## Symmetric 18s

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

### Re: Symmetric 18s

Hi, people!
I've done some experiments with moments caclulations. My goal was to find method of optimal pattern representation, i.e. criterium for selection unique isomorph for given pattern such that it can look fine and be optimal (from mathematical point of view). But I didn't succeed in doing that. The main problem is unicity. For any method I used there was at least one pattern having multiple representations.

First, I produced all possible isomorphs for given pattern. Then I found center of mass for each isomorph (assuming each clue is mass unit). Finally I calculated:
1. Average square distance between clues and center of mass, i.e. sum(distance^2)/number_of_clues. I call square root of this value as "effective pattern's radius" or R_eff.
2. Maximal distance between clues and center of mass, i.e. max(distance). I call this value as "maximal pattern's radius" or R_max.
Both values were calculated for each isomorphs and the least over all isomorphs were treated as given pattern's value.

Let's consider for example 2 simple patterns ("1" denotes clue cell):
Code: Select all
`        P1                    P21 1 1 1 1 1 1 1 1     1 . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . .1 1 1 1 1 1 1 1 1     . . . . . . . . 1`

One can check that R_eff = 3.651, R_max = 5.657 (in cell side's units) for P1 pattern and R_eff = 0.707, R_max = 0.707 for P2 pattern.

I calculated R_eff and R_max for all 121 vertically symmetric 18-clue patterns and found, that

Minimal R_eff: 2.724, maximal R_eff: 3.371 (over all 121 patterns).
Minimal R_max: 3.459, maximal R_max: 4.333 (over all 121 patterns).

To my mind critical values for R_eff and R_max should exist such that R_eff and R_max values for any pattern having valid puzzles must be not less than that critical values.

