## Symmetrical Givens

Everything about Sudoku that doesn't fit in one of the other sections

### Re: Symmetrical Givens

blue wrote:
Code: Select all
`2 3 . . . 7 . . 4                 . . 4 . . 7 2 3 .. . 7 . . 4 . . 1                 . . 1 . . 4 . . 7. . 4 . . 1 8 9 .                 8 9 . . . 1 . . 41 2 3 . . . . . .   swap stacks   . . . . . . 1 2 3. . . . . . . . .     1 and 3     . . . . . . . . .. . . . . . 7 8 9   <--     -->   7 8 9 . . . . . .. 1 2 9 . . 6 . .                 6 . . 9 . . . 1 29 . . 6 . . 3 . .                 3 . . 6 . . 9 . .6 . . 3 . . . 7 8                 . 7 8 3 . . 6 . .`
Nice find, blue.
With all the symmetries (each clue forces 4 or 8) i would not have expected a minimal puzzle at all.

The funny thing is, that we have
- quarter symmetry (4)
- half turn (2)
- diagonal (2)
- antidiagonal (2)
- column sticks (2)
- row sticks (2)

8 automorphisms

but only 2 single symmetries, diagonal and sticks, imply them all.
eleven

Posts: 1559
Joined: 10 February 2008

### Re: Symmetrical Givens

Hi, blue!
blue wrote:Here's a puzzle that can have either, but not both at the same time.
Does one "look more symmetric" than the other ?

Code: Select all
`2 3 . . . 7 . . 4                 . . 4 . . 7 2 3 .. . 7 . . 4 . . 1                 . . 1 . . 4 . . 7. . 4 . . 1 8 9 .                 8 9 . . . 1 . . 41 2 3 . . . . . .   swap stacks   . . . . . . 1 2 3. . . . . . . . .     1 and 3     . . . . . . . . .. . . . . . 7 8 9   <--     -->   7 8 9 . . . . . .. 1 2 9 . . 6 . .                 6 . . 9 . . . 1 29 . . 6 . . 3 . .                 3 . . 6 . . 9 . .6 . . 3 . . . 7 8                 . 7 8 3 . . 6 . .`

Note: column and row stick symmetries, are also present.
I think it's the only (truly) minimal puzzle of its kind.
Too bad it's solvable with singles.

Impressive example! I couldn't imagine such possibility .
I should think better ...

Serg
Serg
2017 Supporter

Posts: 513
Joined: 01 June 2010
Location: Russia

### Re: Symmetrical Givens

I looked for puzzles with DD+JD symmetries (12 automorphisms). There seem to be only 5 grids, which have it, with 12,36,72,108 and 648 (MC grid) automorphisms.
The 169 puzzles i found in 1/2 an hour, all are easy, and only four are minimal.
Code: Select all
` +-------+-------+-------+ | . . . | 1 . . | 7 9 8 | | . . . | 7 . . | . . 6 | | . . . | 4 2 6 | . . 5 | +-------+-------+-------+ | 2 6 4 | . . . | 9 . . | | . . 1 | . . . | 2 . . | | . . 7 | . . . | 3 5 1 | +-------+-------+-------+ | 6 . . | 5 1 3 | . . . | | 5 . . | . . 9 | . . . | | 8 7 9 | . . 2 | . . . | +-------+-------+-------+`

The "hardest" one was
Code: Select all
` +-------+-------+-------+ | . . 7 | . . 4 | . . 2 | | . 3 . | . 5 . | . 1 . | | 6 . . | 9 . . | 8 . . | +-------+-------+-------+ | . . 5 | . . 2 | . . 3 | | . 9 . | . 8 . | . 7 . | | 4 . . | 1 . . | 6 . . | +-------+-------+-------+ | . . 8 | . . 7 | . . 5 | | . 2 . | . 6 . | . 4 . | | 1 . . | 3 . . | 9 . . | +-------+-------+-------+`

This is one of the minimal puzzles
Code: Select all
` +-------+-------+-------+ | . . 7 | . 2 4 | . . . | | . . . | . . 7 | 6 . . | | 6 . . | . . . | 8 9 . | +-------+-------+-------+ | . . . | . . 2 | . 5 3 | | 1 . . | . . . | . . 2 | | 4 6 . | 1 . . | . . . | +-------+-------+-------+ | . 7 8 | . . . | . . 5 | | . . 5 | 9 . . | . . . | | . . . | 3 1 . | 9 . . | +-------+-------+-------+`
eleven

Posts: 1559
Joined: 10 February 2008

### Re: Symmetrical Givens

eleven wrote:I looked for puzzles with DD+JD symmetries (12 automorphisms). There seem to be only 5 grids, which have it, with 12,36,72,108 and 648 (MC grid) automorphisms.

Hi eleven,

Nice work !
There are some additional grid and puzzle types, that I'm guessing you wern't looking for (?).
The JD's in your examples, have 3 cycles of 3 digits for the relabeling part the automorphism.
There are also varieties with 0,1 and 2 cycles of 3 digits.
There's a puzzle for each, shown below (with DD symmetry too).

Something similar happens for Gliding Rows (SR), DBS (D+BS), and BxCxSR (BxCx+SR) symmetries.

For SR, it's 0,1,2 or 3 cycles of 3.
For DBS and BxCxSR, the options are:
1) one cycle of 6, and 3 fixed digits
2) one cycle of 6 and one of 3
3) 3 cycles of 2, and 3 fixed digits
4) 3 cycles of 2, and one of 3
I like to think that it expands the 26 "basic" symmetries to 38, but I'm sure I'ld get a little feedback on that idea.

---

DD+BS puzzles ...

Two cycles of 3, and 3 fixed digits:
Code: Select all
`1 . . | . 5 6 | . . .. . . | . . 4 | 7 . .. . 3 | . . . | 8 9 .------+-------+------. . . | 1 . . | . 6 48 . . | . . . | . . 59 7 . | . . 3 | . . .------+-------+------. 4 5 | . . . | 1 . .. . 6 | 9 . . | . . .. . . | 7 8 . | . . 3`

One cycle of 3, and 6 fixed digits:
Code: Select all
`. . 6 | . 1 7 | . . .. . . | . . 2 | 4 . .3 . . | . . . | 9 5 .------+-------+------. . . | . . 6 | . 1 84 . . | . . . | . . 27 5 . | 3 . . | . . .------+-------+------. 1 9 | . . . | . . 6. . 2 | 4 . . | . . .. . . | 8 5 . | 3 . .`

9 fixed digits:
Code: Select all
`7 . . | . 1 2 | . . .. . . | . . 3 | 4 . .. . 9 | . . . | 5 6 .------+-------+------. . . | 7 . . | . 1 24 . . | . . . | . . 35 6 . | . . 9 | . . .------+-------+------. 1 2 | . . . | 7 . .. . 3 | 4 . . | . . .. . . | 5 6 . | . . 9`
blue

Posts: 573
Joined: 11 March 2013

### Re: Symmetrical Givens

Thank you blue,

i should have known that, at least i remember, that we had JD's (and gliding rows) without digit changes, probably in the MC grid. But i forgot to think about it here.

The 26 "basic symmetries" come from Red Ed's classes, and these are only about cell automorphisms, with cell cycles, not digit cycles.
But it's an option to split them into possible digit cycles, of course.
eleven

Posts: 1559
Joined: 10 February 2008

### Re: Symmetrical Givens

some results on the DD symmetry.

The minimal count on my side confirms blue's results, 50781 puzzles.

I also collected puzzles having a "minimal symmetry", but not minimal stricto sensu and I found 1 906 799 puzzles.

The highest number of clues for such puzzles was 42 with 12 puzzles

Hidden Text: Show
Code: Select all
`523.1.8..4.8.7.95.76..2..14....47...1342.8679...36....69..8..43.51.3.2.6..2.9.785523.1.8..4.8.7.95.76..2..14....63...1348.2679...74....69..8..43.51.3.2.6..2.9.785123.9.8..4.8.7.15.76..2..94....47...9342.8671...36....61..8..43.59.3.2.6..2.1.789123.9.8..4.8.7.15.76..2..94....63...9348.2671...74....61..8..43.59.3.2.6..2.1.789523.6.81.4.8.3..5976..1...4....83...8716.4932...72....6...9..4315..7.2.6.92.4.785523.6.81.4.8.3..5976..1...4....27...8714.6932...38....6...9..4315..7.2.6.92.4.785523.1.8..4.8.7.95.76..2..14...16....1348.2679....49...69..8..43.51.3.2.6..2.9.785523.1.8..4.873.9..76..2..14.3..6....1748.2639....4..7.69..8..43..1.732.6..2.9.785523.1.8..4.873.1..76..2..94.3..6....1748.2639....4..7.61..8..43..9.732.6..2.9.78512637.9..4.7.9.6..83..4..217...8....3926.4817....2...398..6..72..4.1.3.6..1.3748913..9.24.7.2.6.9.8.4.37..16..7.2....9834.6721....8.3..49..37.6.2.1.4.8.3.68.1..7953..1.24.7.2.6.1.8.4.73..96..3.2....1874.6329....8.7..41..73.6.2.9.4.8.3.68.9..75`

Only hard puzzles in that group with a very small number of redundant clues are of interest. I'll look for such puzzles.

The process applied does not use the solution grids, but I have still one test to run to see what is the best process in that family.
champagne
2017 Supporter

Posts: 5675
Joined: 02 August 2007
Location: France Brittany

### Re: Symmetrical Givens

Hi, champagne!
champagne wrote:some results on the DD symmetry ...

Great work, congratulations!
champagne wrote:I also collected puzzles having a "minimal symmetry", but not minimal stricto sensu and I found 1 906 799 puzzles.
The highest number of clues for such puzzles was 42 with 12 puzzles ...

Very interesting! What about "low clue" end of these "minimal symmetric" puzzles?

