The New Sudoku Players' Forum

[m_b_metcalf]

Congratulations to all for having found these sublime Troika so quickly!
May I suggest another challenge : find 4 valid and minimal puzzles forming a grid partition.

This is what my first shot looked like:

Code: Select all

.1......62....9.......4.7.....6...8...5.8.2...6...1.....8.3.....4.7....13......2. 20
5.4.7............8..3....12.2.......1......6.8.....5.37..1.2...6....5.3....8.4... 21
......39.....1.4..9....6...4.....1...3.9.7......42..7.......64...2.......51.6...7 21
...2.8....763...5..8.5.......7.53..9........4..9.......9......5....9.8........9.. 19 9916331

.1......23....8.......5.6.....1...9...5.2.8...9...6.....6.8.....3.2....14......3. 20
..74......5.61......9.....328...5...1..9.3.........12..7......4.......6....5..9.8 21
......5........4.98..7.2.....4...3.6.6......7...84....9..3.1...5.8.9.....2...7... 21
6...39.8...2....7..4.....1.....7...........4.7.3.....5......25......47....1.6.... 19 358381

.1...2..37..4...9...6.......4...5..1.........6..7...8.......4...5...1..29..6...7. 20
....7.6......5...859.1.....3..8.....1....9..6.2...39...6.........7.9.........43.5 21
..49......3...61.......82.7..9.........24.53...5.1....2....7..9.......6..8..2.... 21
8......5...2..........3..4.....6.72..78..............4..358..1.4..3..8....1...... 19 1855212

.1...2..37..5...8...4.......3...1..8.........9..7...5.......2...4...3..15..2...7. 20
.......4..2..3.1...8...7..5..7...4...5...9..7..1.4.3...7...5.39.............1.8.. 21
..5.9.7.......4......16.92.....5....4.8...61..6...8.......8.......97.5....3.....4 21
6..8.......9.....63........2..6...9....32............21.64.....8.2....6..9...6... 19 49572586

.1......62....9.......4.7.....6...8...5.8.2...6...1.....8.3.....4.7....13......2. 20
5.4.7............8..3....12.2.......1......6.8.....5.37..1.2...6....5.3....8.4... 21
...2..39..7..1.....8...6.....7...1.......7..4..94............45..2.9.....51.6...7 21
.....8.....63..45.9..5.....4...53..9.3.9.........2..7..9....6........8........9.. 19 1310972

The large number following the 19 is p4's number of solutions!

Mike

[marek stefanik]

Well, I don't think there's a way to cheese this one with isomorphism.
Good luck, everyone!

Marek

[Mathimagics]

The smallest minimal puzzle in a 4-way partition is 19 or less, but we know that most grids have a relatively small supply of 19's.

I would suggest that it might be better to start with grids that have a 17C, this implies 18's and then 19's, which might then be available in sufficient numbers to merit looking at the larger sets in the partition ...

[m_b_metcalf]

Second attempt shows a big improvement:

Code: Select all

....2.6...1..........9.....23...6.......8..9.5......4....4.13....97.....8........ 17
.578....4...6......8...7.....4.9.8.76............72....25.....91......2.....5..6. 21
9......1......597...6.1..32..........7.5.4......3....6....6.........84.5.43..97.. 22
.....3...3.2.4...84.....5.....1...5...1...2.3.98...1..7......8..6..3.......2....1 21   7044

...2...9..1.....3..5...8...2..9........4..5........1..4....6.2.....3.7......1.... 17
..4.53.......7....9.....26..45..1....3.........73.........9...8..6....1..2.7.4.5. 21
76......1.....94.5..3.4...........73....67.8.68...5......5......9.8.2...3.......6 22
......8..8.26........1....7....8.6..1.9.....2....2..49.71...3..5.......4..8...9.. 21   8910


[Edited with better results.]

[coloin]

Code: Select all

589732614621854379743916825835129746417568293296347158968471532372695481154283967  solution
....3.........4.........825..5.2.7......68....9....1.8.....1.3..72.......5.2...6.  C20
5.9.........8...79..3..6...........64.7......2...47.5.........2...69...1..4...9..  C20
.....2.1..2..5.....4..1....8..1...........29...63.......8.7.5..3....54..1...8....  C20
.8.7..6.46.1...3..7..9......3...9.4..1.5....3.........96.4............8......3..7  C21  15053 sol

[m_b_metcalf]

Code: Select all

....12.5.7.....4......8.......6..3...21.......7..........7....24..3.....3......8. 17
..64......1......9.4.9...1......5.7.6...73.4.5.....2.8.....8.3.....51..6...2..... 22
9..........853....2....76..8...2............5...194....6..4.5........79..57...1.. 21
.3....8.7.....6.2...5.....3.94.....1...8..9....3....6.1.9.......82..........69..4 21   2675 solutions

[Mathimagics]

Ok, now you're getting somewhere. Good luck with the hunt!