Serg
Serg
2018 Supporter

Posts: 651
Joined: 01 June 2010
Location: Russia

### My christmas present

The computation has just finished and here are all 411 vertically symmetric (valid) 18 clue puzzles which are derived from 121 different patterns.
Symmetric 18s: Show
.............1......2...3.............32.45..67.....18...5.3....1.....9..8..7..6.
.............1......1.2.3............4.....5..5.4.6.7...3...2....8...1...7.5.9.4.
.............1......1.2.3............4.....5..6.7.8.4...2...9....3...1...7.4.5.6.
.............1......1.2.3............4.....5..6.7.8.4...3...1....7...2...8.5.4.7.
.............1......1.2.3............4.....5..6.7.8.4...3...2....7...1...8.5.4.7.
.............1......1.2.3............4.....5..6.7.8.4...3...2....9...1...7.4.5.6.
.............1......23.45.....2.6....4.....7..8.....1...3...6....5...2...1..8..4.
.............1......23.45.....2.6....7.....1..8.....4...3...6....5...2...1..7..8.
.............1......23.45.....2.6....7.....1..8.....4...3...6....5...2...4..7..8.
.............1.....23...45....2.4......6.3...7.......8..4...2....6...1..1...5...7
.............1.....23...45....2.4......6.3...7.......8..4...2....6...8..8...5...1
............1.2.....3...4.............4...5...1.6.7.2.....5.....2.....6.78..4..12
............1.2.....3...4.............4...5...1.6.7.2.....5....2.......687..4..12
............1.2.....3...4.............1...5...6.7.8.2.....1....7...5...282.....36
............1.2.....3...4.............4.3.5...6.....1.....5.....2..7..8.68..4..21
............1.2.....3...4.............4.3.5...6.....1.....5.....7..8..2.62..4..71
............1.2.....3...4.............5.3.6...1.....7...8.6.3...2..5..9..7..4..1.
............1.2.....3...4.............5.3.6...1.....7...8.6.3...2..5..9..9..4..1.
............1.2.....3...4.............5.3.6...1..6..2...........2.7.1.8.9.6...3.4
............1.2.....3...4.............5.3.6...1..6..7...........2.7.1.8.9.6...3.4
............1.2.....3...4.............4.3.5...1.6.7.2....854....2.....1..9.....6.
............1.2.....3...4.............4.3.5...1.6.7.2....854....2.....1..9.....7.
............1.2.....3...4............1.5.6.2.2.4...7.1....3......8.4.6...5.....1.
............1.2.....3...4............1.5.6.2.3.4...7.8....7......7.4.9...8.....1.
............1.2.....3...4......3......5.4.6...1.....7...4.8.5...7.....2..2..5..9.
............1.2.....3...4......5.....6.....7..7.....8...1...3....4.3.5...2.8.7.9.
............1.2.....3...4......5.....6.....7..7.....8...4...3....5.6.9...1.8.7.6.
............1.2.....3...4......5.....1.....6..7.....5...8.4.3....46.89...3.....1.
............1.2.....3...4......3.....1.....5..6..7..2....5.6.....1...8..8.7...3.4
............1.2.....3...4......5.....1.....6..7..3..8....6.7.....4...8..9.5...3.4
............1.2.....3...4......3.....5.....1..6..4..7...4...8....7...6...2.5.6.9.
............1.2.....3...4......3.....5.....1..6..4..7...4...8....7...6...2.6.5.9.
............1.2.....3...4......5.....1.....6..6..3..7...4...3....8...5...7.6.9.1.
............1.2.....3...4......5.....1.....6..6..3..7...5...8....8...3...7.6.9.1.
............1.2.....3...4......5.....1.....6..6..4..7...4...2....5...3...7.6.8.1.
............1.2.....3...4......5.....1.....6..6..4..7...4...3....5...8...2.6.9.1.
............1.2.....3...4......5.....1.....6..6..4..7...4...3....5...8...2.9.6.1.
............1.2.....3...4......5.....1.....6..6..4..7...4...8....5...3...2.6.9.1.
............1.2.....3...4......5.....1.....6..6..4..7...4...8....5...3...2.9.6.1.
............1.2.....3...4......5.....1.....6..7..3..2...4...3....8...5...2.6.7.1.
............1.2.....3...4......5.....1.....6..7..3..2...4...3....8...5...2.9.6.1.
............1.2.....3...4......5.....1.....6..7..3..2...4...5....8...3...2.6.7.1.
............1.2.....3...4......5.....1.....6..7..3..2...4...5....8...3...2.9.6.1.
............1.2.....3...4......5.....1.....6..7..4..2...4...8....5...3...2.6.7.1.
............1.2.....3...4......5.....6.....7..1..3..2...4...5....8...3...2.9.7.1.
............1.2.....3...4......5.....6.....7..1..3..6...4...3....8...5...7.6.9.1.
............1.2.....3...4......5.....6.....7..1..3..6...4...5....8...3...7.6.9.1.
............1.2.....3...4......5.....6.....7..1..3..6...4...5....8...3...7.9.6.1.
............1.2.....3...4......5.....6.....7..1..4..6...4...3....5...2...7.6.8.1.
............1.2.....3...4......5.....6.....7..1..4..6...4...3....5...8...2.6.9.1.
............1.2.....3...4......5.....6.....7..1..4..6...4...3....8...5...7.6.9.1.
............1.2.....3...4......5.....6.....7..7..3..8...4...5....9...3...1.7.8.6.
............1.2.....3...4......3.....5.....1.1...4...6...3.5.....6...7..7.4...8.9
............1.2.....3...4......3.....5.....1.12.....67...8.3.....7.4.8...9.....5.
............1.2.....3...4......3.....5.....1.16.....23...3.7.....8.4.9...9.....6.
............1.2.....3...4......3.....5.....1.16.....27...8.3.....9.4.7...7.....6.
............1.2.....3...4.....5.6.....1...7....6...8...2.....5..5..3..6.8...4...2
............1.2.....3...4.....5.6.....4...5....7.4.3...1.....2..6.....8..5..7..1.
............1.2.....3...4.....5.6.....1...5....7.4.8...6.....2..5..8..3.2.......7
............1.2.....3...4.....5.6.....1...7....6.4.8...5.....1..4..7..5.2.......6
............1.2.....3...4.....5.6.....1...7....8.7.6...5.....3..6..4..2.2.......1
............1.2.....3...4.....5.6.....1...7....8.7.6...5.....3..6..9..2.2.......1
............1.2.....3...4.....3.5....2.....1..6..7..8...4...5....9...3...1..6..2.
............1.2.....3.4.5..............6.7....45...89.....8....6.......212.....67
............1.2.....3.4.5..............2.6....45...37...7.5.8..2.......69.......1
............1.2.....3.4.5..............2.6....54...37...7.5.8..2.......69.......1
............1.2.....3.4.5.............4...6...1.2.7.8............5.6.4..92.....17
............1.2.....3.4.5.............4...6...1.7.8.2............5.3.4..12.....87
............1.2.....3.4.5.............4...3...1.6.7.2.....5....7.......186.....72
............1.2.....3.4.5.............4.5.6...1.....2...........7.2.8.1.3.5...4.9
............1.2.....3.4.5.............4.5.6...1.....2...........7.2.8.1.3.5...9.4
............1.2.....3.4.5.............4.5.6...1.....7...........8.2.7.1.3.5...4.9
............1.2.....3.4.5.............4.5.6...1.....7...........8.2.7.1.3.5...9.4
............1.2.....3.4.5.............4.5.6...1.....7...........8.7.1.2.5.6...4.9
............1.2.....3.4.5.............4.5.6...1.....7...........8.7.1.2.5.6...9.4
............1.2.....3.4.5.............4.5.6...7.....1...........1.8.7.2.3.5...4.9
............1.2.....3.4.5.............4.5.6...7.....1...........8.9.1.2.3.5...4.7
............1.2.....3.4.5.............4.6.3...1.7.8.2.....3......6.5.9...8.....1.
............1.2.....3.4.5.............4.6.3...1.7.8.2.....3......9.5.4...8.....1.
............1.2.....3.4.5.............5.3.6...1.7.8.2.....6......6.5.9...2.....7.
............1.2.....3.4.5.............5.6.7...1.8.9.2.....5......4.7.3...2.....8.
............1.2.....3.4.5.............5.6.7...1.8.9.2.....5......7.3.6...2.....8.
............1.2.....3.4.5......3......4.5.6...1.....7.....6.....2.....8.17.....29
............1.2.....1.3.4......5.....6.....2..7.....8...4...1....5...3...2.8.7.6.
............1.2.....1.3.4......5.....6.....2..7.....8...4...1....5...3...2.8.9.6.
............1.2.....1.3.4......5.....6.....2..7.....8...4...1....5...3...2.9.8.6.
............1.2.....1.3.4......5.....6.....2..7.....8...4...1....5...3...2.9.8.7.
............1.2.....3.4.5......1.....2.....6..7.....8...4...9....5...3...6.7.8.2.
............1.2.....3.4.5......6.....1.....7..8.....9...5...6....7...3...9.7.8.1.
............1.2.....3.4.5......6.....1.....7..8.....9...5...6....7...3...9.8.7.1.
............1.2....34...56.............437...8.......2...........5.4.3..1..9.8..7
............1.2....34...56............5...3..7.......8....4......65384..2.......7
............1.2....34...56............5...3..7.......8....4......95384..2.......7
............1.2....34...56............5...3..7.......8....4.....9.538.4.2.......7
............1.2....34...56............3...4..1..7.8..6...........1.4.9..9...6...8
............1.2....34...56............5...3..1..7.8..6...........2.5.4..9...6...7
............1.2....34...56............5...4..1..7.8..6...........2.5.3..9...6...7
............1.2....34...56............5.3.7..1.......8...........3.6.4..7..8.1..9
............1.2....34...56............5.3.7..1.......8...........3.6.4..7..9.1..2
............1.2....34...56............5.3.7..1.......8...........3.6.4..8..7.1..2
