Minimum number of clues in the centre of an X-samurai

For fans of Killer Sudoku, Samurai Sudoku and other variants

Minimum number of clues in the centre of an X-samurai

Postby m_b_metcalf » Mon Jun 11, 2018 9:41 am

Elsewhere it was written:
Hajime wrote:I wonder, because the middle grid is completely empty, that this grid can never be a normal/vanilla Sudoku.
It will never lead to a solution. That's why the X-samurai s posted by m_b_metcalf also have some givens in the middle grid.


Sometime back there was a discussion on what the minimum number of clues in the central puzzle of a standard samurai is (see here). It resolved to finding the minimum number of clues in the non-corner boxes of a standard 9x9 puzzle when the corner boxes are full:
Code: Select all
X X X|. . .|X X X
X X X|. . .|X X X
X X X|. . .|X X X
-----+-----+-----
. . .|. . .|. . .
. . .|. . .|. . .
. . .|. . .|. . .
-----+-----+-----
X X X|. . .|X X X
X X X|. . .|X X X
X X X|. . .|X X X


The least number that could be found was five.

This question arises too for X-samurai. A quick test threw up the answer one, as shown here:

Code: Select all
 8 1 3 . . . 4 7 9
 7 5 9 . . . 2 1 6
 6 4 2 . . . 5 3 8
 . 6 . . . . . . .
 . . . . . . . . .
 . . . . . . . . .
 3 9 6 . . . 1 8 2
 1 8 5 . . . 9 4 7
 2 7 4 . . . 6 5 3


So, is it then possible to have a complete x-samurai with just a single clue in the central puzzle? I surprisingly got the answer on a first test run:

Code: Select all
 . . . . . . 5 2 .       . . 7 . . . . . .
 8 . 1 . . . . . .       . . . . 4 . 8 2 .
 . . . 9 . . . . .       . 9 . . . . . . .
 4 1 . . . . . . 5       4 . 6 . . . . . .
 3 5 . . 8 . . 9 .       . . . 1 8 2 . . .
 . . . . . . 6 . .       . . . . . 5 . . .
 . . . . . . . . . . . . . . . . . . . 5 .
 . . . . 9 7 . . . . . . . . . . . 6 9 . .
 . . . 4 1 . . . . . . . . . . . . . . . .
             . . . . . . . . 7 
             . . . . . . . . . 
             . . . . . . . . . 
 . . . . . 7 . . . . . . . . . . 3 . . . .
 . . 5 . . . . . . . . . . . . 9 . . . . .
 . . . . 9 2 . . . . . . . . . 5 . . . . 7
 5 . . . . . . . .       6 . . . 7 . . . .
 . . . . . . . 7 .       . . . . 6 3 4 . .
 . . . . . . 3 9 8       . 2 3 . . . . . .
 . . 7 . . . . . .       . . . 1 5 . . . 2
 . . . . . . . . .       . . 2 . . . . . .
 . . . 8 . 4 . 5 .       7 . . . 9 . . . 5   Number of clues 63; hard; minimal; only 1 clue in central puzzle.


Regards,

Mike Metcalf

Code: Select all
000000520801000000000900000410000005350080090000000600000000000000097000000410000  TL
007000000000040820090000000406000000000182000000005000000000050000006900000000000  TR
000000000000000000000000000000000007000000000000000000000000000000000000000000000   M
000007000005000000000092000500000000000000070000000398007000000000000000000804050  BL
000030000000900000000500007600070000000063400023000000000150002002000000700090005  BR
Attachments
butterfly.doc
(67.5 KiB) Downloaded 13 times
User avatar
m_b_metcalf
2017 Supporter
 
Posts: 9395
Joined: 15 May 2006
Location: Berlin

Return to Sudoku variants