Serg
Serg
2017 Supporter

Posts: 513
Joined: 01 June 2010
Location: Russia

### Re: Symmetrical Givens

Serg wrote:What about "low clue" end of these "minimal symmetric" puzzles?
Serg

the lowest number of clues for such a symmetry of given was 22.
EDIT something wrong in my search, I gave page 5 of that thread the list of such puzzles or size 20

Hidden Text: Show
98.......6......5.....123.....9..6....1...9....4..1.....789.....5......4.......21 ED=7.1/1.2/1.2
1...4.......2...7....3..5...479.....2.......8.....136...5..7....3...8.......6...9 ED=3.6/3.6/2.6
98......36..........5.12......9..6....1...9....4..1......89.5..........47......21 ED=1.5/1.2/1.2
5..1......9..........24.3..1.4..3.....2...8.....7..6.9..7.68..........1......9..5 ED=2.0/1.2/1.2
98......36......5.....12......9..6....1...9....4..1......89.....5......47......21 ED=1.5/1.2/1.2
98.......6......5.....12......9.36....1...9....47.1......89.....5......4.......21 ED=1.7/1.2/1.2
98.......6......3.....12......9.36....1...9....47.1......89.....7......4.......21 ED=1.7/1.2/1.2
96...1...81.....3......27........6.9.........1.4........38......7.....92...9...41 ED=1.2/1.2/1.2

here the 658 puzzles of size 22

a quick check gives (not checked for ED elementary puzzles, but not likely to come at depth -1)

1460 minimal puzzles oz size 21
546 puzzles of size 20
150 puzzles of size 19
24 puzzles of size 18