............1.2....34...56............5.3.7..1.......8...........3.6.4..8..9.1..2
............1.2....34...56............5.3.7..1.......8...........3.6.4..9..7.1..2
............1.2....34...56............5.3.7..8.......1...........3.6.4..2..7.1..8
............1.2....34...56............5.3.7..8.......1...........3.6.4..2..9.1..8
............1.2....34...56............5.4.7..1.......8...........3.6.4..7..8.1..9
............1.2....34...56............5.4.7..1.......8...........3.6.4..7..9.1..2
............1.2....34...56............5.4.7..1.......8...........3.6.4..8..7.1..2
............1.2....34...56............5.4.7..1.......8...........3.6.4..8..9.1..2
............1.2....34...56............5.4.7..1.......8...........3.6.4..9..7.1..2
............1.2....34...56............5.4.7..8.......1...........3.6.4..2..7.1..8
............1.2....34...56............5.4.7..8.......1...........3.6.4..2..9.1..8
............1.2....34...56............5.3.4..1.......7....5......8...3..9..7.1..8
............1.2....34...56............6.4.3..1.......7....5......8...4..9..7.1..8
............1.2....34...56............6.4.3..7.......8....5......9...4..1..7.8..9
............1.2....34...56............5.6.7..8.......1....4....2.......391.....48
............1.2....34...56............5.6.7..8.......1....4....9.......321.....48
............1.2....34...56............5.6.7..8.......3....4....9.......121.....98
............1.2....34...56.....3.......7.8.....1...2...5.....8.8.......73...4...1
............1.2....34...56.....3....7.......18.......7...7.8.....2...3....6.4.9..
............1.2....34...56.....3....7.......18.......7...7.8.....2...3....6.5.4..
............1.2....34...56.....3....7.......18.......7...7.8.....2...4....6.5.3..
............1.2....34...56.....3....7.......18.......7...7.8.....9...3....6.4.2..
............1.2....34...56.....4....1.......77.......8...7.8.....2...3....5.6.4..
............1.2....34...56.....4....1.......77.......8...7.8.....9...3....5.6.4..
............1.2....34...56.....4....5.......78.......1...9.7.....6...8....9.5.4..
............1.2....34...56.....4....6.......78.......1...9.7.....5...8....9.6.4..
............1.2....31...45.....4....6.......78...3...2...7.6.....4...9....5...1..
............123.....4...5..............456....1.....3....2.1.....6...4..4.5...7.8
............123.....4...5.............2...4...1.6.7.3.....5....8.......393.....61
............123.....4...5.............5.4.6...7.....1.....6.....3..8..2.71.....39
............123.....4...5.............5.4.6...7.....1.....6.....3..8..9.79.....31
............123.....4...5.............5.4.6...7.....1.....6.....8..9..3.73.....81
............123.....4...5.............5.6.7...1.....2.....7.....3..4..8.89.....31
............123.....4...5.............5.4.6...2.....1...7.6.4...1.....3..8..5..2.
............123.....4...5.............5.6.7...1.7.8.3.....4.....3.....6..2..5..1.
............123.....4...5.............54.67...2..8..3....5.7....1.....2..8.....1.
............123.....4...5.....4.6....2.....3..7.....1...5...8....6...4...1..8..7.
............123.....4...5.....4.6....2.....3..7.....1...5...8....6...4...8..3..7.
............123.....4...5.....4.6....3.....1.2.......7..5...4....6...1...1..8..2.
............123.....4...5.....4.5....3.....1.2.......6..5...7....7.8.9...1.....2.
...........1...2....3.4.5......3......41.53...6.....2....2.6....7.....6.8.......7
...........1...2....3.4.5.....1.6......7.5.....6...8...9.....6..2..9..7.7.......1
...........1...2....3.4.5.....1.6......7.8.....6...8...9.....6..2..9..7.7.......1
...........1...2....21.34......5......6...1...7..8..5....4.2....5.....6..8.....7.
...........1...2....21.34......5......6...1...7..8..5....4.2....5.....7..8.....6.
...........1...2....21.34......5......6...1...7..8..5....4.2....5.....7..8.....9.
...........1...2....21.34......5......6...1...7..8..5....4.2....5.....9..8.....7.
...........1...2....23.41......5......3...6...5..7..8....6.1....8.....5..9.....7.
...........1...2....23.41......5......3...6...5..7..8....6.1....8.....7..9.....5.
...........1...2....23.41......5......3...6...5..7..8....6.2....8.....5..9.....7.
...........1...2....23.41......5......3...6...5..7..8....6.2....8.....7..9.....5.
...........1...2....23.45......3......6...1...7..8..3....2.5....3.....9..8.....6.
...........1...2....23.45......6......3...1...6..7..8....2.5....4.....6..8.....7.
...........1...2....23.45......6......3...1...6..7..8....2.5....4.....7..8.....6.
...........1...2....23.45......6......3...1...7..8..6....2.5....6.....3..8.....9.
...........1...2....23.45......6......3...1...7..8..6....2.5....6.....4..8.....7.
...........1...2....23.45......6......3...1...7..8..6....2.5....6.....7..8.....4.
...........1...2....23.45......6......3...1...7..8..6....2.5....6.....9..8.....3.
...........1...2....23.45......6......5...3...7..8..6....1.5....6.....7..8.....9.
...........1...2....23.45......6......5...3...7..8..6....1.5....6.....9..8.....7.
...........1...2....23.45......6......5...3...7..8..6....5.1....6.....7..8.....9.
...........1...2....23.45......6......5...3...7..8..6....5.1....6.....9..8.....7.
...........1...2....23.45......6......7...1...6..8..9....2.5....3.....6..9.....8.
...........1...2....23.45......6......7...1...6..8..9....2.5....3.....8..9.....6.
...........1...2....23.45......6......7...1...6..8..9....2.5....4.....6..9.....8.
...........1...2....23.45......6......7...1...6..8..9....2.5....4.....8..9.....6.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....3..9.....7.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....3..9.....8.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....4..9.....7.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....4..9.....8.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....7..9.....3.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....7..9.....4.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....8..9.....3.
...........1...2....23.45......6......7...1...8..9..6....2.5....6.....8..9.....4.
...........1...2....31.45......6......2...3...7..8..6....2.5....6.....4..8.....7.
...........1...2....31.45......6......2...3...7..8..6....2.5....6.....7..8.....4.
...........1...2....31.45......6......5...3...7..8..6....5.2....6.....4..8.....7.
...........1...2....31.45......6......5...3...7..8..6....5.2....6.....7..8.....4.
...........1...2....34.56......4......2...1...4..7..8....1.3....6.....4..9.....7.
...........1...2....34.56......4......2...3...4..7..8....3.1....6.....4..9.....7.
...........1...2....34.56......7......2...1...7..8..9....1.3....4.....7..6.....8.
...........1...2....34.56......7......2...1...7..8..9....1.3....4.....8..6.....7.
...........1...2....34.56......7......2...1...7..8..9....1.3....5.....7..9.....8.
...........1...2....34.56......7......2...1...7..8..9....1.3....5.....8..9.....7.
...........1...2....34.56......7......2...3...7..8..9....1.3....4.....7..9.....8.
...........1...2....34.56......7......2...3...7..8..9....1.3....4.....8..9.....7.
...........1...2....34.56......7......2...3...7..8..9....1.3....5.....7..6.....8.
...........1...2....34.56......7......2...3...7..8..9....1.3....5.....8..6.....7.
...........1...2....34.56......7......2...3...8..9..7....2.6....7.....5..9.....8.
...........1...2....34.56......7......2...3...8..9..7....2.6....7.....8..9.....5.
...........1...2....34.56......7......6...3...7..8..9....1.3....4.....7..9.....8.
...........1...2....34.56......7......6...3...7..8..9....1.3....4.....8..9.....7.
...........1...2....23.45......6......3...1...6..7..8....5.2....8.....7.9.......6
...........1...2...3..4..5............2...6..4...5...7...2.6.....68.13...7.....9.
...........1...2...3..4..5............21.67...5.....3............62.71...8..3..4.
...........1...2...3..4..5............21.67...5.....3............67.21...4..5..8.
...........1...2...3..4..5............21.67...5.....3............67.21...8..3..4.
...........1...2...3..4..5............21.67...5.....3............76.21...4..5..8.
...........1...2...3..4..5............31.26...4.....7...........5..8..4.6.9...1.2
...........1...2...3..4..5............31.26...4.....7...........5..8..4.9.6...1.2
...........1...2...3..4..5............61.78...5.....4.....3....7.......62.8...1.7
...........1...2...3..4..5............61.27...4.....3....5.8.....2...4...8..3..9.
...........1...2...3..4..5............61.27...4.....3....5.8.....7...4...8..3..9.
...........1...2...3..4..5............61.27...4.....3....8.5.....2...4...8..3..9.
