SudokuPXD (8-clue)

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SudokuPXD (8-clue)

Postby Mathimagics » Sun Sep 23, 2018 9:29 am

.
We know that 9-clue SudokuPX and SudokuWX puzzles exist, but 8-clue puzzles seem unlikely.

hkociemba1 uncovered this grid, which is a SudokuPX grid, but also has the DP property, ie no pair of diagonally-adjacent cells have the same value, thus allowing specification of clues in SudoKakuro form.
Code: Select all
 +-------+-------+-------+
 | 8 6 4 | 3 2 7 | 5 1 9 |
 | 3 2 7 | 5 9 1 | 8 4 6 |
 | 5 1 9 | 8 4 6 | 3 2 7 |
 +-------+-------+-------+
 | 2 4 6 | 1 7 5 | 9 8 3 |
 | 1 5 3 | 9 6 8 | 2 7 4 |
 | 9 7 8 | 2 3 4 | 6 5 1 |
 +-------+-------+-------+
 | 4 3 1 | 6 5 2 | 7 9 8 |
 | 6 8 5 | 7 1 9 | 4 3 2 |
 | 7 9 2 | 4 8 3 | 1 6 5 |
 +-------+-------+-------+


Remarkably, HK found a SudoKakuroPX puzzle on this grid with just 6 clues:
Code: Select all
 +----------+----------+----------+
 |  .  .  . |  .  .  . |  .  .  . |
 |  .  .  . |  .  .  . |  .  .  . |
 |  .  . 22 |  . 30  . |  .  .  . |
 +----------+----------+----------+
 |  .  .  . |  .  .  . |  .  .  . |
 |  .  . 28 |  .  .  . | 30  .  . |
 |  .  .  . |  .  .  . |  .  .  . |
 +----------+----------+----------+
 |  .  . 22 |  . 12  . |  .  .  . |
 |  .  .  . |  .  .  . |  .  .  . |
 |  .  .  . |  .  .  . |  .  .  . |
 +----------+----------+----------+


I took this grid, and tested it with a HS engine configured for SudokuPXD (ie SudokuPX + DP), and although I found no 8-clue puzzle, I did find a large number (2498) of 9-clue puzzles.

So I morph-tested all of these via {+1, -2} and found an 8-clue puzzle:

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