Hidden Text: Show
1..2..4...9..6..3...5.....84..1......8.....2......9..62.....5...7..4..1...6..8..9
1..2....3.9...68....5....4.4..1...2...........8...9..6.6....5....24...1.7....8..9
1..2..4......1..3...9..6..84.....2...1.....9...8.....62..4..1...7..9......6..8..9
5..1..2....6....3..8..9...61....7.....9...1.....3....94...1..2..7....4....8..9..5
1..2..4.....3...7...5.9...847.........9...1.........362...1.5...3...7.....6..8..9
1..2..4.....1...3...9.7...841.........3...7.........962...3.1...7...9.....6..8..9
5..2..1......63.....1.....94....7.3..8.....2..7.3....61.....9.....74......9..8..5
5..2..1......36.....1.....94....7.2..7.....3..8.3....61.....9.....47......9..8..5
12.......4..8...3.....96....6...72....9...1....83...4....41.....7...2..6.......89
12.......4..8...3.....96....6...32....9...1....87...4....41.....7...2..6.......89
12.......4..3...5.....98....7.1..4....9...1....6..9.3....21.....5...7..6.......89
12.......4..8...3...5.6.7...6...7.....8...2.....3...4...3.4.5...7...2..6.......89
13.......7..2...3.....45....4.1..5....2...8....5..9.6....56.....7...8..3.......79
12.......4..3..1....5..6.9..7.1..2.............8..9.3..1.4..5....9..7..6.......89
12.......4..3..1....5..8.9..7.1..4.............6..9.3..1.2..5....9..7..6.......89
12.......4..8...3...96..7...681...................924...3..41...7...2..6.......89
1..2..4...5.1...3...9.....841...7...............3...962.....1...7...9.5...6..8..9
52.......4..8...3...14..5...62..7...............3..84...5..69...7...2..6.......85
52.......4..8...3...14..5...62..3...............7..84...5..69...7...2..6.......85
12.......4..8.........963...6...72....9...1....83...4...741.........2..6.......89
12.......4..3.........987...7.1..4....9...1....6..9.3...321.........7..6.......89
1..2.4....9.....3.....6.7..4..1....8..8...2..2....9..6..3.4.....7.....1....6.8..9
1..26.....9.....3......47..4....38..8.......2..27....6..36......7.....1.....48..9
1..2....5..39......7..6....49...3.....8...2.....7...16....4..3......17..5....8..9
1..2....5..39......7..6....49...7.....8...2.....3...16....4..3......17..5....8..9
1..2.......39......7..6.5..49...7.....8...2.....3...16..5.4..3......17.......8..9
5..2....3.1.7.......9.4....43.1.......2...8.......9.76....6.1.......3.9.7....8..5
5..1....3.9.2.......1.6....14...7.....8...2.....3...69....4.9.......8.1.7....9..5
1..3....5..72......3..4....74.1.......2...8.......9.63....6..7......83..5....7..9
1..3....5.9.2.........6.7..74.1.......8...2.......9.63..3.4.........8.1.5....7..9
1..3....5.9.2.........4.7..74.1.......2...8.......9.63..3.6.........8.1.5....7..9
1..3....5.9.2.........6.7..74.9.......8...2.......1.63..3.4.........8.1.5....7..9
1..2....3.9.7.......5.4....43.1.......2...8.......9.76....6.5.......3.1.7....8..9
1...2.8.....4...3....7....4.231.....4.......6.....978.6....3....7...6.....2.8...9
1..2....5..31......7..6....41...3.....8...2.....7...96....4..3......97..5....8..9
1..2....3.9.3.......5.4....47.1.......2...8.......9.36....6.5.......7.1.7....8..9
1..2..4......6..5....1....84.1..3....8.....2....7..9.62....9....5..4......6..8..9
52.......4..6.8.......1.3...8...7.4...1...9...6.3...2...7.9.......2.4..6.......85
52.......4..6.8.......1.3...8...3.4...1...9...6.7...2...7.9.......2.4..6.......85
12.......4..21.........65...4...32...1.....9...87...6...54.........98..6.......89
12.......4..3.......5.98....7.1..4....9...1....6..9.3....21.5.......7..6.......89
1..2..4...5.....3.....36..84.....2....7...3....8.....62..47.....7.....5...6..8..9
1..2..4...5...6.....9..3..84.....32...........87.....62..7..1.....4...5...6..8..9
1..2..4.......6.3...9..3..84.....32...........87.....62..7..1...7.4.......6..8..9
1..2..4...5.4.......9..3..842....3.............7....862..7..1.......6.5...6..8..9
1..2..4.....4...3...9..3..842....3.............7....862..7..1...7...6.....6..8..9
1..2..6...5...........36..24..1..2....7...3....8..9..68..47...........5...4..8..9
52.6.....4............183..8....74....1...9....63....2..729............6.....4.85
52.6.....4............183..8....34....1...9....67....2..729............6.....4.85
12..9....4...........5.63....5..32..9.......1..87..5....74.5...........6....1..89
52......34..6.........81....8.1..9....6...4....1..9.2....92.........4..67......85
52......34...6.......1.8.....1..74...8.....2...63..9.....2.9.......4...67......85
52......34..8.........16....6...32....1...9....87...4....49.........2..67......85
12......54...3.......4.9.....2..71...7.....3...93..8.....1.6.......7...65......89
12.......4..6...5....98.....89..3.....6...4.....7..12.....21....5...4..6.......89
52.......4..6...3....78.....831.......6...4.......972.....23....7...4..6.......85
1..3....5.9.....3....24....7.49.......2...8.......16.3....68....7.....1.5....7..9
1..3....5.9.....7....24....7.49.......2...8.......16.3....68....3.....1.5....7..9
52.......4..8...3....67.....681.......3...7.......924.....34....7...2..6.......85
52.......4..6...3....87.....861.......3...7.......942.....32....7...4..6.......85
12.......4..6...3....98.....89..7.....6...4.....3..12.....21....7...4..6.......89
12.......4..2...3....96.....49..7.....8...2.....3..16.....41....7...8..6.......89
12.......4..6........38.5...879.......6...4.......132...5.27........4..6.......89
1..2......9.....5....34.7..4.71.......2...8.......93.6..3.67....5.....1......8..9
12.......4..6........89.3...86..7.....9...1.....3..42...7.12........4..6.......89
12.......4..5...3....96.....59..7.....8...2.....3..15.....41....7...5..6.......89
52......34..7.........617...3....9....8...2....1....7...394.........3..67......85
5..2......1....2.....34..6.4.79.......2...8.......13.6.4..67.....8....9......8..5
1..2......5....4.....34..8.4.79.......2...8.......13.6.2..67.....6....5......8..9
52....6..4..3........81...2.76........1...9........43.8...92........7..6..4....85
12......54..3........96.....79..3.....8...2.....7..13.....41........7..65......89
12.9.....4......3....7..5..9.3.4.......2.8.......6.7.1..5..3....7......6.....1.89
12.8.....4.....9.....6..31.6.81...................92.4.97..4.....1.....6.....2.89
52......34..6.......1.89....8....1....6...4....9....2....12.9.......4..67......85
12......34..8.......5.94....6....8....9...1....2....4....61.5.......2..67......89
53.2.....7.....4....1.9..8.4....7.....9...1.....3....6.2..1.9....6.....3.....8.75
12..9....4......3...58..7....6..7...9.......1...3..4....3..25...7......6....1..89
13....2..7..4.......52....6.24..3...............7..68.4....85.......6..3..8....79
53.2.....7.....4....1..4.8.4....78.............23....6.2.6..9....6.....3.....8.75
12.6.....4......3...5.12...8.....6....1...9....4.....2...89.5...7......6.....4.89
13.2.....7.....4....54...8.4.2..7...............3..8.6.2...65....6.....3.....8.79
52......34..6.8.......1.....8...7.4...1...9...6.3...2.....9.......2.4..67......85
12......54..63.........8....8.9..4...7.....3...6..1.2....2.........74..65......89
12......54..3.8.......6.....7.1...4...8...2...6...9.3.....4.......2.7..65......89
12......54..3.8.......6.....7.9...4...8...2...6...1.3.....4.......2.7..65......89
12......54..38.........6....7...32...6.....4...87...3....4.........27..65......89
12......54..38.........6....7.1..2...6.....4...8..9.3....4.........27..65......89
13......57..24.........6....4.1..2...2.....8...8..9.6....4.........68..35......79
13......57..2.4.......6.....4...7.8...8...2...2.3...6.....4.......6.8..35......79
13......57..2.9.......6.....4...3.1...8...2...9.7...6.....4.......1.8..35......79
13......57..2.9.......6.....4...7.1...8...2...9.3...6.....4.......1.8..35......79
12.8.....4...........36.5..6.79.......8...2.......13.4..5.47...........6.....2.89
52.8.....4...........16.3..6.1..3.....8...2.....7..9.4..7.49...........6.....2.85
12.8.....4...........69.3..6.8..7.....9...1.....3..2.4..7.14...........6.....2.89
13....2..7..54.......9....6.59.......2.....8.......15.4....1.......65..3..8....79
12.3.....4.....67......8.2.7..9..4.............6..1..3.8.2......34.....6.....7.89
12.9.....4......3....63.5..9.8........7...3........2.1..5.74....7......6.....1.89
13.2.....7.....85....6...4.4.8..7...............3..2.6.6...4....52.....3.....8.79
13.2.....7.....85....6...4.4.8..3...............7..2.6.6...4....52.....3.....8.79
12......54..63......5..8....8....4...7.....3...6....2....2..5......74..65......89
13.2.....7.....8.....7..54.4.31...................97.6.65..3.....2.....3.....8.79
12.......4..2...5....98.....49..3.....6...4.....7..16.....21....5...8..6.......89
12.......4..1...5....89.....16..3.....9...1.....7..49.....12....5...9..6.......89
12.......4..2...3....98.....49..7.....6...4.....3..16.....21....7...8..6.......89
12.......4..1...3....89.....16..7.....9...1.....3..49.....12....7...9..6.......89
13.2.....7......3.....65...4..1..5....8...2....5..9..6...54.....7......3.....8.79
13.2.....7......3.....45...4..1..5....2...8....5..9..6...56.....7......3.....8.79
12.......4..93.......6..3...98..3....7.....3....7..21...7..4.......71..6.......89
13.2.....7.....45....4...8.4.29...................18.6.2...6....56.....3.....8.79
13.2.....7.....45....4...8.4.2..7...............3..8.6.2...6....56.....3.....8.79
12.9.....4......3....61.7..9.8........1...9........2.1..3.94....7......6.....1.89
13....2..7..83......5.....6.6...7....7.....3....3...4.4.....5......72..3..8....79
52......34..8........41.7...62........1...9........84...3.96........2..67......85
12......34..1........89.7...16........9...1........49...3.12........9..67......89
52......34..8........41.5...62........1...9........84...5.96........2..67......85
52.3.....4......5.....18...7....34....1...9....67....3...29.....5......6.....7.85
12.......4..38.......6..3...789......6.....4......123...7..4.......27..6.......89
1.......5..298.....4.6......98..3....6.....4....7..21......4.6.....218..5.......9
5.......3..278.....4.6......381......6.....4......927......4.6.....238..7.......5
5..26.....1.3.......9...7..47.9.....8.......2.....1.36..3...1.......7.9.....48..5
1.......5..324.....7.9......49..3....2.....8....7..16......1.3.....687..5.......9
1..26.....9.3.......5...7..47.9.....8.......2.....1.36..3...5.......7.1.....48..9
1.......5..324.....7.9......49..7....2.....8....3..16......1.3.....687..5.......9
5..1.......326.....7....5..14...7....8.....2....3...69..5....3.....487.......9..5
5..24.....1.3.......9...3..47.1.....2.......8.....9.36..7...1.......7.9.....68..5
12......34......5....84.7....6..3.....2...8.....7..4....3.62....5......67......89
13..2....7..9........6..5...98..7...4.......6...3..21...5..4........1..3....8..79
13..2....7..9........4..5...92..3...4.......6...7..81...5..6........1..3....8..79
52..8....4..3........6..7...781.....6.......4.....923...3..4........7..6....2..85
52..8....4..3........6..3...781.....6.......4.....923...7..4........7..6....2..85
52......34..3.......189.....76........9...1........43.....129.......7..67......85