...........1...2...3..4..5............61.27...4.....3....8.5.....7...4...8..3..9.
...........1...2...3..4..5............61.27...4..8..3.....5.......734.....2...6..
...........1...2...3..4..5............61.78...4..2..3.....5.......439.....8...1..
...........1...2...3..4..5............21.67...4..5..8.....3......7...1...5..8..3.
...........1...2...3..4..5............21.67...4..8..3.....5......6...1...8..3..9.
...........1...2...3..4..5............21.67...4..8..9.....5......6...1...8..3..4.
...........1...2...3..4..5............21.67...5..3..4.....8......7...1...8..5..3.
...........1...2...3..4..5............21.67...8..5..4.....3......7...1...5..8..3.
...........1...2...3..4..5............61.27...4..8..3.....5......7...1...8..3..7.
...........1...2...3..4..5............61.27...5..8..3.....6......7...1...8..3..4.
...........1...2...3..4..5............61.78...4..5..9.....3......7...1...5..9..3.
...........1...2...3..4..5............61.78...4..5..9.....3......8...1...5..9..3.
...........1...2...3..4..5............61.78...4..9..3.....5......8...1...9..3..6.
...........1...2...3..4..5............61.78...5..3..4.....9......7...1...9..5..3.
...........1...2...3..4..5............61.78...5..3..4.....9......8...1...9..5..3.
...........1...2...3..4..5............61.78...9..5..4.....3......7...1...5..9..3.
...........1...2...3..4..5............61.78...9..5..4.....3......8...1...5..9..3.
...........1...2...3..4..5............21.67...5..8..4.....3.....4..5..8.7.......1
...........1...2...3..4..5............61.27...4..8..3.....5.....8..3..4.1.......6
...........1...2...3..4..5............61.27...4..8..3.....5.....8..3..4.1.......9
...........1...2...3..4..5............61278...5.....3.....8.......359.....7...1..
...........1...2...3..4..5............61278...5.....3.....8.......539.....7...1..
...........1...2...3..4..5............61278...5.....3.....8.......935.....7...1..
...........1...2...3..4..5............61278...5.....3.....8.......953.....7...1..
...........1...2...3..4..5............61278...9.....3.....8.......395.....7...1..
...........1...2...3..4..5............61278...9.....3.....8.......539.....7...1..
...........1...2...3..4..5............61278...9.....3.....8.......593.....7...1..
...........1...2...3..4..5............61278...9.....3.....8.......935.....7...1..
...........1...2...3..4..5...........4.....6.5..2.7..1....6......7...3..1.2...8.7
...........1...2...3..4..5...........2.....4.6..1.7..8....3....7.......68.9...1.7
...........1...2...3..4..5.....6.......1.7....4.....8.....3......6.8.7..7.2...1.9
...........1...2...3..4..5.....6.......1.7....4.....3.....5....2...8...79.7...1.2
...........1...2...3..4..5.....3.......1.6....7.....4...26.89...4.....3..8.....7.
...........1...2...3..4..5.....6.......178....9.....2...2...7....8...1..6...3...8
...........1...2...2..3..4.....5.......142...6.......7...6.4...8.......17...9...6
...........1...2...3..4..5.....3......2...6....7...1.....1.6....5.....8..8.4.7.3.
...........1...2...3..4..5.....3......2...6....72.81.....6.1....5.....3..8.....4.
...........1...2...3..4..5.....3......2...6....72.81.....6.1....8.....3..9.....4.
...........1...2...3..4..5.....3......2...6....72.81.....6.1....8.....4..9.....3.
...........1...2...3..4..5.....3......2...6....78.91.....1.6....5.....3..8.....4.
...........1...2...3..4..5.....3......2...6....78.91.....1.6....5.....3..9.....4.
...........1...2...3..4..5.....3......6...7....26.81.....1.7....5.....3..8.....4.
...........1...2...3..4..5.....3......6...7....26.81.....1.7....5.....3..9.....4.
...........1...2...3..4..5.....3......6...7....26.81.....1.7....5.....4..9.....3.
...........1...2...3..4..5.....3......6...7....26.81.....7.1....5.....3..8.....4.
...........1...2...3..4..5.....3......6...7....26.81.....7.1....5.....3..9.....4.
...........1...2...3..4..5.....3......6...7....26.81.....7.1....5.....4..8.....3.
...........1...2...3..4..5.....3......6...7....26.81.....7.1....5.....4..9.....3.
...........1...2...3..4..5.....3......6...7....27.81.....6.2....5.....3..9.....4.
...........1...2...3..4..5.....3......6...7....28.91.....1.7....8.....4..9.....3.
...........1...2...3..4..5.....6......2...4....75.81.....2.7....4.....6..9.....3.
...........1...2...3..4..5.....6......2...5....74.81.....7.2....5.....3..9.....6.
...........1...2...3..4..5.....6......2...5....74.81.....7.2....5.....4..9.....3.
...........1...2...3..4..5.....6......2...7....83.91.....2.8....5.....3..6.....4.
...........1...2...3..4..5.....6......2...7....85.91.....8.2....4.....6..7.....3.
...........1...2...3..4..5.....3......2...6...4..5..7.....7......82.61...1.....3.
...........1...2...3..4..5.....3......6...1...5..7..3.....5......21.68...4.....7.
...........1...2...3..4..5.....3......6...1...5..7..3.....5......21.68...7.....4.
...........1...2...3..4..5.....3......6...1...5..7..3.....5......81.69...4.....7.
...........1...2...3..4..5.....3......6...1...5..7..3.....5......81.69...7.....4.
...........1...2...3..4..5.....3......6...1...7..5..8.....8......21.79...5.....4.
...........1...2...3..4..5.....3......6...1...7..5..8.....8......21.97...5.....4.
...........1...2...3..4..5.....6......2...3...4..5..7.....8......71.29...9.....4.
...........1...2...3..4..5.....6......2...7...4..5..8.....8......91.23...6.....4.
...........1...2...3..4..5.....6......2...7...5..3..6.....5......82.71...4.....3.
...........1...2...3..4..5.....6......2...7...5..3..6.....5......82.79...4.....3.
...........1...2...3..4..5.....3......2...5...6..7..4.....6......81.29..9.......3
...........1...2...3..4..5.....1......2...6..6...3...4....5......72.68...8.....3.
...........1...2...3..4..5.....6......2...7..8...3...4....5......92.81...1.....3.
...........1...2...3..4..5.....3......4...6..2.6...1.3...1.2......6.7....8.....9.
...........1...2...3..4..5.....3......4...6..6.2...1.3...1.6......7.2....8.....9.
...........1...2...3..4..5.....3......6...7..2.8...1.6...1.8....4.....8..5.....2.
...........1...2...3..4..5.....3......6...7..2.8...6.1...1.8....4.....2..5.....8.
...........1...2...3..4..5.....6......7...8..2.8...9.1...2.9....1.....4..9.....3.
...........1...2...3..4..5.....6......7...8..2.9...1.7...1.9....4.....9..5.....2.
...........1...2...3..4..5.....3......6...1..7.2...8.6...6.8....9.....4.5.......3
...........1...2...3..4..5.....3......6...7..8.2...1.6...1.6....9.....4.5.......3
...........1...2...3..4..5.....3......6...7..8.2...6.1...1.2....9.....4.5.......3
...........1...2...3..4..5.....3......6...7..8.2...6.1...1.6....9.....4.5.......3
...........1...2...3..4..5.....6......4...7..3.8...1.2...1.2....4.....6.5.......3
...........1...2...3..4..5.....6......7...8..2.8...1.3...2.8....5.....6.3.......4
...........1...2...3..4..5.....3......21.67..8.......1...2.3......7.1....5.....8.
...........1...2...3..4..5.....6.....5.....7.87.....95...3.5.....6...4..2.......1
...........1...2...3..4..5....4.6.....2...1....3.7.8.....1.8....4.....9..5.....6.
...........1...2...3..4..5....4.6.....2...7....6.1.8.....7.2....4.....1..9.....3.
...........1...2...3..4..5....1.4.....2...6....7.8.1.....2.6....8.....4.9.......3
...........1...2...3..4..5....1.5.....6...7..2.......4...726....1.....3..5.....8.
...........1...2...2..3..4....4.5.....6...7..3...8...1...7.9....3.....2..4.....8.
...........1...2...2..3..4....4.5.....6...7..3...8...1...7.9....3.....8..4.....2.
...........1...2...3.4.5.6............2.1.7...8..9..4.....7......9.2.3...4.....8.
...........1...2...3.4.5.6............7.1.8..4..5.6..3....7......2...1...6.....3.
...........1...2...3.4.5.6............2.7.8..36.....45....2.......3.6.....8...7..
...........1...2...3.4.5.6...........4..1..3..5..7..4....3.6.....2...1..8.......7
...........1...2...3.4.5.6.....7.......3.6.....2...8......1.....7.....3.36.....49
...........1...2...3.4.5.6.....1.......3.6.....2...7......7....4.......686.....53
...........1...2...3.4.5.6.....7.......128....9.....5...2...1....4...7...5.....4.
...........1...2...3.4.5.6.....7......7...8...6.....4.....2.....1.....5.45.....36
...........1...2...3.4.5.6.....7......7...8...9.....4.....2.....1.....5.45.....96
...........1...2...3.4.5.6.....7......7...8...9.....4.....2.....1.....5.65.....94
...........1...2...3.4.5.6.....7......7...8...9.....3.....2....3.......454.....93
...........1...2...3.4.5.6.....1......7...8...6.....3....3.4.....2...9....8.2.1..
...........1...2...3.4.5.6.....7......7.1.8...4.....3....3.9.....2...7...5.....4.
...........1...2...3.1.4.5.....6.....7.....8..5..2..3....3.7.....2...1....9...6..
...........1...2...3.4.5.6.....7.....4.....3..5..1..4....3.6.....2...8....7...1..
