . 3 5 . 4 1 . .
2 5 . 6 1 . . . 4
. 3 6 8 4 2 7 . 1
1 . 2 7 . . 4 3 .
. 7 . 1 . . . 6 .
. 2 . 4 7 . . . 3
. . . . 5 6 . . 7
. . 7 2 . . 5 4 . #74 diagonal (main)

. 2 . . 1 . . . .
5 . 4 2 . 3 1 . .
. 1 . . . 4 . 2 3
. 5 . . 3 1 6 . .
4 . . 6 . 2 . . 1
. 6 1 4 5 7 3 . 2
. 4 . 3 . 6 8 . .
. . 5 . . . . . 6
. . 6 . 4 5 . 3 7 #75 diagonal (main)

. 2 . . 1 . . . .
5 . 4 3 . 2 1 . .
. 1 . . . 4 . 2 3
. 6 . . 5 1 3 . .
4 . . 2 . 6 . . 1
. 5 1 4 3 7 6 . 2
. 4 . 6 . 3 8 . .
. . 5 . . . . . 6
. . 6 . 4 5 . 3 7 #76 diagonal (main)

. 1 . 2 . . . . .
4 . 3 5 1 . 2 . .
. 2 . . 6 4 1 . 3
3 5 . . . 6 7 . .
. 4 7 . . . 6 . 8
. . 1 7 . . . 3 5
. 3 4 6 7 . 8 . 2
. . . . . 2 . . 7
. . 2 . 8 5 3 6 . #77 diagonal (main)

. 1 . 2 . . . . .
4 . 3 6 5 . 2 . .
. 2 . . 1 4 7 . 3
3 6 . . . 1 8 . .
. 5 4 . . . 1 . 7
. . 1 4 . . . 3 6
. 3 8 7 4 . 6 . 2
. . . . . 2 . . 8
. . 2 . 8 6 3 7 . #78 diagonal (main)

. . 1 . . 2 . . .
. . . . 4 . . . 3
5 . 4 . 3 6 . 2 .
. . . . 7 8 . . 4
. 4 7 3 6 5 2 . 8
8 . 6 2 1 . 3 7 .
. . . . 8 7 . 3 2
. . 8 . . 3 7 . .
. 7 . 4 2 . 8 . 6 #79 diagonal (main)

1 . . . . 2 . . .
. . 4 5 6 . . . 3
. 3 . . 4 1 . 2 5
. 7 . . . 3 . . .
. 6 3 . 1 5 . 7 8
8 . 1 4 7 . 3 . 2
. . . . . 4 . 8 7
. . 8 . 5 . 2 . .
. 4 7 . 2 8 5 . 1 #80 diagonal (main)

1 . . . . 2 . . .
. . 4 5 6 . . 3 .
. 3 . . 4 1 . 5 2
. 7 . . . 3 . . .
. 6 3 . 1 5 . 7 8
8 . 1 4 7 . 3 2 .
. . . . . 4 . 8 7
. 4 7 . 5 8 2 1 .
. . 8 . 2 . 5 . . #81 diagonal (main)

. . . . . 1 . 2 .
. 2 1 . 3 . 4 . 5
. 4 6 5 2 . 7 1 .
. . 7 2 . . . . 4
. 6 2 . 7 . 5 3 .
4 . . . . 5 2 . .
. 1 3 . 5 7 8 4 .
6 . 4 . 8 . 1 5 .
. 8 . 1 . . . . . #82 antidiagonal (+ pattern main diagonal symmetry bonus)

. . 1 2 . . . . .
. . . . 4 . . 1 3
3 . 5 . 6 1 4 2 .
7 . . . 8 . . . .
. 8 6 4 . . . 3 7
. . 3 . . 6 8 4 .
. . 8 . . 4 . 7 1
. 3 7 . 1 8 2 6 4
. 1 . . 2 . 3 8 . #83 diagonal (main)

. 2 . . . . . 1 .
5 . 4 . 1 3 2 . .
. 1 6 . 2 4 3 . .
. . . . . 7 . . 1
. 4 5 . 8 1 7 3 .
. 7 1 3 4 . 8 . 6
. 5 7 . 3 8 6 . .
4 . . . 7 . . . 3
. . . 4 . 6 . 7 . #84 diagonal (main)