52......34..6.......192.....89........4...6........12.....819.......4..67......85
12......34..1.......589.....16........9...1........49.....125.......9..67......89
13..2....7..4........3..5...279.....4.......6.....138...5..7........6..3....8..79
12.8.....4...9.......3..5..6.79......9.....1......13.4..5..7.......1...6.....2.89
13.......7..26......54......42..3....8.....2....7..86......65......48..3.......79
12......54..83......96......68.......7.....3.......24......41......72..65......89
12......54..63......98......86.......7.....3.......42......21......74..65......89
12......54..83......56......68.......7.....3.......24......45......72..65......89
12.3.....4....8.....9..53..7.....54...........65.....3..75..1.....2....6.....7.89
1..92.......3...5......47..97....8..4.......6..2....31..36......5...7.......81..9
1..29.......3...5......47..47....8..9.......1..2....36..36......5...7.......18..9
1..29.......3...5......67..47....2..9.......1..8....36..34......5...7.......18..9
13.2.....7...8......56.....4.8..3....6.....4....7..2.6.....45......2...3.....8.79
13.2.....7...4......56.....4.8..3....2.....8....7..2.6.....45......6...3.....8.79
53.1.....7...2......96.....1.8..3....4.....6....7..2.9.....41......8...3.....9.75
13.2.....7...4......93.....4.71......2.....8......93.6.....71......6...3.....8.79
53.1.....7...2......16.....1.8..3....4.....6....7..2.9.....49......8...3.....9.75
1..2..8...5..........36...44.79.......8...2.......13.66...47..........5...2..8..9
1..26.....5.9...3.......7..49...3...8.......2...7...16..3.......7...1.5.....48..9
1..26.....5.9...3.......7..49...7...8.......2...3...16..3.......7...1.5.....48..9
52......34..8...7.....16....6....2....1...9....8....4....49.....3...2..67......85
12......34..8...5.....94....6....8....9...1....2....4....61.....5...2..67......89
12.......4..1...5....89.3...16........9...1........49...7.12....5...9..6.......89
13.2.....7....94........78.4....7.1...........9.3....6.23........61....3.....8.79
52......34..8...7....41.....62........1...9........84.....96....3...2..67......85
52......34..7..1.......8.9..3.1..4.............6..9.7..1.2.......9..3..67......85
52.....1.4...6...9...2..3....49......8.....2......16....7..8...1...4...6.9.....85
52......34..7..1.....8...9..369...................147..1...2.....9..3..67......85
52......34..7..1.....8...9..361...................947..1...2.....9..3..67......85
52....1..4..6...3....8....9.869...................142.1....2....7...4..6..9....85
52.....1.4..6....9...8..3...869...................142...7..2...1....4..6.9.....85
13......57..2..4.....9...8..49..3...............7..16..2...1.....6..8..35......79
52.....1.4..6....9...2..3...849...................162...7..8...1....4..6.9.....85
5..3..1.........5....24...97.41.......2...8.......96.31...68....5.........9..7..5
12......34..7...5.....8.7...3.9.......6...4.......1.7...3.2.....5...3..67......89
1..2....5..496.....2.......49...3....8.....2....7...16.......8.....416..5....8..9
1..2....5..496.....2....3..49........8.....2........16..7....8.....416..5....8..9
52......34...1.6.....8...2...6..3....1.....9....7..4...8...2.....4.9...67......85
13..2....7..8...5....4......62..7...4.......6...3..84......6....5...2..3....8..79
13..2....7..4...5....8......26..7...4.......6...3..48......2....5...6..3....8..79
12.6.....4....8.5......93..8.....14...........69.....2..71......5.2....6.....4.89
52.......4..6...3...187.....86........3...7........42.....329...7...4..6.......85
52.......4..8...3...167.....68........3...7........24.....349...7...2..6.......85
12.9.....4.....13...56...9.9.8.....................2.1.1...45...79.....6.....1.89
13.....2.7...4...6..58.......6..3....2.....8....7..4.......25..4...6...3.8.....79
12.6.....4....8.3...9...7..8....3.4...........6.7....2..3...1...7.2....6.....4.89
12......34...6..7...98.......6..3....8.....2....7..4.......21...3..4...67......89
12......34..8...7...9..6....6...32.............87...4....4..1...3...2..67......89
12.8.....4...6..3...9...5..6..1......8.....2......9..4..5...1...7..4...6.....2.89
52......34..8...7...1..6....6...32.............87...4....4..9...3...2..67......85
12......54..8...3...9.6.....6...7.....8...2.....3...4.....4.1...7...2..65......89
53.2.....7....48....1....4.4....7.8...........2.3....6.6....9....26....3.....8.75
13.2.....7....48....5....4.4....7.8...........2.3....6.6....5....26....3.....8.79
12......34..8...7...9.7.....6.9.......3...7.......1.4.....3.1...3...2..67......89
13.....2.7...4...6..52.......4..3....2.....8....7..6.......85..4...6...3.8.....79
12......54..8...3...96......681...................924......41...7...2..65......89
12......34..8...7...97......631...................974......31...3...2..67......89
52......34..8...7...17......631...................974......39...3...2..67......85
12......34..8...7...94......62..7...............3..84......61...3...2..67......89
52......34..8...7...14......62..7...............3..84......69...3...2..67......85
52......34..8...7...14......621...................984......69...3...2..67......85
13.....2.7..4....6..52......24..3...............7..68......85..4....6..3.8.....79
12......34..8...7...91......61..7...............3..94......91...3...2..67......89
12......54..8...3...91......61..7...............3..94......91...7...2..65......89
52.6.....4...3..7...1..8...8.....4...7.....3...6.....2...2..9...3..7...6.....4.85
52.6.....4....8.3...1.7....8......4...3...7...6......2....3.9...7.2....6.....4.85
13......57...2.6....9....2....1.3....4.....6....7.9....8....1....4.8...35......79
52.6.....4...3..7...12.....8.4.......7.....3.......6.2.....89...3..7...6.....4.85
52......34...1..7...18.......6..3....1.....9....7..4.......29...3..9...67......85
5..24.....1.3...5...9......47.1.....2.......8.....9.36......1...5...7.9.....68..5
5..1....3..326.....7.......14...7....8.....2....3...69.......3.....487..7....9..5
52....1..4..2...3....6....9.489...................126.1....4....7...8..6..9....85
12.......4..83......96..3...68.......7.....3.......24...7..41......72..6.......89
12.......4..63......98..3...86.......7.....3.......42...7..21......74..6.......89
12......34..1...5....89.....16........9...1........49.....12....5...9..67......89
1..24.....5.3..2....9....6.47.......2.......8.......36.4....1....8..7.5.....68..9
52.6.....4...38.........3..8..1...4..7.....3..6...9..2..7.........27...6.....4.85
12..3....4..2..........65...4.1..2..7.......3..8..9.6...54..........8..6....7..89
5.......3..274.....4.1......319......2.....8......197......9.6.....638..7.......5
1..2..4...5..........34...84.79.......2...8.......13.62...67..........5...6..8..9
12..9....4..2...3....4..5...42......9.......1......86...5..6....7...8..6....1..89
12.9.....4......3...561....9.8........1...9........2.1....945...7......6.....1.89
13..2....7..1........6..5...18..7...4.......6...3..29...5..4........9..3....8..79
12.8.....4...1.......3..5..6.71......1.....9......93.4..5..7.......9...6.....2.89
52......34..6...5.....81....8....9....6...4....1....2....92.....5...4..67......85
52......34..7...5.....61....3....9....8...2....1....7....94.....5...3..67......85
52......34..8...5.....14....6....8....1...9....2....4....69.....5...2..67......85
13.2.....7....14........58.4....7.9...........1.3....6.25........69....3.....8.79
52......34..8...5....41.....62........1...9........84.....96....5...2..67......85
13.2.....7....18........74.4....7.9...........1.3....6.63........29....3.....8.79
1..29......26...5..4....3..48.......9.......1.......26..7....6..5...48......18..9
12......54..3..1.......6.9..7.1..2.............8..9.3..1.4.......9..7..65......89
52.......4..3..8.....81..4..76........1...9........43..6..92.....2..7..6.......85
13.2.....7..6..4........38.48...7...............3...26.27........6..4..3.....8.79
13.2.....7..6..4....5....8.48...7...............3...26.2....5....6..4..3.....8.79
52......34..8...7....6.1....68...9.............1...24....9.4....3...2..67......85
52......34..8...7....1.4....61...8.............2...94....6.9....3...2..67......85
52......34..1...7....8.4....16...8.............2...49....6.2....3...9..67......85
12......34..8........9.47...69...8.............2...14...36.1........2..67......89
12.......4..8...5....9.43...69...8.............2...14...76.1....5...2..6.......89
12......34..8.......59.4....69...8.............2...14....6.15.......2..67......89
52....1..4..2.3......6....9.48....3...........7....26.1....4......7.8..6..9....85
12.8.....4..36..........3..67.9......8.....2......1.34..7..........47..6.....2.89
52.8.....4..36......1......67.9......8.....2......1.34......9......47..6.....2.85
12.8.....4..36......5......67.9......8.....2......1.34......5......47..6.....2.89
53.1.....7..24......9......14...3....2.....8....7...69......1......68..3.....9.75
12.3.....4..2..1........39.74.1...................9.63.17........9..8..6.....7.89
13....2..7..5.4......9....6.59....8...........2....15.4....1......6.5..3..8....79
52.......4..3.6......81.....76....2...1...9...8....43.....92......4.7..6.......85
12.......4.3....5..7.89......69.......9...1.......14......12.3..5....7.6.......89
13.......7.26......4...85...8...34.............67...2...52...6......48.3.......79
13.......7.26......4...87...8...34.............67...2...32...6......48.3.......79
13......57.26......4...8....8...34.............67...2....2...6......48.35......79
13.2.....7.8..4....65......4....7.8...........2.3....6......54....6..2.3.....8.79
1..26.....5.....3....4.1...4.2...9..8.......2..1...8.6...9.6....7.....5.....48..9
12.......4.32......7...65...4.1..2.............8..9.6...54...3......87.6.......89
12......349..........84.7....6..3.....2...8.....7..4....3.62..........167......89
52.8.....41..........36....6.79.......8...2.......13.4....47..........96.....2.85
53....2..7..4.......1...3.6.2.9.3...............7.1.8.4.7...9.......6..3..8....75
13....2..7..8.......5...3.6.6.9.7...............3.1.4.4.7...5.......2..3..8....79
12......34.....1....58...9...69.3...............7.14...1...25....9.....67......89
13.2.....7.....4....9...78.4..1.7...............3.9..6.23...1....6.....3.....8.79
52......34..8.......1.4.....6.1.3.....2...8.....7.9.4.....6.9.......2..67......85
12......54..8.........4.3...6.9.7.....2...8.....3.1.4...7.6.........2..65......89
52..6....4..2.......1...3...4.9.7...8.......2...3.1.6...7...9.......8..6....4..85
12..8....4......3....6..5....89.3...6.......4...7.12....5..4....7......6....2..89
53.2.....7.....1......6..9.4..9.7.....8...2.....3.1..6.1..4......9.....3.....8.75
53.2.....7.....1......6..9.4..1.7.....8...2.....3.9..6.1..4......9.....3.....8.75
52..6....41..........8..3....61.7...8.......2...3.94....7..2..........96....4..85
52..6....41..........8..3....61.3...8.......2...7.94....7..2..........96....4..85
12..8....45..........6..3....81.3...6.......4...7.92....7..4..........56....2..89
12......54..8.........6.3...6.9.7.....8...2.....3.1.4...7.4.........2..65......89
