## 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

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 XX X X|. . .|X X XX X X|. . .|X X X-----+-----+-----. . .|. . .|. . .. . .|. . .|. . .. . .|. . .|. . .-----+-----+-----X X X|. . .|X X XX X X|. . .|X X XX 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  TL007000000000040820090000000406000000000182000000005000000000050000006900000000000  TR000000000000000000000000000000000007000000000000000000000000000000000000000000000   M000007000005000000000092000500000000000000070000000398007000000000000000000804050  BL000030000000900000000500007600070000000063400023000000000150002002000000700090005  BR`
Attachments
butterfly.doc

m_b_metcalf
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### Re: Minimum number of clues in the centre of an X-samurai

Hi Mike,

This issue interests me - but I have to make an X-Samurai solver to check out your example. Could you post the solution please?

There are in fact no valid 36-clue Sudoku puzzles in which all the clues are in the corner boxes (B1, B3, B7 and B9) [*], so your 1-clue is definitely the lowest possible, unless you make it a SudokuP (or a SudokuW).

[*] Some guy on Stack Exchange apparently has a proof of this, see first answer in this post.

Cheers!

Mathimagics
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### Re: Minimum number of clues in the centre of an X-samurai

The single clue avoiding corner boxes has to be in the 4th or 6th line otherwise the empty c46 & r46 free swap would guarantee multiple solutions

tarek

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### Re: Minimum number of clues in the centre of an X-samurai

Nice one, tarek, I'll buy that!

Speaking of proofs (and I know, this is off-topic) I keep hearing on the news here in Oz people saying "the proof is in the pudding". This makes me so mad! The pudding's providence is the unknown quality, the proof lies in the eating! The only proof in a pudding is if it's laced with rum, and you still wouldn't know for sure until you ate the thing!

This concludes today's rant

Mathimagics
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### Re: Minimum number of clues in the centre of an X-samurai

It should be 'the proof of the pudding is in the eating'

And, sorry, I'm away from base for 10 days, so maybe someone else could kindly provide the solution you request.
Mike

m_b_metcalf
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### Re: Minimum number of clues in the centre of an X-samurai

Here is the solution, maybe not in the best format.
Solved with SAT in < 2 seconds.

Hidden Text: Show
Code: Select all
`6 9 3 1 7 8 5 2 4       2 4 7 8 5 9 6 3 18 2 1 3 4 5 9 6 7       5 6 3 7 4 1 8 2 95 7 4 9 6 2 1 3 8       8 9 1 6 2 3 5 7 44 1 6 7 3 9 2 8 5       4 2 6 9 3 7 1 8 53 5 2 6 8 4 7 9 1       9 7 5 1 8 2 4 6 37 8 9 5 2 1 6 4 3       3 1 8 4 6 5 7 9 29 4 7 8 5 6 3 1 2 5 6 4 7 8 9 2 1 4 3 5 61 3 8 2 9 7 4 5 6 9 7 8 1 3 2 5 7 6 9 4 82 6 5 4 1 3 8 7 9 1 2 3 6 5 4 3 9 8 2 1 7            2 9 8 4 3 1 5 6 7                        7 6 5 2 8 9 4 1 3                        1 4 3 7 5 6 9 2 8            2 9 1 3 5 7 6 8 4 3 1 7 2 9 5 7 3 4 8 6 17 3 5 4 6 8 9 2 1 8 4 5 3 7 6 9 8 1 5 2 46 8 4 1 9 2 5 3 7 6 9 2 8 4 1 5 2 6 9 3 75 4 8 7 3 9 2 1 6       6 8 9 4 7 5 2 1 39 6 3 2 8 1 4 7 5       1 5 7 2 6 3 4 9 81 7 2 6 4 5 3 9 8       4 2 3 8 1 9 7 5 68 5 7 9 2 6 1 4 3       9 6 4 1 5 8 3 7 24 1 9 5 7 3 8 6 2       5 3 2 6 4 7 1 8 93 2 6 8 1 4 7 5 9       7 1 8 3 9 2 6 4 5`
creint

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### Re: Minimum number of clues in the centre of an X-samurai

m_b_metcalf wrote:It should be 'the proof of the pudding is in the eating'

Thanks Mike, you've been nominated for a Sybil Fawlty award ...
Is this degree of major metaphor-mangling not, in fact, restricted to Australia?
Do I need to ramp up my rant-level?