...........1...2...3.4.5.6.....7.....4.....3..5..2..4....3.6.....2...1....8...7..
...........1...2...3.4.5.6.....7.....4.....3..5..2..4....3.6.....2...7....8...1..
...........1...2...3.4.5.6.....7.....4.....3..5..2..4....3.8.....2...7....9...1..
...........1...2...3.4.5.6.....7.....4.....3..6..1..8....8.6.....2...5....7...1..
...........1...2...3.4.5.6.....7.....4.....3..8..1..4....3.6.....2...1....7...5..
...........1...2...3.4.5.6.....7.....4.....3..8..1..4....3.6.....2...5....7...1..
...........1...2...3.4.5.6.....7.....4.....3..8..1..4....6.3.....2...5....7...1..
...........1...2...3.4.5.6.....7.....4.....3..8..2..4....8.6.....2...7....5...1..
...........1...2...3.4.5.6.....7.....4.....8..6..1..3....6.8.....2...5....7...1..
...........1...2...3.4.5.6.....7.....4.....8..6..1..3....6.8.....2...9....7...1..
...........1...2...3.4.5.6.....7.....4.....8..6..1..3....8.6.....2...4....7...1..
...........1...2...3.4.5.6.....7.....4.....8..6..1..3....8.6.....2...5....7...1..
...........1...2...3.4.5.6.....7.....4.....8..6..1..4....3.8.....2...5....7...1..
...........1...2...3.4.5.6.....7.....4.....8..6..1..9....6.8.....2...1....7...9..
...........1...2...3.4.5.6.....7.....4.....8..6..9..5....8.6.....2...5....7...1..
...........1...2...3.4.5.6.....7.....6.....3..4..2..8....8.6.....2...7....5...1..
...........1...2...3.4.5.6.....7.....6.....8..8..1..4....3.6.....2...5....7...1..
...........1...2...3.4.5.6.....7.....6.....8..8..1..4....6.3.....2...1....7...5..
...........1...2...3.4.5.6.....7.....6.....8..8..1..4....6.3.....2...5....7...1..
...........1...2...3.4.5.6.....7.....8.....9..6..1..3....6.9.....2...1....7...4..
...........1...2...3.4.5.6.....7.....8.....9..6..1..3....6.9.....2...1....7...5..
...........1...2...3.4.5.6.....7.....8.....9..6..1..3....6.9.....2...4....7...1..
...........1...2...3.4.5.6.....7.....4.....3..6..1..8....8.6.....2...5..7.......1
...........1...2...3.4.5.6.....7.....4.....3..8..1..4....8.3.....7...1..2.......5
...........1...2...3.4.5.6.....7.....4.....3..8..1..4....8.3.....7...1..2.......8
...........1...2...3.4.5.6.....7.....4.....8..6..1..3....8.6.....2...5..7.......1
...........1...2...3.4.5.6.....7.....4.....8..9..1..4....9.3.....7...1..2.......9
...........1...2...3.4.5.6.....7.....6.....3..4..2..8....8.3.....2...7..5.......1
...........1...2...3.4.5.6.....7.....6.....3..8..4..9....3.8.....7...1..2.......5
...........1...2...3.4.5.6.....1.....4.....3.6.......4...3.7.....2...1....8.9.7..
...........1...2...32...45.....6......3...1..6...7...8...2.4...7.......68.......9
...........1...2...32...45.....6......7...1..3...8...9...2.4...6.......89.......3
...........1...2...32...45.....6......7...1..3...8...9...2.4...6.......89.......7
...........1...2...32...45.....6......7...1..6...8...9...2.4...8.......69.......3
...........1...2...34...56.....7......2...1..7...8...9...2.5...8.......79.......3
...........1...2...34...56.....7......2...4..8...9...7...1.4...6.......87.......9
...........1.2.3...4..5..6....5.7.....2...1....3...8.....8.2....7.....4..9.....5.
...........1.2.3...4.5.6.7.....8.......4.7.....2...1......1......8.3.2...7.....9.
...........1.2.3...4.5.6.7.....1.......4.7.....3...8......8.....5.....4..7..3..6.
...........1.2.3...4.5.6.7.....1.......4.7.....3...8......8.....7.....9..9..3..5.
...........1.2.3...4.5.6.7.....1.......7.8.....3...9......9.....7.....6..6..3..8.
...........1.2.3...3.4.5.6.....7.....4.....3..6.....8....8.6.....2...1....9...7..
...........1.2.3...3.4.5.6.....7.....4.....3..6.....8....8.6.....2...7....9...1..
...........1.2.3...3.4.5.6.....7.....4.....3..6.....8....8.6.....2...9....7...1..
...........1.2.3...4.1.5.6.....7.....5.....4..6.....8....6.8.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....5.....8..6.....4....8.6.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....5.....8..6.....9....6.8.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....5.....8..6.....9....6.9.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....6.....4..8.....5....6.8.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....6.....4..8.....5....8.6.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....6.....4..8.....9....6.8.....3...1....7...2..
...........1.2.3...4.1.5.6.....7.....6.....4..8.....9....8.6.....3...1....7...2..
...........1.2.3...4.5.6.7.....5.....6.....4..7.....8....4.8.....2...1....3...5..
...........1.2.3...4.1.5.6.....7.....6.....4..8.....5....4.8.....7...2..3.......1
...........1.2.3...4.1.5.6.....7.....6.....4..8.....5....8.4.....7...2..3.......1
...........12.34..25.....67....5........6......3...1.....1.4....6.....8..7.....5.
...........12.34..25.....67....6........8......4...3.....1.4....6.....5..9.....8.
...........12.34..52.....67....6........8......4...3.....4.1....6.....5..9.....8.
...........12.34..56.....78....5........7......2...1.....1.4....4.....5..7.....6.
...........12.34..56.....78....5........7......2...1.....4.1....4.....5..7.....6.
...........12.34..56.....78....5........7......4...2.....1.4....3.....5..7.....6.
...........12.34..56.....78....5........7......4...2.....4.1....3.....5..7.....6.
...........12.34..56.....78....5........7......4...2.....4.1....7.....3..8.....5.
...........12.34..56.....78....5........7......4...2.....4.1....7.....6..8.....3.
...........12.34..56.....78....5........7......2...1.....1.4....4.....5.7.......6
...........12.34..56.....78....5........7......2...1.....4.1....4.....5.7.......6
...........12.34..56.....78....5........8......2...1.....1.4....7.....5.8.......3
...........12.34..56.....78....5........8......2...1.....1.4....8.....6.9.......5
...........12.34..56.....78....5........8......2...1.....4.1....4.....6.7.......5
....1........2......3...4......5......43.67...1.....8....4.7....8.....3..2..8..1.
....1........2......3...4.....3.5....1.....6..2.....7...5...8....68.43...9.....1.
....1........2......3...4.....3.5....1.....6..2.....7...5...8....68.43...9.....2.
....1.......2.3.....4...5......2......5...4...16...78....6.5....3.....7.2.......1
....1.......2.3.....4...5.....3.4.....5...6....7...8.....587....2.....1..4.....3.
....1.......2.3.....4...5.....3.4.....5...6....7...8.....765....2.....1..4.....3.
....1.......2.3.....4...5.....4.6....2.....1..3.....2...5...7....6.8.4...1.....5.
....1.......2.3.....4...5.....4.6....2.....3..3.....1...6...7....5.8.4...1.....6.
....1.......234.....5...6......2......7...8....6.9.5.....7.6....2.....9..4.....8.
....1.......234.....5...6......5......7...5...3..2..8....8.6....2.....3..4.....9.
....1.......234.....5...6......7......6...8...2..4..3....8.5....7.....4..9.....2.
....1......1...2....2...3.....2.3......4.5....6.....7...3...5...8.....6..7..8..1.
....1......1...2....2...3.....2.4......5.3....6.....7...3...4...8.....6..7..8..1.
....1......2...3....3...4.....2.3......5.6....4.....7...5...2...7.....1..1..8..9.
....1......2...3....3...4.....2.3......5.6....7.....8...5...2...8.....7..1..4..9.
....1......2...3....3...4.....2.4......3.5....6.....7...4...8...9.....1..7..8..6.
....1......2...3....3...4.....2.5......6.3....4.....7...6...2...7.....1..1..8..9.
....1......2...3....3...4.....2.5......6.3....7.....8...6...2...8.....7..1..4..9.
....1......2...3....4...5.....2.4......5.6....7.....8...5...2...3.....1..8..9..5.
....1......2...3....4...5.....1.6.....3...4...6..7..8....3.2....7.....1..8.....6.
....1......2...3....4...5.....6.7.....3...4...1..8..6....5.4....6.....7..8.....1.
....1......2...3....4...5.....6.7.....3...4...6..8..7....2.3....1.....8..7.....6.
....1......2...3....4.5.6.....2.4.....3...4...7.....8....3.6....1.....9.8.......5
....1......2...3....4.5.6.....2.4.....6...4...7.....8....3.6....1.....9.8.......5
....1......2...3...4.....5.....6......3...7..1.5...2.8...2.7....6.....1..9.....4.
....1......2...3...4.....5.....6......7...8..1.8...2.7...2.8....1.....6..8.....4.
....1......2...3...4.....5.....6......7...8..1.8...9.2...8.9....1.....6..8.....4.
....1......2...3...4.....5....4.6.....7...1....8...2.....2.1....6.....9..9..7..4.
....1......2...3...4.....5....6.7.....3...6....8...9...6.....7..5..8..4.4.......1
....1......2...3...4.....5....2.5...1.......67.......8...761.....4...9....5...2..
....1......2...3...4.....5....2.6...7.......18.......9...971.....3...5....6...2..
....1......2...3...4.....5....4.6...7.......18.......9...179.....3...5....6...4..
....1......2...3...4.....5....4.6...7.......18.......9...971.....3...5....6...4..
....1......2...3...1..4..5.....6......7...8...4..5..1....2.8.....3...2...6.....4.