1...2......84...3..6....5...2.9.7...4.......6...3.1.8...5....4..7...62......8...9
1...2......84...3..6....5...2.1.3...4.......6...7.9.8...5....4..7...62......8...9
5..2..4.........3...1.6...84..9.3.....8...2.....7.1..62...4.9...7.........6..8..5
1..2..4.........3...9.6...84..1.3.....8...2.....7.9..62...4.1...7.........6..8..9
53.......7..2..1......6..9..4.1.7.....8...2.....3.9.6..1..4......9..8..3.......75
53.......7..2..1......6..9..4.9.7.....8...2.....3.1.6..1..4......9..8..3.......75
12......34..8...5.....4.....6.1.3.....2...8.....7.9.4.....6.....5...2..67......89
53.2.....7.....1......4..9.4..1.7.....2...8.....3.9..6.1..6......9.....3.....8.75
53.2.....7.....1......4..9.4..9.7.....2...8.....3.1..6.1..6......9.....3.....8.75
52..6....4..........18..3....69.7...8.......2...3.14....7..29..........6....4..85
12..8....4..........96..3....81.3...6.......4...7.92....7..41..........6....2..89
12.8.....4......3.....6.5..6..9.7.....8...2.....3.1..4..5.4.....7......6.....2.89
12.......4...2..5....6..3....81.7....4.....6....3.92....7..4....5..8...6.......89
52..6....4..........12..3....49.7...8.......2...3.16....7..89..........6....4..85
53.2.....71....4.......4.8.4....78.............23....6.2.6.......6....93.....8.75
52..1....49..........6..3....81.7...1.......9...3.92....7..4..........16....9..85
53....2..7..4.........1...6.2.1.3.....1...9.....7.9.8.4...9.........6..3..8....75
53....2..7..4.........1...6.2.1.7.....1...9.....3.9.8.4...9.........6..3..8....75
1..26.....5.....3...9...7..4..1.7...8.......2...3.9..6..3...1...7.....5.....48..9
1..24.....5.....3...9...7..4..1.7...2.......8...3.9..6..3...1...7.....5.....68..9
12.6.....45.....3....2..7..8.49...................16.2..3..8....7.....56.....4.89
52......34......5....61......81.7.....1...9.....3.92......94....5......67......85
12......54......3....89......69.3.....9...1.....7.14......12....7......65......89
12......54......3....69......89.7.....9...1.....3.12......14....7......65......89
5.....2....347.....7......6.2.1.3....3.....7....7.9.8.4......3.....367....8.....5
1.....2....347.....7......6.2.1.3....3.....7....7.9.8.4......3.....367....8.....9
12......54...........89.3....69.7.....9...1.....3.14....7.12...........65......89
52......34...........61.7....81.7.....1...9.....3.92....3.94...........67......85
12......54...........89.3....69.3.....9...1.....7.14....7.12...........65......89
12.......4......3....69.7....89.7.....9...1.....3.12....3.14....7......6.......89
5.....2...1.....3....23...6..49.7.....7...3.....3.16..4...78....7.....9...8.....5
12.......4.3.......7.89......69.7.....9...1.....3.14......12.3.......7.6.......89
12.......4......3...569......89.7.....9...1.....3.12......145...7......6.......89
12.9.....4.....13....6..79.9.8.....................2.1.13..4....79.....6.....1.89
12.......4..3........96.3...79..3.....8...2.....7..13...7.41........7..6.......89
1..24.....9.....3....1..7..4.1..7...2.......8...3..9.6..3..9....7.....1.....68..9
1..24.....9.....5....1..3..4.1..7...2.......8...3..9.6..7..9....5.....1.....68..9
12..4....4......3....8..7....61.3...2.......8...7.94....3..2....7......6....6..89
12....4..4..3..........5..8.7..6.5.....8.2.....5.4..3.2..5..........7..6..6....89
12.......4..3...5....81.....761.......1...9.......943.....92....5...7..6.......89
12.3.....4......3....89....7.69.......9...1.......14.3....12....7......6.....7.89
12.3.....4......7....89....7.69.......9...1.......14.3....12....3......6.....7.89
5..1.......263.....4....5..18...7....7.....3....3...29..5....6.....748.......9..5
12.......4..9........23.5...94..7.....7...3.....3..61...5.78........1..6.......89
5..1..2.....43..5.........612...7....7.....3....3...894.........5..76.....8..9..5
12.3.....4..........569....7.89.......9...1.......12.3....145..........6.....7.89
12.......4..81.......3..5...671......1.....9......934...5..7.......92..6.......89
12.3.....4..21..........3..74.1......1.....9......9.63..7..........98..6.....7.89
12.3.....4..21..........7..74.1......1.....9......9.63..3..........98..6.....7.89
12.8.....4..13..........5..61...7....7.....3....3...94..5..........79..6.....2.89
12.3.....4...........61.7..7.81.......1...9.......92.3..3.94...........6.....7.89
12.3.....4...........89.7..7.69.......9...1.......14.3..3.12...........6.....7.89
12.3.....4...........69.7..7.89.......9...1.......12.3..3.14...........6.....7.89
12.......4..93.......6..3...98..7....7.....3....3..21...7..4.......71..6.......89
5..1..2...1..........43...61.2..7.....7...3.....3..8.94...76..........9...8..9..5
5..1..2...9..........43...61.2..7.....7...3.....3..8.94...76..........1...8..9..5
52.......4..3........61.3...781.......1...9.......923...7.94........7..6.......85
12.5.....4...........89.3..5.69.......9...1.......14.5..7.12...........6.....5.89
12.4.....4......5....98....2.9..3.....6...4.....7..1.8....21....5......6.....6.89
12.4.....4......3....98....2.9..7.....6...4.....3..1.8....21....7......6.....6.89
52..1....4.....83....6...4...8..3...1.......9...7..2...6...4....72.....6....9..85
12.3.....4.....87....2...4.7.41...................96.3.6...8....32.....6.....7.89
12.3.....4...........89.3..7.69.......9...1.......14.3..7.12...........6.....7.89
12....4..4..9.3......5....8.95....3...........7....51.2....5......7.1..6..6....89
52.4.....4.....13......8.9.2..1..4.............6..9..8.1.2......79.....6.....6.85
12......34.7.......3.89......69.......9...1.......14......12.7.......3.67......89
52.4.....4...3.......1..7..2.1..7....7.....3....3..9.8..3..9.......7...6.....6.85
53.......7..26......17......431......8.....2......976......39......48..3.......75
1..26.....5.1...3.......7..41...3...8.......2...7...96..3.......7...9.5.....48..9
13......77..2...5.....6.....4.9.3.....8...2.....7.1.6.....4.....5...8..33......79
1..24.....5.1...3...9......41...7...2.......8...3...96......1...7...9.5.....68..9
1..3....5..326.....7.......74.1......8.....2......9.63.......3.....487..5....7..9
12.....4.4..6....8...9..3...89..7...............3..12...7..1...2....4..6.6.....89
5..2..6.........5....13...24.1..7.....7...3.....3..9.68...79....5.........4..8..5
12....4..4..8...5......9..8.6...31.............97...4.2..1......5...2..6..6....89
12....4..4..8...3....9....8.69..7...............3..14.2....1....7...2..6..6....89
12....4..4..8...5....9....8.69..3...............7..14.2....1....5...2..6..6....89
12.4.....4....8.5......93..2.....14...........69.....8..71......5.2....6.....6.89
12.3.....4.....87...52...4.7.4.....................6.3.6...85...32.....6.....7.89
52.......4.37......7.81.....36........1...9........47.....92.3......37.6.......85
12.4.....4...2.......3..5..2.79......4.....6......13.8..5..7.......8...6.....6.89
52.4.....4..3..6........32.27.1...................9.38.87........4..7..6.....6.85
12.4.....4..3..6........32.27.1...................9.38.87........4..7..6.....6.89
52.4.....4..3..6....1...32.27.......................38.87...9....4..7..6.....6.85
12.4.....4..3..1........79.27.1...................9.38.13........9..7..6.....6.89
12.4.....4..3..1........39.27.1...................9.38.17........9..7..6.....6.89
12.....9.4..3....1...4.5....72...5.............5...83....5.6...9....7..6.1.....89
12.4.....4..38..........3..27.9......6.....4......1.38..7..........27..6.....6.89
12.4.....4..39..........3..27.9......9.....1......1.38..7..........17..6.....6.89
12.4.....4..31..........3..27.1......1.....9......9.38..7..........97..6.....6.89
1.....2.....63.......8.5..6.86...5...7.....3...5...42.4..5.2.......74.....8.....9
12.......4.3...1...7..8..9....1.7.....6...4.....3.9....1..2..3...9...7.6.......89
12.4.....4.8.......6.9..3..2.9..7...............3..1.8..7..1.4.......2.6.....6.89
52....1..4......3....6....9..816.......8.2.......492..1....4....7......6..9....85
13....2..7..8...........5.6.6..23......4.6......78..4.4.5...........2..3..8....79
53.2.....71....8.........4.4...87......6.4......32...6.6.........2....93.....8.75
52....1..4......3....6....9..8.43......2.8......76.2..1....4....7......6..9....85
52..4....41..........8..3....61.3...2.......8...7.94....7..2..........96....6..85
13.2.....75.....3.....6....4..1.7.....8...2.....3.9..6....4.....7.....53.....8.79
13.2.....75.....3.....4....4..1.7.....2...8.....3.9..6....6.....7.....53.....8.79
12..9....49.....3....8..7....6..7...9.......1...3..4....3..2....7.....16....1..89
52.8.....41....2......3..6.6..9.......7...3.......1..4.4..7......8....96.....2.85
12.9.....49.....5....6..3..9.8..7...............3..2.1..7..4....5.....16.....1.89
12.9.....49.....5....6..3..9.8..3...............7..2.1..7..4....5.....16.....1.89
52.1.....49....2........36.1...6.......8.2.......4...9.47........8....16.....9.85
52.1.....49.....3.....8....1..9.3.....6...4.....7.1..9....2.....7.....16.....9.85
12......34..7...5.....8.....3.9.7.....6...4.....3.1.7.....2.....5...3..67......89
12.4.....45..8..........3..2..1.7....6.....4....3.9..8..7..........2..56.....6.89
12.4.....45..8..........3..2..1.3....6.....4....7.9..8..7..........2..56.....6.89
12.9.....49..3.......6..7..9.8.......7.....3.......2.1..3..4.......7..16.....1.89
13.......7..52.6.........2..5.9.7....4.....6....3.1.5..8.........4.85..3.......79
13.......7..52.6.........2..5.9.3....4.....6....7.1.5..8.........4.85..3.......79
12.......4..68..3...9.......8.1.3....6.....4....7.9.2.......1...7..24..6.......89
13....2..7..83..5.........6.6...7....7.....3....3...4.4.........5..72..3..8....79
12......34..67..5......8....8....4...3.....7...6....2....2......5..34..67......89
5..21.9....6..3....8......14......3.1.......9.7......69......2....7..4....1.98..5
1..2..4.............9.36..84....72....7...3....83....62..47.1.............6..8..9
1..2..4...5.........9.36..84.....2....7...3....8.....62..47.1.........5...6..8..9
1..2..4.........3...5.96..84.....2....9...1....8.....62..41.5...7.........6..8..9
1..2..4.........3...9.36..84.....2....7...3....8.....62..47.1...7.........6..8..9
12.4.....4....8........93..2....314...........697....8..71........2....6.....6.89
.2.6..1..4....3......8....98.69...3...........7...14.21....2......7....6..9..4.8.
.2.8..1..4....3......6....96.89...3...........7...12.41....4......7....6..9..2.8.
13.9.....7..2..........65..94...32.............87...61..54..........8..3.....1.79
1..3..2.....4.......5..8..672.1..4.............6..9.834..2..5.......6.....8..7..9
12.4.....4....8.3....3.....2.71...4...........6...93.8.....7....7.2....6.....6.89
1..2..4.....4.......9..3..842.9..3.............7..1.862..7..1.......6.....6..8..9
12.4.....4..3..........83..27.1..4.............6..9.38..72..........7..6.....6.89
12..9....4......3....6..5....8.6....9..8.2..1....4.2....5..4....7......6....1..89
1..2..4.........3...5.16..84.....2....1...9....8.....62..49.5...7.........6..8..9
12.6.....4....8.3......5...8....354...........657....2...5......7.2....6.....4.89
12.6.....4....8........53..8....354...........657....2..75........2....6.....4.89