. 2 . . . . . 1 .
5 . 4 . 1 3 2 . .
. 1 6 . 2 4 3 . .
. . . . . 7 . . 1
. 4 5 . 8 1 7 3 .
. 7 1 3 4 . 6 . 8
. 5 7 . 3 6 8 . .
4 . . . 7 . . . 3
. . . 4 . 8 . 7 . #85 diagonal (main)

. 1 . 2 . . . . .
4 . . . . 5 . 1 3
. . 7 1 3 . 2 4 6
8 . 4 6 5 . . . .
. . 5 3 . 2 . 8 4
. 3 . . 8 . 6 . .
. . 8 . . 6 . . 1
. 4 1 . 2 . . . 8
. 5 6 . 1 . 4 2 9 #86 diagonal (main)

. . 2 1 . . . . .
. . 1 . . 4 2 3 .
6 4 . . 3 . . 5 1
4 . . . 2 7 . 1 .
. . 7 6 . 1 . . 3
. 1 . 3 4 8 . 2 7
. 6 . . . . . 7 .
. 7 5 4 . 6 3 8 .
. . 4 . 7 3 . . 5 #87 diagonal (main)

. . 2 1 . . . . .
. 5 1 . 2 . . 4 3
4 7 8 6 . 3 1 2 .
7 . 3 . . 6 . 5 .
. 4 . . 8 . 3 . .
. . 6 3 . . 4 7 .
. . 7 . 6 2 . . 4
. 2 4 5 . 1 . . 7
. 6 . . . . 2 1 . #88 diagonal (main)

. . 2 1 . . . . .
. 5 1 . 2 . 4 . 3
4 7 8 6 . 3 1 2 .
7 . 3 . . 6 5 . .
. 4 . . 8 . . 3 .
. . 6 3 . . 7 4 .
. 2 7 5 . 1 . . 4
. . 4 . 6 2 . . 7
. 6 . . . . 2 1 . #89 diagonal (main)

. 3 2 1 . . . . .
5 . . . . 4 . 3 2
4 . 6 3 2 . 1 5 .
8 . 5 7 4 . . . .
. . 4 2 . 1 . 8 5
. 2 . . 8 . 7 . .
. . 8 . . 7 . . 3
. 5 3 . 1 . . . 8
. 4 . . 3 . 5 1 9 #90 diagonal (main)


#82 has an extra pattern symmetry.

[blue]

Below are 12 minimal puzzles with 38 symmetric givens.

Below are 90 minimal puzzles with 37 symmetric givens.

Nice !

I have a couple of questions:
Do those lists include every puzzle in your 37+ lists, that can be transformed to show certain kinds of symmetries ?
If so, did you check them for "lesser known" symmetry types, as well ?
(In other words, would those include every puzzle in your lists, that has a non-trivial automorphism group ?)

[dobrichev]

I used automorphism for search criteria.
Saying that, now I realized that I am not sure whether all reported symmetries are correct. Glenn's tool seemingly reports the pattern symmetries and converts the puzzle to its "most symmetric" form.

Do "less known" symmetries that cannot be represented as (rotations and/or transposition) + re-labelling exist?

[blue]

Do "less known" symmetries that cannot be represented as (rotations and/or transposition) + re-labelling exist?

They do. This one, with 3 automorphisms, has "Gliding Rows" symmetry.
It isn't minimal, or even difficult ... but it does have the symmetry.

Code: Select all

+-------+-------+-------+
| . 5 2 | . 8 1 | . 6 3 |
| . 4 3 | . 6 2 | . 9 1 |
| . 7 1 | . 5 3 | . 4 2 |
+-------+-------+-------+
| 3 . 6 | . 2 5 | 4 1 . |
| 5 1 . | 3 . 4 | . 2 6 |
| . 2 4 | 6 1 . | 3 . 5 |
+-------+-------+-------+
| . . . | 5 3 8 | 1 7 4 |
| 1 8 5 | . . . | 6 3 9 |
| 4 3 7 | 1 9 6 | . . . |
+-------+-------+-------+

[eleven]