Of course a validation would be appreciated but it's unlikely due to the length of the computation.
Afmob

Posts: 130
Joined: 28 June 2011

### Re: Symmetric 18s

Oh, thanks for this christmas present !

Great work, so finally you have resolved this long term open question. Congratulations.

I am also happy, that the number of possible patterns is eleven square To verify it is far beyond my cpu capabilites, though i had a program, which i think worked similar to yours. If i remember right, the point was to exclude the automorphs, when testing.
What i can say is, that all 404 formerly known puzzles are in your list.
I had calculated the hammond distance of the 5 you already posted to the others. It was 7-9, and only one of them was inside {-4+4} to known puzzles.
This result always allows a new speculation, how many 17 clues might be unknown. Here 1.7% of the puzzles have not been found without exhaustive search.
eleven

Posts: 2077
Joined: 10 February 2008

### Re: Symmetric 18s

Hi, Afmob!
Great work! Maybe it's the first time we know full list of the puzzles of certain class.
Good news for eleven (eleven^2) concerning variants of patterns arrangement.
When exhaustive search was started I expected to see more newfound patterns.

It is hard to believe that this problem has been closed.

Serg
Serg
2018 Supporter

Posts: 651
Joined: 01 June 2010
Location: Russia

### Re: Symmetric 18s

Hi, Afmob!
Just for your tool's benchmarking - how much time does it take you to find all 7 valid puzzles for Fractal Pattern (see http://forum.enjoysudoku.com/valid-puzzles-for-fractal-pattern-t30237.html)?

Serg
Serg
2018 Supporter

Posts: 651
Joined: 01 June 2010
Location: Russia

If your estimation of 650 billion essentially different Sudokus is correct then there is no point in checking the "Fractal pattern". On average I can check about 140,000 puzzles per second (per core) but this number can vary depending on the number of pattern automorphisms or specific properties of the pattern which might produce a lot of puzzles with no solution.
Afmob

Posts: 130
Joined: 28 June 2011

### Re: My christmas present

Afmob wrote:The computation has just finished and here are all 411 vertically symmetric (valid) 18 clue puzzles which are derived from 121 different patterns.

Reaction here implies that we know that the 121 patterns is a closed set. Is this true?
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

### Re: Nxt challenge ?

ronk wrote: Reaction here implies that we know that the 121 patterns is a closed set. Is this true?

I would tend to believe it !

This corespondence amused me at the time - you may have missed it !
Afmob wrote:
coloin wrote:I may be way off here if i havent fully understood how you are iterating those patterns so fast !

I use parallel computing (31+1 cores) and the modified version of ZhouSolver provided by Jason.

My unsubtle approach by generating symmetric 19s .... a superfluos clue be removed from the c5 to give a symmetric 18 .
When I stopped I had a substancial collection of symmetric 19s ....I stopped because i knew Afmob was going to complete a search
and there was no guarrentee that there actually were any more puzzles to find !
Finding them was rewarding ... in a masochistic sort of way !

Well done in finding the remaining puzzles.

Next challenge ?