12.6.....4..8..........53..86...75.............53...42..75..........2..6.....4.89
1..52.........4.3......7...5..1..78.4.......6.23..9..5...3......7.6.........85..9
1..2..4.....4.......9..3..842...73.............73...862..7..1.......6.....6..8..9
12......34......5....98......9.6......68.24......4.1......21....5......67......89
12.......4......5....98.3....9.6......68.24......4.1....7.21....5......6.......89
12......34..........598......9.6......68.24......4.1......215..........67......89
12.9....34............8.5..9..1.3.....6...4.....7.9..1..5.2............67....1.89
52.3....74............18...7....34....1...9....67....3...29............63....7.85
12.......4..38..5....6......789......6.....4......123......4....5..27..6.......89
52.6....34...........81....8.6..7.....1...9.....3..4.2....92...........67....4.85
12.3....54...........69....7.89.......9...1.......12.3....14...........65....7.89
12......34..7.6.5....8......36....2...........8....47......2....5.4.3..67......89
13.2.....7..6..45........8.48...7...............3...26.2........56..4..3.....8.79
13.2.....7..6..45........8.48...3...............7...26.2........56..4..3.....8.79
13.2.....7..6..85........4.48...7...............3...26.6........52..4..3.....8.79
12.......4..92..3....6......98..7....4.....6....3..21......4....7..81..6.......89
12..3....4..8..15........9..6.9.....7.......3.....1.4..1........59..2..6....7..89
12..3....4..2..15........9..4.1.....7.......3.....9.6..1........59..8..6....7..89
52.6....34..........1..87..8....74.............63....2..32..9..........67....4.85
13..2.4..7..........9...7.8...1.7...4.......6...3.9...2.3...1..........3..6.8..79
12.6....34..........92..5..8.4..7...............3..6.2..5..81..........67....4.89
12.6....34..........92..5..8.4..3...............7..6.2..5..81..........67....4.89
13.9...2.7.......6....8....9..1.7.....6...4.....3.9..1....2....4.......3.8...1.79
12.3....74...8..........5..7..9.3....6.....4....7.1..3..5..........2...63....7.89
12.3....54......7....6.9...7.8...1.............9...2.3...1.4....3......65....7.89
13.2..6..7....4.........5.24....7.8...........2.3....68.5.........6....3..4..8.79
12.9....54......3....61....9.8........1...9........2.1....94....7......65....1.89
12.6....34......7....21....8.4........1...9........6.2....98....3......67....4.89
5..26.1.........3....7....94.39.....8.......2.....17.61....3....7.........9.48..5
5..24...3.1.7.......9......43.1.....2.......8.....9.76......1.......3.9.7...68..5
1..26.4.........3....7....84.31.....8.......2.....97.62....3....7.........6.48..9
13.2....57....9.......6....4....7.1...8...2...9.3....6....4.......1....35....8.79
1..2.......843..7..6....5..42........7.....3........86..5....4..3..762.......8..9
12.6....34......7......85..8....74.............63....2..52......3......67....4.89
12..9...34......7....6..5....8..7...9.......1...3..2....5..4....3......67...1..89
52.6...1.4.......9.....83..8....34.............67....2..72.....1.......6.9...4.85
52.6..1..4......3.....2...98..1.......4...6.......9..21...8.....7......6..9..4.85
52.8..1..4......3....6....96.89...................12.41....4....7......6..9..2.85
12.6....34......7....3..5..8.79...................13.2..5..7....3......67....4.89
12.6....34......7....2..5..8.4..7...............3..6.2..5..8....3......67....4.89
12.6....34......7....2..5..8.49...................16.2..5..8....3......67....4.89
12.6....34....8........97..8.....14...........69.....2..31........2....67....4.89
12.6....54....8......9..3..8.9....4...........6....1.2..7..1......2....65....4.89
12.......4..83..7...56......68.......7.....3.......24......45...3..72..6.......89
52.6....34....8.....1...7..8....7.4...........6.3....2..3...9.....2....67....4.85
52.6....34......7...1..8...8....74.............63....2...2..9...3......67....4.85
52..1...34......7...96.......8..7...1.......9...3..2.......41...3......67...9..85
53.1..2..7....4.....9.....61....7.8...........2.3....94.....1.....6....3..8..9.75
52.6....34......7...19.....8.9..3...............7..1.2.....19...3......67....4.85
12.3....54......7...96.....7.89...................12.3.....41...3......65....7.89
12.3....54......7...96.....7.81...................92.3.....41...3......65....7.89
13.2...8.7.......4..54.....4.2..7...............3..8.6.....65..6.......3.2...8.79
12.6....34......7...92.....8.49...................16.2.....81...3......67....4.89
52.6....34......7...12.....8.4..3...............7..6.2.....89...3......67....4.85
12.6....54......3...92.....8.49...................16.2.....81...7......65....4.89
5..1.24.........3...96....81.8.....6.........4.....2.92....41...7.........68.9..5
13.2....57....94.........8.4....7.1...........9.3....6.2.........61....35....8.79
52.1....34....8.7......4...1.....84...........62.....9...6......3.2....67....9.85
12.6....34....8.7...9......8..9...4...........6...1..2......1...3.2....67....4.89
52.6....34....8.7...1......8..9...4...........6...1..2......9...3.2....67....4.85
12.6....34...98.7..........8......4..9.....1..6......2..........3.21...67....4.89
1..24.9.............53....14.71.....2.......8.....93.69....75.............1.68..9
12.3....74...2..........5..7..9.3....4.....6....7.1..3..5..........8...63....7.89
12.3...9.4.......1...6.5...7.8...5.............5...2.3...5.4...9.......6.1...7.89
5..3.12....34......7......672......9.........1......834......3......67....89.7..5
12.9....34.....1.....6...9.9.8..7...............3..2.1.1...4.....9.....67....1.89
12.9....54.....1.....6...9.9.8..3...............7..2.1.1...4.....9.....65....1.89
12.9....54.....1.....6..39.9.8.....................2.1.17..4.....9.....65....1.89
52.6....34......5...19.....8.9..7...............3..1.2.....19...5......67....4.85
53.2..1..7....4.5.........94....7.8...........2.3....61.........5.6....3..9..8.75
12.9....54.....13....6...9.9.8.....................2.1.1...4....79.....65....1.89
12.9....54..8.........6.3..96.........8...2.........41..7.4.........2..65....1.89
13.2..8..7..6.......5.....448...3...............7...266.....5.......4..3..2..8.79
53.1..2..7..4.......9.....612...3...............7...894.....1.......6..3..8..9.75
52.6....34..8...7...1......86.9...................1.42......9...3...2..67....4.85
13.2...4.7......58.....6...4..9..2.............8..1..6...4.....25......3.6...8.79
13.2...8.7......54...6.....4.8..7...............3..2.6.....4...65......3.2...8.79
13.2...8.7......54...4.....4.2..7...............3..8.6.....6...65......3.2...8.79
13.2...8.7......54...4.....4.29...................18.6.....6...65......3.2...8.79
12.9.....4..83..7...5......96........7.....3........41......5...3..72..6.....1.89
52.6.....4..83..7...1......86........7.....3........42......9...3..72..6.....4.85
13......57..2........9.6....49..32.............87..16....4.1........8..35......79
12.......4..8...3....9.6....69..72.............83..14....4.1....7...2..6.......89
12.......4..8...3....9.6....69..32.............87..14....4.1....7...2..6.......89
12.......4..8...3....1.6....61..72.............83..94....4.9....7...2..6.......89
1..2..4.....4.3.....9.....842.1...3...........7...9.862.....1.....7.6.....6..8..9
1..3.24.............56....87.89....6.........4....12.32....45.............68.7..9
.2.4.3...4.....1.....8...9.2.69....3.........7....14.8.1...2.....9.....6...7.6.8.
12.......4..8........9.63...69..72.............83..14...74.1........2..6.......89
12..8..6.4......32......7.....1.3...6.......4...7.9.....3......87......6.4..2..89
13.2..4..7..1.......9.....841...7...............3...962.....1.......9..3..6..8.79
13.2..4..7..1.......9...7.841.......................962.3...1.......9..3..6..8.79
52..1....4......3....6.......8.63...1..8.2..9...74.2.......4....7......6....9..85
5..2......1....8......3..4.4..98......76.43......21..6.6..7......2....9......8..5
13.......7.26...3..4.8......869...................142......2.6..7...48.3.......79
12.......4......5....98......9.63.....68.24.....74.1......21....5......6.......89
12.......4......3....98......9.67.....68.24.....34.1......21....7......6.......89
52.6....341............87..8....74.............63....2..32............967....4.85
12.6.....45...8.3.......7..8....3.4...........6.7....2..3.......7.2...56.....4.89
12.6....345.....7......8...8....74.............63....2...2......3.....567....4.89
13.2..6..75...4...........24....7.8...........2.3....68...........6...53..4..8.79
12.6....345.....7....2.....8.4..7...............3..6.2.....8....3.....567....4.89
12.6....345.....7....2.....8.49...................16.2.....8....3.....567....4.89
12.3....549.....7....6..3..7.8.....................2.3..7..4....3.....165....7.89
1..2.......3..68...7.....4.4..9.3.2...........8.7.1..6.6.....3...24..7.......8..9
52......34..8...7......6....6.1.32.............87.9.4....4......3...2..67......85
12.......4..8...3...5..6....6.9.72.............83.1.4....4..5...7...2..6.......89
52.6....34.............87..8..1.74.............63.9..2..32.............67....4.85
53.2.....7.....4.......4.8.4..1.78.............23.9..6.2.6.......6.....3.....8.75
52.......4..8...3...1..6....6.9.72.............83.1.4....4..9...7...2..6.......85
12.......4..2...3......65...4.1.72.............83.9.6...54......7...8..6.......89
12.6....349............8...8..1.74.............63.9..2...2............167....4.89
12.6....34.............87..8..1.74.............63.9..2..32.............67....4.89
1..2..4.......6.3...9.....84..1.7.2...........8.3.9..62.....1...7.4.......6..8..9
52.6....34....8.........7..8..1.7.4...........6.3.9..2..3.........2....67....4.85
12.6.....4....8.3...5......8..1.3.4...........6.7.9..2......5...7.2....6.....4.89
52.......4.38......7...6....6.1.72.............83.9.4....4...3......27.6.......85
53.2.....7....48.........4.4..1.7.8...........2.3.9..6.6.........26....3.....8.75
13.2.....7....48.........4.4..1.7.8...........2.3.9..6.6.........26....3.....8.79
13.......7..2.........65....4.1.35....8...2....57.9.6....54.........8..3.......79
13.2....579.....3.....6....4....7.....8...2.....3....6....4.....7.....135....8.79
12.3..8..45.2...........3.474.......................636.7...........8.56..2..7.89
13.2.....7..4..85........4.42.9...................1.86.6........52..6..3.....8.79
52.4....34...........1.75..2.1...7.............3...9.8..53.9...........67....6.85
12.8..4..4.............93.86....71.............93....42.71.............6..6..2.89
13.2.....7..6..25........6.48...3...............7...26.4........58..4..3.....8.79
52.4....34..........19..7..2.9..7...............3..1.8..3..19..........67....6.85
52.4...1.4.......9....8....2..1.3.....6...4.....7.9..8....2....1.......6.9...6.85
1..23...5..61......8....3..41.......7.......3.......96..7....2......94..5...78..9
12.6...4.4.......8...3..5..8.71...................93.2..5..7...2.......6.6...4.89
12.4....34..7..........8...23.1..4.............6..9.78...2..........3..67....6.89