[m_b_metcalf]

Code: Select all

.46.5.2.....8......5.......7..1.8...8..7...........4...2..4....1......3.........7 17
...3.7....7..9..4.........6....6.95..95.2......2.....8.....381...9......46...1... 21
.......8.2.1......3...1.79.........2......1.3...5.9.7...76......8.2.56.4..3.8.... 22
9.......1.....63.5..84.2....34...........4.6.61..3....5.......9....7.......9..52. 21    (only) 516 solutions

Gasp!!!

[Mathimagics]

Good!

[m_b_metcalf]

Well, I don't think there's a way to cheese this one with isomorphism.

Well, I'm tantalisingly close with what Marek describes as 'a cheese' (in which language is that an understood expression?). My trick is to make p1 an x-sudoku:

Code: Select all

.................1..2........3...........4....25...67.81.......4...........5..... 12   <----  p1 is an x-sudoku
5.4........86.5.......39..76....7..2.......39...18.........64....78.....23.....9. 22
.9.71..6...........6.....5.....5.1..78.2.....9.......4...3......5..92.....6...7.8 20
.....82.337..2.94.1..4..8...4.9...8...1.6.5.......3.....9.7..25......316....41... 27   2 solutions


Mike

[coloin]

An improvement for the purists....

Code: Select all

589732614621854379743916825835129746417568293296347158968471532372695481154283967  solution
....3.........4.........825..5.2.7......68....9....1.8.....1.3..72.......5.2...6.  C20
5.9.........8...79..3..6...........64.7......2...47.5.........2...69...1..4...9..  C20
.8.7..6........3..74.91.....3...9.4..1.5...9...6......9..4..........5.8......3...  C20
.....2.14621.5.............8..1...........2.3...3......68.7.5..3.....4..1...8...7  C21 8289 sols

Code: Select all

589732614621854379743916825835129746417568293296347158968471532372695481154283967  solutiom
.......1..2..5.....4.......8..1.9.........2.....3......6..7.5.23.....4..1...8....  C17
.8.7..6.4......3..7..91.....3.....4..1.5...9...6......9..4..........5.8......3...  C19
....3.........4.........825..5.2.7......68....9....1.8.....1.3..72.......5.2...6.  C20
5.9..2...6.18...79..3..6...........64.7.....32...47.5...8.........69...1..4...9.7  C25   2 solutions

[Mathimagics]

coloin: Fantastic effort, well done!!

The Magic Quadroika can't be far away ...

[coloin]

It has to be there !
advancing on the same grid

Code: Select all

589732614621854379743916825835129746417568293296347158968471532372695481154283967
.8.7..6.4......3..7..91.....3.....4..1.5...9...6......9..4..........5.8......3...  C19
....3.........4.........825..5.2.7......68....9....1.8.....1.3..72.......5.2...6.  C20
5.9.........8...79..3..6...........64.7......2...47.5.........2...69...1..4...9..  C20
.....2.1.621.5.....4.......8..1.9.........2.3...3......68.7.5..3.....4..1...8...7  C22   365 solutions

[m_b_metcalf]

It has to be there !

I think so too:

Code: Select all

.......1..2..5.....4.......8..1.9.........2.....3......6..7.5.23.....4..1...8....  17
.8.7..6.4......3..7..91.....3.....4..1.5...9...6......9..4..........5.8......3...  19
5.9.3.......8...79..3..6...........64...68...2....715........3...2.9...1...2..9..  22
.....2...6.1..4.........825..5.2.7....7.....3.9..4...8..8..1....7.6......54....67  23, 2 solutions

M

[m_b_metcalf]

Almost there:

Code: Select all

.......1..2..5.....4.......8..1.9.........2.....3......6..7.5.23.....4..1...8.... 17   1.5/1.2/1.2
.8.7..6.4......3..7..91.....3.....4..1.5...9...6......9..4..........5.8......3... 19   2.0/1.2/1.2
..9......6.18.4.........825....2.7.64.7..........47.5...8..1......6......5....9.7 22   7.1/1.2/1.2
5...32..........79..3..6.....5..........68..329....1.8.......3..72.9...1..42...6. 23   1.5/1.2/1.2, 2 clues redundant

Mike

[Edit: for the record, the two redundant clues are r6c7 and r7c8. removing them raises p4's rating to 2.0.]