C
coloin

Posts: 1790
Joined: 05 May 2005

### Re: Symmetric 18s

Now i have tried different things to get a nice representation of the patterns (different ways to select equivalents, order them in a spiral starting from the center, color the borders after the finders, etc.), but i am not satisfied with any of them.
So i leave you with 2 black/white pictures, showing the patterns, as they were posted, and the patterns sorted in minlex form.
Happy new year (i wanted to finish that last year ).
posted, minlex.
This is the list of puzzles with new patterns, as published:
Hidden Text: Show
.............1......1.2.3............4.....5..5.4.6.7...3...2....8...1...7.5.9.4. # 1. mauricio
.............1......23.45.....2.6....4.....7..8.....1...3...6....5...2...1..8..4. # 2. mauricio
............1.2.....3...4.............4...5...1.6.7.2.....5.....2.....6.78..4..12 # 3. mauricio
............1.2.....3...4.............4...5...1.6.7.2.....5....2.......687..4..12 # 4. mauricio
............1.2.....3...4.............4.3.5...1.6.7.2....854....2.....1..9.....6. # 5. mauricio
............1.2.....3...4............1.5.6.2.2.4...7.1....3......8.4.6...5.....1. # 6. mauricio
............1.2.....3...4......5.....6.....7..7.....8...1...3....4.3.5...2.8.7.9. # 7. mauricio
.5.....9..3.....6...47.82....2.3.4....7...1......5.......2.4....9.....3.......... # 8. olimpia
............2.3...8...1...9.32...56...6...3......4.......6.5...91.....74......... # 9. Rico Alan,R828-S11
...........1.3.5.....2.4....2.....8...3.9.1......5.....4.6.8.2...5.7.9........... # 10. olimpia
............1.2.....1.3.4....4...1....5...3...2.6.7.8.....5.....7.....6..8.....2. # 11. mauricio,heart
..........9..1..3...6.2.7.....3.4...21.....98...........25.64...8.....1.......... # 12. Vic Reinhart,Raphael
..........7..1..6...9...2.....2.3...56.....18...........34.29...1..5..8.......... # 13. Vic Reinhart,Odyssey
....1.....5.....7...72.36...........81.....95...4.2.....3...2...9..5..1.......... # 14. Vic Reinhart,Electra
............1.2....34...56.....3....1.......77.......8...7.8....2.....4..6..5..3. # 15. mauricio
.........12.....84.3.....7...4...6.....2.3.....5...9....6.9.5...7.....2.....5.... # 16. Rico Alan,R828-S09
....1.....9..2..7...8...4.....3.4....1.....9.....7......45.63...2..9..1.......... # 17. Vic Reinhart,El Morro
...........8...3...4.2.7.6...........7.....4...3.1.8...2.6.4.7...5.3.1........... # 18. tarek
..........3..1..2...6...5...8..2..4...5...8......3......85.69...2..4..1.......... # 19. olimpia
...........1...2...3..4..5.....3......2...6....7...1.....1.6....5.....8..8.4.7.3. # 20. mauricio
....8....21.....343.......2.6.....5...8...9.....3.7.....5...6...........7...9...3 # 21. Rico Alan,R828-S12
...........1...2...3..4..5.....3......6...7..2.8...1.6...1.8....4.....8..5.....2. # 22. mauricio
1.2...5.3....8....3.......4..9...2.....3.5....8.....6...........6..9..2.5.......1 # 23. Rico Alan,R828-S10
86.....145.......6....1.......8.2.....7...3.....6.4.....1.5.9...........4.......8 # 24. Vic Reinhart,Lulin
..........3..1..4...7...5......2......5...6...4..3..2.....4......85.69...1.....3. # 25. olimpia
..........6.....5...7.1.2......2.....7.....6...1.3.4......4.....8.5.6.9.2.......8 # 26. olimpia
....1....5...2...4.6.....5...7...2...8.5.6.7.....3......2.4.3...1.....6.......... # 27. olimpia
1...2...3.4.....5.............6.4...3.......12.......7....3.....8.....6..5.8.9.4. # 28. mauricio
1...2...3.4.....5.............6.4...2.......7..3...8.....2.5....5..7..9..6.....4. # 29. mauricio
1...2...3.4.5.6.7.............8.4.......3....9.......1....9.....5.....4..8..1..5. # 30. mauricio
....1........2......3...4......5......43.67...1.....8....4.7....2..8..1..8.....3. # 31. mauricio
1...2...3.4.....5.....1.......6.7...3.......18.......2.7.....4...........6.4.9.7. # 32. mauricio
...........1...2....23.45......6......3...1...6..7..8....5.2....8.....7.9.......6 # 33. mauricio
1.......2.3.4.5.6.............7.3...2.......58.......1....8.....6.....3..7..1..4. # 34. mauricio
1.......2.3.4.5.6.............7.6.....2...5..8.......1....8.....6.....3..7..1..4. # 35. mauricio
...........1...2...3.4.5.6.....1.....7.....3.36.....48....7.......3.6.....2...9.. # 36. mauricio
...........1...2...3.4.5.6.....1.....4.....3.6.......4...3.7.....2...1....8.9.7.. # 37. mauricio
....1......1...2....2...3.....2.3......4.5....6.....7...3...5...7..8..1..8.....6. # 38. mauricio
1...2...3.4.5.6.7.............8.7...3.......26.......1....6.....8.....4..7.....5. # 39. mauricio
1...2...3.4.....5....3.6......7.5...2.......13.......6....1.....8.....4..5.....7. # 40. mauricio
1.2...3.4.5.6.7.8.............5.8...3.......74.......1....3.....6.....5.....1.... # 41. mauricio
1.2...3.4.5.6.7.8.............5.8...3.......2..8...1......3.....6.....5.....1.... # 42. mauricio
....1.....4..2..3...7...5......3.....3..4..2...6...8.....5.6.....8...6...1.....4. # 43. olimpia
...........1.2.3...4.1.5.6.....7.....6.....4..8.....5....4.8.....7...2..3.......1 # 44. mauricio
...4.5...2.......1.3.....7...6...5.....2.3.......1........2......5...6...14...79. # 45. Vic Reinhart,Oaxaca
....1......2...3...4.....5....167.....3...4....8...5.....4.3...6.......19.......7 # 46. mauricio,spaceship
....8....13.....57.2.....3...4.6.9..............3.7.....8.4.6...5.....7.....9.... # 47. Rico Alan, R747-S13
81.....492.......1...5.7.....2.8.7....5...6......9.......6.3............1.......8 # 48. Rico Alan, Jade
..........3..1..2...6...5..7.......5.1..2..4.....3......85.67...2..4..1.......... # 49. olimpia
....1....4...2...883.....47..9...6.....3.4.......5......1.6.9...8.....3.......... # 50. olimpia
....1....84.....32.3.....8...1...2.....3.4.....5...3....6...5...7.....4.....2.... # 51. olimpia
...2.3...7...1...46.......55.......6..........7..2..9..2.....8..9.....3....5.4... # 52. olimpia
....1....5.......343.....65..7...1...5.3.4.8............2...9...6.....5.....2.... # 53. olimpia
....1....1.......92.......33.......7.7..6..5..8.5.2.6..6.....7....4.3............ # 54. olimpia
...7.3....1.....2....5.9.....5...9...6..4..1...3...6......6....9.7...5.4......... # 55. Ano1
............1.5.....7...3...9..3..1...4.2.7...5..6..8...6.7.2...1.....9.......... # 56. Ano1
...........4...6.....275.....6...1...3..5..9...9...4.....6.1....2.....3..5.....7. # 57. Ano1
...........1...9...4.2.7.8.....9.......8.4.....5...1..87.....464.......2....5.... # 58. Ano1
...........6...7.....1.3.....4.6.2.....9.8.....8...3...2.....1.1.......97...4...8 # 59. Ano1
...9.6.....5...4.....4.8.....9.1.2.............7...8...4.....3.8.......6.6..3..7. # 60. Ano1
............6.3.....5.7.2...9.....6...4.2.1...3.....8.7.2...5.4...3.8............ # 61. Ano1
............4.2.....6.7.3.............1.6.7...9.....4.7.8...6.3.4.2.9.5.......... # 62. Ano1
...........8...3...9.1.2.6.1...3...626.....79....8.......8.6.....7...4........... # 63. Ano1
...........6.4.3...2.....7.97.....51.1.....9.....8.......5.1...4.3...8.2......... # 64. Ano1
.........8.9...6.4...5.1.......8.....5.....7...23948...7.....3.4.......9......... # 65. Ano1
............2.1.....7...8...6.....5...5.7.3.....4.8.......8.....1.....9.82.....61 # 66. Ano1
..........9.671.2...8...4.....845.......2.....1.....6...4.3.8...6.....7.......... # 67. Ano1
...........7.3.5...8..7..2.9.3...7.1.6.4.2.8.............2.8.....9...3........... # 68. Ano1
....5......3.8.9...5.....7...2...6.....3.1.....9...8.....896.......4.....1.....5. # 69. Ano1
..........2..3..9...7.8.6......7......1.6.3...9.....2...6...8...4.5.2.1.......... # 70. Ano1
............954....6.....7...........2.6.8.9...3.4.5.....7.2...5.4...3.1......... # 71. Ano1
...428.............3.....7.4...9...1....5....1.8...6.4.5..7..3.8.......6......... # 72. Ano1
..........3.....7....4.8.....4.9.1......5....8.1.3.4.6.5..7..3...8...6........... # 73. Ano1
..........4.9.3.8...5...6...8.....5...2.6.7.....1.2.....6.5.1...9.....3.......... # 74. Ano1
...........6...1...4.....2...32167...9.....4.....7.......9.5...1.7...6.3......... # 75. Ano1
...........1...5...6.8.4.2.94.....828.......6....7.......2.6.....5.1.7........... # 76. Ano1
...........8.7.6.....3.4.....2...7...3.9.5.4..........59.....34..7.6.8........... # 77. Ano1
..........1.....2....678....2.....1...6.3.7...5.....6...8...6.....1.5...7.......9 # 78. Ano1
...........2.4.6...8.....1.5.......331.....98....6.....9.1.3.5...4...2........... # 79. Ano1
.1.....6...4...3.....5.7.......3.....6.....5...3.2.9...5.6.1.8...2...4........... # 80. Ano1
.1.....5....3.7.......6.....8.....4...6...8...5..9..1...3...7.....1.8.....7...6.. # 81. Ano1
..........3.1.7.5...2.6.8...7.....3.....9.......582.....8.2.9...1.....7.......... # 82. Ano1
.2.....5.....3.......1.6.....3.5.1......9....1.8...6.7.9..2..4...6...8........... # 83. Ano1
.8.....4...5.7.9...2.....3...9...5.....2.8.....1.6.7.....9.5.............4.....2. # 84. Ano1
............785.....2...6...7.....5...4.9.2...1..6..8...6.2.9...8.....7.......... # 85. Ano1
.........3..1.2..5..8...6....7...4......6.......5.3.....6.4.7..53.....12......... # 86. olimpia
.........4.......7.3..1..6...2...5......3....9.8...2.4..52.48...7.....1.......... # 87. olimpia
............2.4...7.......86...5...7..1.6.2....2...3....3...4..9...7...6....8.... # 88. olimpia
56.....78...........2.1.3....4.3.2............7.....8..4.7.8.5............3...9.. # 89. olimpia
...9.6...1.......32.......43.......5.6.8.7.1..7.....9..8.....6.....3........4.... # 90. olimpia
.1.2.3.4.5.......6.........6...5...8...4.2...7.......5....6....8.......9.4.....1. # 91. olimpia
.........5..1.3..6.4..2..8...3...6......4.....7.....2...65.13...2.....7.......... # 92. olimpia
...2.3...5...1...66.......47.......5....2.....8.....3..2.....1..3.....9....4.5... # 93. olimpia
3.......21...6...5.7.....3...5...4.....2.3.......1.......5.4....16...78.......... # 94. olimpia
....1.....7.....1...3...6.............62.38...4.....5.6.8...2.3....4......2...5.. # 95. olimpia
...3.4...7.......58.......1.9.....3...7.5.2.....8.6...1...7...8.2.....1.......... # 96. olimpia
...1.2...4.......65.......36...3...9..........7..1..8..1.....7..8.....2....4.3... # 97. olimpia
..........5..1..7...6...3...........7..3.4..6.1.....2...4...5......2....6.3...8.4 # 98. olimpia
..1...8....8...3.....2.7.............7.....4....518....2.6.4.7...5.3.1........... # 99. eleven
...........1...9...4.3.7.8....498..............5...1..87.....464.......2....5.... # 100. eleven
...4.5....2.....8..3.....7...6...5.....293.......1........2......5.8.6....4...7.. # 101. eleven
...1.2...9.......48.......15.......6............3.4.....3.5.8...1..6..2..2.....3. # 102. olimpia
1.2...3.4.5.7.6.8.............5.8...3...1...24.......9....4.....6.....5.......... # 103. eleven
...........1.2.9...4.3.9.8.............874.....5...1...7..1..4..8.....2.....5.... # 104. eleven
...1.2..............3...4....7...3....4.3.5.....7.6....1..5..7..2.....1..9.....6. # 105. eleven
1.......32.......4..78.46....6...9...........5...3...13.......2....1.......9.8... # 106. olimpia
...........3...5..6..7.9..1.........9...4...71...5...9...6.1.....5...3...4.....8. # 107. eleven
....1......2...3....4.5.6.....6.3.....6...2...7.....8....4.2....9.....1.5.......7 # 108. eleven
............123.....4...5.....4.5....3.....1.2.......6..5...7....7.8.9...1.....2. # 109. eleven
...........1...2...3..4..5....1.5.....6...7..2.......4...726....5.....8..1.....3. # 110. eleven
............1.2....32...45.....4....6.......78...3...1...6.7.....5...2....4...9.. # 111. coloin
....1......2...3...1.4.5.6...........2.....5..4..6..1....3.2.....3...7..7.......8 # 112. coloin
..........1.2.3.4...5...6.....657....3.....8.....9......9...2....6...5...2.....3. # 113. coloin
.1.....2..3.....4....567.....1...7.....3.2.......4.......2.1.....7...6....5...8.. # 114. coloin
....1......2...3...4.....5...6...7.....8.9.....9...2...5..7..4..8.....9.1.......5 # 115 coloin
1.......2............345......162..............3...4...34...78...5...3.....6.1... # 116 coloin
............1.2.....3...4......3.....5.....1.1...4...6...3.5.....6...7..7.4...8.9 # 117 afmob
...........1...2...3..4..5............2...6..4...5...7...2.6.....68.13...7.....9. # 118 afmob
...........1...2...3..4..5.....6.......178....9.....2...2...7....8...1..6...3...8 # 119 afmob
...........1...2...2..3..4.....5.......142...6.......7...6.4...8.......17...9...6 # 120 afmob
...........1...2...3..4..5.....3......21.67..8.......1...2.3......7.1....5.....8. # 121 afmob
eleven