12.4....34..7..1.........9.23.1...................9.78.1.........9..3..67....6.89
12..6..4.4......38......7.....1.3...8.......2...7.9.....3......27......6.6..4..89
52.6...4.4......38..17.....8.3.....................7.2.....39..27......6.6...4.85
12.3...4.4......78..56.....7.8.....................2.3.....45..23......6.6...7.89
12.......4..8...3....4.5....62..35.............57..84....5.6....7...2..6.......89
12.......4..8........4.53...62..75.............53..84...75.6........2..6.......89
12.4....54..31.............27.1......1.....9......9.38.............97..65....6.89
1..2..4.............9..3..84..12.3.....4.6.....7.89..62..7..1.............6..8..9
12.......4..2.........3.5...4..87.....76.43.....32..6...5.7.........8..6.......89
12.......45..........83......6.87.....76.43.....32.4......72..........56.......89
12.......4...........83.5....6.87.....76.43.....32.4....5.72...........6.......89
12.9....349..........6..7..9.8..7...............3..2.1..3..4..........167....1.89
13.2.....75...14.........8.4....7.9...........1.3....6.2.........69...53.....8.79
12.6....545...8.3..........8....3.4...........6.7....2..........7.2...565....4.89
12......34...8........3.5.....1.3....67...34....7.9.....5.7........2...67......89
1..2..4......6..5.....3...84..9......87...32......1..62...7.....5..4......6..8..9
1..2......9..4..5.....3.7..4..9......27...38......1..6..3.7.....5..6..1......8..9
5..2......1..4.8......3..4.4..9......27...38......1..6.6..7......2.6..9......8..5
52..1....4...2..3....6.......8..3...14.....69...7..2.......4....7..8...6....9..85
12..4....4..2.........3.5...4...7...2.7...3.8...3...6...5.7.........8..6....6..89
12..9....4..8.........3.5...6...7...9.7...3.1...3...4...5.7.........2..6....1..89
1..2......9..3........4.7..4..1.7....72...83....3.9..6..3.6........7..1......8..9
5..2......1..3........4.7..4..1.7....72...83....3.9..6..3.6........7..9......8..5
12.86....4......3.......5..6..1.7...8.......2...3.9..4..5.......7......6....42.89
52.61....4...........8..3..8.6..7...1.......9...3..4.2..7..2...........6....94.85
13.92....7...........6..5..9.8..7...4.......6...3..2.1..5..4...........3....81.79
5..26......2.1.....4....3..4....3...81.....92...7....6..7....6.....9.8......48..5
5..26.....1..9..5.......3..4....3...89.....12...7....6..7.......5..1..9.....48..5
13....2.57..8..........4..6.6.9..8.............2..1.4.4..6..........2..35.8....79
1..26.....9..1..5.......3..4....3...81.....92...7....6..7.......5..9..1.....48..9
1..24.....9..1..5.......3..4....3...21.....98...7....6..7.......5..9..1.....68..9
13....2.57..6.9...........6.8...7.1...........9.3...2.4...........1.4..35.8....79
5..24.....1..3..7...9......4....7...27.....38...3....6......1...3..7..9.....68..5
52.......4..86........1.3...6.1......81...92......9.4...7.9........42..6.......85
5..24.....1...........3.7..4..1.7...2.7...3.8...3.9..6..3.7...........9.....68..5
1..26........3..5.......7..4..9.7...87.....32...3.1..6..3.......5..7........48..9
12..3....4......5....89......69.....7.9...1.3.....14......12....5......6....7..89
12..3....4......3....89......69.....7.9...1.3.....14......12....7......6....7..89
12..3....4......7....89......69.....7.9...1.3.....14......12....3......6....7..89
12..3....4...........89.7....69.....7.9...1.3.....14....3.12...........6....7..89
12.63....4......5.....8....8....7...7.6...4.3...3....2....2.....5......6....74.89
52.16....4....8.3..........1....3.4.8.......2.6.7....9..........7.2....6....49.85
12.93....4..8...........5..96...7...7.......3...3...41..5...........2..6....71.89
52.13....4..6...........5..18...7...7.......3...3...29..5...........4..6....79.85
52.....134..6....9....8.....8.1.......6...4.......9.2.....2....1....4..679.....85
1..29.....9.3.........4.7..47.......9.2...8.1.......36..3.6.........7.1.....18..9
1..26.......3...5.....1.7..47.......8.1...9.2.......36..3.9.....5...7.......48..9
1..26.......5...3.....1.7..45.......8.1...9.2.......56..3.9.....7...5.......48..9
13.2.....7..9...........5..49..23......4.6......78..16..5...........1..3.....8.79
52.63....41..........8..3..8.6......7.......3......4.2..7..2..........96....74.85
12.63....49.....5....8.....8.6......7.......3......4.2.....2....5.....16....74.89
5.....2....38......7.4....6.621.7...............3.984.4....6.3......27....8.....5
12.8.....4......3....4..5..6.29.7...............3.18.4..5..6....7......6.....2.89
5..2.......34..6...7.....2.42.1.7...............3.9.86.8.....3...4..67.......8..5
5..2..6....34......7......242.1.7...............3.9.868......3......67....4..8..5
52......34..8...7....4......621.7...............3.984......6....3...2..67......85
12.6....34......5....2.....8.41.3...............7.96.2.....8....5......67....4.89
1..2..4...9.6...3.........848.1.3...............7.9.262.........7...4.1...6..8..9
52.......4..8...3...16......681.3...............7.924......49...7...2..6.......85
52......34..8...7....4......621.3...............7.984......6....3...2..67......85
12.......4..8...3...54......629.7...............3.184......65...7...2..6.......89
52.......4..8...3...14......629.7...............3.184......69...7...2..6.......85
1..2..4...5.4.......9.....842.1.3...............7.9.862.....1.......6.5...6..8..9
1..2..4...5.4...3.........842.1.3...............7.9.862.........7...6.5...6..8..9
12.......4..8.......54..3...629.7...............3.184...7..65.......2..6.......89
12.6....34......7....2.....8.41.7...............3.96.2.....8....3......67....4.89
1..2..4.........3...96....84.81.3...............7.92.62....41...7.........6..8..9
52.6....34..8...7..........86.1.7...............3.9.42..........3...2..67....4.85
12.6....34..8...7..........86.9.3...............7.1.42..........3...2..67....4.89
12.6....54..8...3..........86.1.3...............7.9.42..........7...2..65....4.89
1..2.......348.....7.......42.1.7....6.....4....3.9.86.......3.....267.......8..9
1..........326.....7.4......421.7....8.....2....3.986......6.3.....487..........9
1...2.......4...3....5..7...259.3...4.......6...7.158...3..5....7...6.......8...9
1..26.......5...3.......7..45.9.7...8.......2...3.1.56..3.......7...5.......48..9
13.......7..2........56.....451.7.....8...2.....3.956.....45........8..3.......79
13.2.....7...........54....4.51.7.....2...8.....3.95.6....65...........3.....8.79
52.63....41.....5....8.....8.6......7.......3......4.2.....2....5.....96....74.85
52.14....4...........6..3..1.8..3...2.......8...7..2.9..7..4...........6....69.85
52.14....4...........8..3..1.6..3...2.......8...7..4.9..7..2...........6....69.85
52.41....4...........9..3..2.9..3...1.......9...7..1.8..7..1...........6....96.85
52.41....4..........1...3..2..1.3...1.......9...7.9..8..7...9..........6....96.85
13.26....7...........3..5..4.79.....8.......2.....13.6..5..7...........3....48.79
12....4.54...3.......9....8..9..3....7.....3....7..1..2....1.......7...65.6....89
12.49....4......3....8..5..2.6......9.......1......4.8..5..2....7......6....16.89
12.3.....49....87....2...4.7.4.....................6.3.6...8....32....16.....7.89
13.27....7.....8....5....4.4....3...3.......7...7....6.6....5....2.....3....38.79
12....4.34..8.......59....8.69.....................14.2....15.......2..67.6....89
52.14....4....8.3..........1....3.4.2.......8.6.7....9..........7.2....6....69.85
12....4.34..8...5....9....8.69.....................14.2....1....5...2..67.6....89
52.16....4..2...........3..14...3...8.......2...7...69..7...........8..6....49.85
52.14....4..8...........3..16...3...2.......8...7...49..7...........2..6....69.85
12.39....4..2...5..........74.9.....9.......1.....1.63..........5...8..6....17.89
12.....634..8....2..97......63.....................74......31..8....2..674.....89
12.34....4..2...........7..74.1.....2.......8.....9.63..3...........8..6....67.89
52.41....4..3...........7..27.1.....1.......9.....9.38..3...........7..6....96.85
52.41....4..3...........3..27.1.....1.......9.....9.38..7...........7..6....96.85
13.......7..9........2..5...94.83......6.4......72.61...5..8........1..3.......79
52.41....49.............3..2..1.3...1.......9...7.9..8..7.............16....96.85
12.9.3...49..........6..7..9.8.....3.........7.....2.1..3..4..........16...7.1.89
13.27....79....8.........4.4....3...3.......7...7....6.6.........2....13....38.79
13.......7..2........5.6....459.32.............87.156....4.5........8..3.......79
12.6..9.54...3............18..1......7.....3......9..29............7...65.1..4.89
12.4..9.54...3............12..9......7.....3......1..89............7...65.1..6.89
1..9..2.....3...5...58....6976.....................4314....25...5...7.....8..1..9
1..2..4.....3...7...59....8479.....................1362....15...3...7.....6..8..9
12.9.....4..3.......58..7..976.....................431..3..25.......7..6.....1.89
12.9.....4..3...5....8..7..976.....................431..3..2....5...7..6.....1.89
12.9.....4..3...5....6..7..978.....................231..3..4....5...7..6.....1.89
1..2..8...9.3.......59....4479.....................1366....15.......7.1...2..8..9
12.3....54..6........9..3..789.....................123..7..1........4..65....7.89
12.9.....4..3...5....2..3..974.....................631..7..8....5...7..6.....1.89
12.9....34..3.......58.....976.....................431.....25.......7..67....1.89
12.9....34..3...5....8.....976.....................431.....2....5...7..67....1.89
12.8.....4..3...5....1..3..671.....................934..7..9....5...7..6.....2.89
52.4.....4..3.......19..7..279.....................138..3..19.......7..6.....6.85
12.3.....49.1........8..3..716.....................493..7..2........9.16.....7.89
12.3.....49.1........8..7..716.....................493..3..2........9.16.....7.89
12.5.....49.1........8..3..516.....................495..7..2........9.16.....5.89
52.......4..3........815....76...5....1...9....5...43....592........7..6.......85
128......4..9...3.6..2......94..7...............3..61......8..4.7...1..6......289
5..1..2.....435...........612.....5..7.....3..5.....894...........576.....8..9..5
128......4..8...5.6..9..3...69.....................14...7..1..4.5...2..6......289
1..265......3...........7..47.9....58.......25....1.36..3...........7......548..9
5236.....4....8...7........8..1.7.4...........6.3.9..2........3...2....6.....4785
1326.....7....8...4........8..1.7.4...........6.3.9..2........6...2....3.....4879
128.....34..9...7.6..2......94.....................61......8..4.3...1..67.....289
1236.....45.......7....8...8....74.............63....2...2....3.......56.....4789
5326.....71.......4....8...8....74.............63....2...2....6.......93.....4875
5236.....41...8...7........8....7.4...........6.3....2........3...2...96.....4785
1324.....7....8...4........2..1.7.4...........6.3.9..8........6...2....3.....6879
13.2.....7.....8.........4.4..187......6.4......329..6.6.........2.....3.....8.79
1..2.........3..5.......7..4..927....7.4.6.3....381..6..3.......5..7.........8..9