I have no tools to verify it, but for me it is very unprobable, that high clue puzzles have a symmetry other than pi or diagonal.
I let gsf's program list the number of automorphisms of the solution grids in Mladen's 38 collection (2 mio). Out of 162 (103 ed) with more than the trivial automorphisms there was a single one with 3 - and the puzzle can't have it (single box with 6 givens).
I quickly found a grid with sticks symmetry manually (don't know how much there are), but the puzzle was far away from having the symmetry too:

Code: Select all

38 clue with 2 automorphisms in the grid
 +-------+-------+-------+
 | . . 3 | 4 5 6 | 7 . 9 |
 | . 5 6 | 7 8 . | . . 3 |
 | . . . | . . . | . . . |
 +-------+-------+-------+
 | . 4 . | . 1 7 | 8 9 . |
 | 6 . . | . . . | . . 1 |
 | 8 . 1 | 6 9 5 | . 7 4 |
 +-------+-------+-------+
 | . 6 4 | . 7 1 | 9 . 8 |
 | . . 8 | 9 . 4 | . . . |
 | . . . | . 6 8 | . 4 7 |
 +-------+-------+-------+
Switch B12,c12,c45,c78 to get the grid in normal sticks symmetry
 +-------+-------+-------+
 | 4 . x | 1 . 7 | 9 8 . |
 | x 6 . | x x . | . . 1 |
 | . 8 1 | 9 6 5 | 7 . 4 |
 +-------+-------+-------+
 | x . 3 | 5 4 6 | x 7 9 |
 | 5 . 6 | 8 7 x | 2 . 3 |
 | . . . | . . . | . . . |
 +-------+-------+-------+
 | 6 . 4 | 7 . 1 | . 9 8 |
 | . x 8 | . 9 4 | x . . |
 | 1 x . | 6 x 8 | 4 . 7 |
 +-------+-------+-------+
12 givens missing for sticks symmetry in the puzzle (marked x)
+-------+-------+-------+
 | 4 2 5 | 1 3 7 | 9 8 6 |
 | 9 6 7 | 4 8 2 | 3 5 1 |
 | 3 8 1 | 9 6 5 | 7 2 4 |
 +-------+-------+-------+
 | 2 1 3 | 5 4 6 | 8 7 9 |
 | 5 4 6 | 8 7 9 | 2 1 3 |
 | 8 7 9 | 2 1 3 | 6 4 5 |
 +-------+-------+-------+
 | 6 3 4 | 7 2 1 | 5 9 8 |
 | 7 5 8 | 3 9 4 | 1 6 2 |
 | 1 9 2 | 6 5 8 | 4 3 7 |
 +-------+-------+-------+
(1)(4)(7)(23)(56)(89)

[champagne]

Hi eleven,

using a collection of minimal puzzles, I would agree

We have in the file of potential hardest a non minimal puzzle from mauricio very hard with a high number of clues.

Here is another example of double diagonal symmetry of given of size 40

Code: Select all

.43192...
2..3.81..
7...4..9.
17..8..46
9.26.48.1
46..2..39
.1..6...3
..92.7..8
...81976.

That one should be "symmetrically minimal" but can not be minimal (blues list does not go over 37 clues)

in fact, as in mauricio's puzzle, you can clear many clues and still have a valid puzzle.

here is a progressive reduction to a 25 clues minimal puzzle

Hidden Text: Show

Code: Select all

.43192...2..3.81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
..3192...2..3.81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
...192...2..3.81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....92...2..3.81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9....2..3.81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.......3.81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7...4..9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.17..8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1...8..469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1......469.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1.......69.26.48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1.......69.2..48.146..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1.......69.2..48..46..2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1.......69.2..48..4...2..39.1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1.......69.2..48..4...2..3..1..6...3..92.7..8...81976. ED=1.2/1.2/1.2
....9.........81..7......9.1.......69.2..48..4...2..3..1..6...3..92.7..8....1976. ED=2.0/1.2/1.2


[dobrichev]

A 15 hours scan over about 1e10 minimal 36-given puzzles gave 475 symmetrical ones, none with more than one symmetry.