Posts: 2077
Joined: 10 February 2008

### Re: Symmetric 18s

As a by-product of some tests I'm doing, here are some diagonally-symmetric 18s:

Code: Select all
` 1 . . . . . . . 2 . . . . . 3 . 4 . . . 5 . . 6 . . . . . . . . . . 7 . . . . . . 8 3 . . . 9 2 . 5 . . . . . . . . 2 . . . 5 . 7 . 9 . . . . . 3 . . . . . 6 . .   ED=1.5/1.5/1.5 1 . . . . . . . 2 . . . . . 3 . 4 . . . 5 . . 6 . . . . . . . . . . 7 . . . . . . 8 3 . . . 7 2 . 5 . . . . . . . . 2 . . . 5 . 4 . 9 . . . . . 3 . . . . . 6 . .   ED=1.7/1.5/1.5 1 . . . . . . . 2 . . . . . 3 . 4 . . . 5 . . 6 . . . . . . . . . . 1 . . . . . . 7 8 . . . 8 2 . 5 . . . . . . . . 2 . . . 5 . 7 . 9 . . . . . 3 . . . . . 6 . .   ED=7.2/1.5/1.5`

Code: Select all
` . . . . . . . . 1 . . . . . . . 2 3 . . . . 4 5 . . . . . . 2 . 1 . 6 . . . 4 . . . . . . . . 7 8 . . 5 . . . . . . . 7 9 . . . 1 . 6 . . . . . 3 2 . . . . . . .   ED=2.6/1.2/1.2`

Regards,

Mike Metcalf m_b_metcalf
2017 Supporter

Posts: 10018
Joined: 15 May 2006
Location: Berlin

### Re: Symmetric 18s

Well ...... and i thought the topic was finished !
So how many patterns are there with 0,2,4,6 or 8 clues on the diagonal ?
C
coloin

Posts: 1790
Joined: 05 May 2005

### Re: Symmetric 18s

coloin wrote:Well ...... and i thought the topic was finished !
So how many patterns are there with 0,2,4,6 or 8 clues on the diagonal ?
C

No idea, and I'm certainly not going to try to find out! However, another test gave one pattern with zero on the diagonal:
Code: Select all
` . . 9 . . . . . 2 . . . . . 8 . 7 . 3 . . . . 4 . . . . . . . . . . 8 . . . . . . 3 4 . . . 6 2 . 1 . . . . . . . . 9 . . . 1 . 7 . 2 . . . . . 4 . . . . . 9 . .  zero on diagonal`

which I found remarkable because this was the first 18-clue puzzle I've ever produced ab initio.

Regards,

Mike Metcalf m_b_metcalf
2017 Supporter

Posts: 10018
Joined: 15 May 2006
Location: Berlin

### Re: Symmetric 18s

Hi, coloin!
coloin wrote:So how many patterns are there with 0,2,4,6 or 8 clues on the diagonal ?

Obviously number of 18-clue diagonally symmetric patterns is the same as number of 18-clue vertically symmetric patterns. i.e. 2,402,330,656 patterns (see my combinatorial calculations on Page 4 of this thread).

But I don't know how to calculate number of essentially different 18-clue diagonally symmetric patterns. Crude estimate of lower bound can be obtained by dividing 2.4 x 10^9 by 6^4 - there should be not less than 2 x 10^6 18-clue essentially different diagonally symmetric patterns.

Serg
Serg
2018 Supporter

Posts: 651
Joined: 01 June 2010
Location: Russia

### Re: Symmetric 18s

Serg wrote:Obviously ..........

Thanks for that - it is very elegant - and i hadnt appreciated the similarity - the difference between the 6^5 and 6^4 must contribute to

14 d-symmetrical 17-puzzles which probably have 80 or so different non-minimal 18-puzzles / patterns

at the time of writing i have an ever expanding clotch of over 6000 diagonally symmetrical 18-puzzles - i will update with a representative puzzle of each pattern.

C
coloin

Posts: 1790
Joined: 05 May 2005

### Re: Symmetric 18s

Well it seems that diagonally symmetric 18s are not that rare ....... well over 18000 churned out and 250 different patterns......

here is one with 6 clues in the diagonal
Code: Select all
`+---+---+---+|1..|...|...||.2.|...|.34||..5|.67|...|+---+---+---+|...|8..|.4.||..7|...|...||..6|..5|9..|+---+---+---+|...|..6|5..||.4.|2..|...||.8.|...|...|+---+---+---+`

and here is one which is symmetric in both diagonals
Code: Select all
`+---+---+---+|1..|...|2..||...|.34|...||...|..5|..4|+---+---+---+|...|...|67.||.2.|...|.8.||.94|...|...|+---+---+---+|7..|1..|...||...|78.|...||..3|...|..5|+---+---+---+`

although this pattern
Code: Select all
`+---+---+---+|...|21.|...||..7|3..|...||.58|...|...|+---+---+---+|43.|...|...||2..|...|..8||...|...|.76|+---+---+---+|...|...|25.||...|..7|3..||...|.98|...|+---+---+---+`

has been previoussly shown in this thread .......
coloin

Posts: 1790
Joined: 05 May 2005

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