champagne
champagne
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### Re: Symetrical Givens

Hi, blue!
blue wrote:There are 23391 canonical grids, that can be made to show diagonal symmetry ...

I encountered problems trying to confirm your number of e-d solution grids which can be morphed to main diagonal symmetric form. It's very likely that I missed something ...

First I searched for all main diagonal symmetric solution grids having following digits in places:
Code: Select all
`1 . . 5 . . . . .. 5 . . . . . . .. . 9 . . . . . .5 . . 1 2 3 . . .. . . 4 5 6 . . .. . . 7 8 9 . . .. . . . . . 1 . .. . . . . . . 5 .. . . . . . . . 9`

I got 173992 solution grids. Then I canonicalized them by GridChecker deleting duplicated grids. And I got 22682 e-d solution grids. It is less than your 23391 number ... Where is my mistake? Could you publish your 23391 canonical grids, that can be made to show diagonal symmetry?

Serg
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### Re: Symetrical Givens

Serg wrote:First I searched for all main diagonal symmetric solution grids having following digits in places:
Code: Select all
`1 . . 5 . . . . .. 5 . . . . . . .. . 9 . . . . . .5 . . 1 2 3 . . .. . . 4 5 6 . . .. . . 7 8 9 . . .. . . . . . 1 . .. . . . . . . 5 .. . . . . . . . 9`

I got 173992 solution grids. Then I canonicalized them by GridChecker deleting duplicated grids. And I got 22682 e-d solution grids. It is less than your 23391 number ... Where is my mistake?

Hi Serg,

You aren't getting results for grids containing this arrangement of 5's:

Code: Select all
`1---------5 . . .. 5 . . . | . . .. . 9 . . | 5 . .. . . 1 . | . . 5. . . . 5 | . . .5 . . . . 9 . . .. . 5 . . . 1 . .. . . . . . . 5 .. . . 5 . . . . 9`

For each of the off-diagonal 5's, the row and column containing the 5, intersect the main diagonal at cells containing distinct digits (1 & 9).

Best Regards,
Blue.

Added: I think the difference in counts is as low as it is, because the 1's and 9's would need to be in a similar situation. It does happen, though. Each of these, has an extension to a complete grid with diagonal symmetry:

Code: Select all
`1 . . . 9 5 . . .. 5 . . . 1 9 . .. . 9 . . . 5 1 .. . . 1 . . . 9 59 . . . 5 . . . 15 1 . . . 9 . . .. 9 5 . . . 1 . .. . 1 9 . . . 5 .. . . 5 1 . . . 9`

Code: Select all
`1 . . | . . 5 | . 9 .. 5 . | 9 . . | . . 1. . 9 | . 1 . | 5 . .------+-------+------. 9 . | 1 . . | . . 5. . 1 | . 5 . | 9 . .5 . . | . . 9 | . 1 .------+-------+------. . 5 | . 9 . | 1 . .9 . . | . . 1 | . 5 .. 1 . | 5 . . | . . 9`
Last edited by blue on Mon May 25, 2015 4:03 pm, edited 2 times in total.
blue

Posts: 573
Joined: 11 March 2013

### Re: Symmetrical Givens

an example of "hard puzzles" not far from a "symmetry of given", and the solution has a symmetry of given.
How to solve them ?

12.3.....4....5.7.....4.3..7..1...5...2...8...5...9..3....6.....3.5....6.....7.89 _1
5...2.4...1.....5....3....8..714....4..2.8..6....693..2....7..........9...6.8...5 _2
12.3.....4....2.3...5.8.7..7..1...6...6...4...4...9..3..3.2.5.....8....6.....7.89 _3

Code: Select all
`12.3.....4....5.7.....4.3..7..1...5...2...8...5...9..3....6.....3.5....6.....7.89 5...2.4...1.....5....3....8..714....4..2.8..6....693..2....7..........9...6.8...5 12.3.....4....2.3...5.8.7..7..1...6...6...4...4...9..3..3.2.5.....8....6.....7.89 `
champagne
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### Re: Symmetrical Givens

champagne wrote:some results on the DD symmetry.

The minimal count on my side confirms blue's results, 50781 puzzles.

I also collected puzzles having a "minimal symmetry", but not minimal stricto sensu and I found 1 906 799 puzzles.

The highest number of clues for such puzzles was 42 with 12 puzzles

Nice Work !

champagne wrote:Only hard puzzles in that group with a very small number of redundant clues are of interest.

Are you sure ? eleven's 27-clue "hardest" non-minimal puzzle with DD+JD symmetry (here), has 27 redundant clues.

Code: Select all
` +-------+-------+-------+ | . . 7 | . . 4 | . . 2 | | . 3 . | . 5 . | . 1 . | | 6 . . | 9 . . | 8 . . | +-------+-------+-------+ | . . 5 | . . 2 | . . 3 | | . 9 . | . 8 . | . 7 . | | 4 . . | 1 . . | 6 . . | +-------+-------+-------+ | . . 8 | . . 7 | . . 5 | | . 2 . | . 6 . | . 4 . | | 1 . . | 3 . . | 9 . . | +-------+-------+-------+`

Regards,
Blue.
blue

Posts: 573
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### Re: Symmetrical Givens

blue wrote:
champagne wrote:Only hard puzzles in that group with a very small number of redundant clues are of interest.

Are you sure ? eleven's 27-clue "hardest" non-minimal puzzle with DD+JD symmetry (here), has 27 redundant clues.

Code: Select all
` +-------+-------+-------+ | . . 7 | . . 4 | . . 2 | | . 3 . | . 5 . | . 1 . | | 6 . . | 9 . . | 8 . . | +-------+-------+-------+ | . . 5 | . . 2 | . . 3 | | . 9 . | . 8 . | . 7 . | | 4 . . | 1 . . | 6 . . | +-------+-------+-------+ | . . 8 | . . 7 | . . 5 | | . 2 . | . 6 . | . 4 . | | 1 . . | 3 . . | 9 . . | +-------+-------+-------+`

Regards,
Blue.

Hi blue,

This was a "general sentence".

having no clear definition of what is "a hard puzzle", the door is open.

I selected puzzles with a rating over 9.0 but up to 10.x in the small lot posted above.

another point is the number of redundant clues.

In my understanding, it is the number of clues you can simultaneously kill and still have a valid puzzle. You are clearly using another definition. In the lot I posted, I did not pass 2 redundant clues in depth.

One missing information about that work, the final code piling all information we shared could find all puzzles in the range 18_37 clues (all your valid puzzles) in less than 4 hours. I did not test the final code in the range 38_42 clues.
champagne
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### Re: Symmetrical Givens

Champagne wrote :
an example of "hard puzzles" not far from a "symmetry of given", and the solution has a symmetry of given.
How to solve them ?

12.3.....4....5.7.....4.3..7..1...5...2...8...5...9..3....6.....3.5....6.....7.89 _1
5...2.4...1.....5....3....8..714....4..2.8..6....693..2....7..........9...6.8...5 _2
12.3.....4....2.3...5.8.7..7..1...6...6...4...4...9..3..3.2.5.....8....6.....7.89 _3

Not sure what you mean here. All 3 puzzles have anti-diagonal symmetry and only the first one is still hard after the symmetry move.

Leren
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### Re: Symetrical Givens

Hi, blue!
Thank you for clarification. One should consider 2 variants of "outer" digits "5" placement:
Code: Select all
`1 . . 5 . . . . .   Variant 1. 5 . . . . . . .. . 9 . . . 5 . .5 . . 1 2 3 . . .. . . 4 5 6 . . .. . . 7 8 9 . . 5. . 5 . . . 1 . .. . . . . . . 5 .. . . . . 5 . . 91 . . . . 5 . . .   Variant 2. 5 . . . . . . .. . 9 . . . 5 . .. . . 1 2 3 . . 5. . . 4 5 6 . . .5 . . 7 8 9 . . .. . 5 . . . 1 . .. . . . . . . 5 .. . . 5 . . . . 9`
blue wrote:There are 23391 canonical grids, that can be made to show diagonal symmetry ...

Now I am able to confirm your number of e-d solution grids which can be morphed to main diagonal symmetric form.

Serg
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