[m_b_metcalf]

here is a progressive reduction to a 25 clues minimal puzzle

Or even lower (but I couldn't manage 21). Examples:

Code: Select all

.431.2...2..3.........4..9..7..8..4...26....1.6........1..6...3..9.........8.976. 23
..3192......3.8...7.........7.....4.9....48.146..2...9....6...3..92....8.......6. 23

.4.......2....81..7......9.17......6.....48..4...2..3.....6...3..92....8....197.. 22
.4.1.....2....81..7............8...6..2..4..146.....3..1..6...3..92.7........9.6. 22

Regards,

Mike

[Leren]

dobrichev wrote : #82 has an extra pattern symmetry.

OK I'll ask the dumb question. What's the extra pattern symmetry in your puzzle # 82 ?

It looks like the anti-diagonal digits 379 can't occupy the main diagonal for some (symmetry) reason, but I'm just guessing.

Leren

[dobrichev]

#82 has anti-diagonal symmetry for pattern AND givens, which is the subject of this thread. Additionally, by coincidence, it has diagonal symmetry for the pattern but not for the givens.
The first symmetry always helps in the solving process. The re-labelling cycles, once obtained from the givens, are applicable to all pencilmarks. The solver can tailor additional constrains in the form "if the value of cell X is N, then the value of its symmetric cell X' is the corresponding value N'". The same is applicable to eliminated values of course.
The secondary symmetry is less constrained and is not discussed here. It might help in the solving process in some different way, or might not help at all.

[eleven]

A 15 hours scan over about 1e10 minimal 36-given puzzles gave 475 symmetrical ones, none with more than one symmetry.

Does it mean, that in 10 mio 36's there was not one with symmetrical givens, but in 2 mio 38's there are 12 ? Interesting.
Btw. as i use it, there is a slight difference between digital symmetry (puzzle/grid with symmetrical givens) and automorphism. A symmetry has at least 2 automorphisms (beyond the trivial one).

[diagonal symmetry for the pattern but not for the givens] ...might help in the solving process in some different way, or might not help at all.

No, never even heard of an idea, how it could.

[dobrichev]

In 10 billions 36s there are 475 symmetrical puzzles. All the symmetries are diagonal or 180 deg rotational. All symmetrical puzzles have exactly 2 automorphisms.

The irreducible symmetrical puzzles of size 36, 37 and 38 can be downloaded from here.

The sizes of the scanned collections are respectively
36s - about 1e10 puzzles (+/- 20%), 475 of them symmetric
37s - 423 774 106 puzzles, 110 of them symmetric
38s - 2 334 099 puzzles, 12 of them symmetric

[Leren]

dobrichev wrote: The irreducible symmetrical puzzles of size 36, 37 and 38 can be downloaded from here.

I've just downloaded your puzzles and the 36 clue ones all have the same sequence number. Is it possible for you to correct this ?

Leren

<edit> In the meantime I manually edited your 36 clue .niced file and came up with 464 puzzles. Did i get it right ?

Leren

[eleven]

<edit> In the meantime I manually edited your 36 clue .niced file and came up with 464 puzzles. Did i get it right ?

Hm, my python script gives only 462.

Hidden Text: Show

Code: Select all

#!/usr/bin/python
import sys

def main():
    f = open('36symmetric.niced', 'rU')
    o = open('36symmetric.numbered.txt', 'w')
    nr = 0
    lines = f.readlines()
    for line in lines:
        if line[:2] == ' 2':
            nr += 1
            o.write('#'+ str(nr)+ line)
        else:
            o.write(line)

if __name__ == '__main__':
    main()

The missing 13 (?) puzzles seem to have sticks symmetry, this is line 14 in 36symmetric, in row sticks normal form:

Code: Select all

 +-------+-------+-------+
 | 1 2 . | . 8 . | 4 . 6 |
 | . 9 . | . . . | . 3 . |
 | 4 . 6 | . 2 . | 7 8 . |
 +-------+-------+-------+
 | . 8 9 | . 1 2 | 6 . 7 |
 | 2 . . | . 6 . | 8 . . |
 | 6 . 1 | . 7 8 | . 2 3 |
 +-------+-------+-------+
 | . 1 . | . 9 . | 3 . 4 |
 | . . . | . . 4 | . . . |
 | 9 . 4 | . 3 . | . 7 . |
 +-------+-------+-------+
(4)(5)(6)(1,7)(2,8),(3,9)