More importantly, can this post be rescued from being totally off-topic?

Indeed! Thanks very much to creint for that solution to Mike's puzzle. (The format was convenient, BTW)

PS: Somebody has found a practical use for SAT

PPS: Responses to my SAT jibe should be directed to my complaints department which is located in the Coffee Bar

Mathimagics
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### Re: Minimum number of clues in the centre of an X-samurai

Houston, we have a problem!

If we take the puzzle for Mike's top-left corner, and fill in the bottom-right corner of that from creint's solution grid, we get:
Code: Select all
` +-------+-------+-------+ | . . . | . . . | 5 2 . | | 8 . 1 | . . . | . . . | | . . . | 9 . . | . . . | +-------+-------+-------+ | 4 1 . | . . . | . . 5 | | 3 5 . | . 8 . | . 9 . | | . . . | . . . | 6 . . | +-------+-------+-------+ | . . . | . . . | 3 1 2 | | . . . | . 9 7 | 4 5 6 | | . . . | 4 1 . | 8 7 9 | +-------+-------+-------+`

Now this should have one solution, should it not?

But it doesn't, so either the Samurai is dodgy, or the proposed solution is dodgy, or my setup is dodgy(always a possibility), or some combination of these!

[EDIT] creint's solution grid appears to be perfectly valid, so he is exonerated! I tested all 4 corner grids of his solution the same way as I did for the TL corner above, and they all seem kosher. Multiple solutions in all 4 cases though.

m_b_metcalf wrote:I'm away from base for 10 days, so maybe someone else could kindly provide the solution you request.

… and our other suspect is on the run. That looks awfully guilty to me … we need to send out a BOLO or an APB or something!

Mathimagics
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### Re: Minimum number of clues in the centre of an X-samurai

The Samurai X solution (all 5 grids have both main diagonals as constraints)
solution: Show
693178524821345967574962138416739285352684791789521643947856312138297456265413879
247859631563741829891623574426937185975182463318465792789214356132576948654398217
312564789456978132879123654298431567765289413143756928684317295921845376537692841
291357684735468921684192537548739216963281475172645398857926143419573862326814759
295734861376981524841526937689475213157263498423819756964158372532647189718392645

TL partially solved (which is a sudoku x) has indeed one unique solution

tarek

tarek

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### Re: Minimum number of clues in the centre of an X-samurai

.
Come on, guys, are you seriously suggesting that it's the JUDGE who is dodgy?

Ok, I just might have been testing using standard Sudoku, not SudokuX … and perhaps I thought "X-Samurai" simply referred to the puzzle's X-shape … and it is entirely possible, I concede, that when I retest it all with SudokuX it will all have turned out to be a trumped-up charge …

And maybe I'll just take my gavel and go home …

PS: Mike is (reluctantly) discharged, call off the dogs!

Mathimagics
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### Re: Minimum number of clues in the centre of an X-samurai

Mathimagics wrote:.
P.S. Mike is (reluctantly) discharged, call off the dogs!

Absolved, but you'll be hearing from my lawyers (Bondi Beach branch) re libel, character assassination, defamation, etc
Mike

m_b_metcalf
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### Re: Minimum number of clues in the centre of an X-samurai

Well, I built a DLX solver for SamuraiX, and it reports that Mike's puzzle is valid.

PS: good luck with that Bondi law firm, they are dodgy, just like I am

Mathimagics
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### Re: Minimum number of clues in the centre of an X-samurai

tarek wrote:The Samurai X solution (all 5 grids have both main diagonals as constraints)

tarek, thanks for confirming that result. Did you also check my claim that the puzzle is minimal (I have my doubts)?

Thanks,

Mike

m_b_metcalf
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### Re: Minimum number of clues in the centre of an X-samurai

m_b_metcalf wrote:
tarek wrote:The Samurai X solution (all 5 grids have both main diagonals as constraints)

tarek, thanks for confirming that result. Did you also check my claim that the puzzle is minimal (I have my doubts)?

I'm hopefully looking into that depending on my solver capability within the next 24h!

tarek

tarek

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### Re: Minimum number of clues in the centre of an X-samurai

m_b_metcalf wrote:tarek, thanks for confirming that result. Did you also check my claim that the puzzle is minimal (I have my doubts)?

Confirmed as minimal (with my DLX solver).

Mathimagics
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