## Minimum number of clues in middle Su-Doku of a Samurai?

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Minimum number of clues in middle Su-Doku of a Samurai?

The subject explains it all. The intersecting boxes must be empty. The upper bound is 8 because I know that there was a (very old) Times Samurai Su-Doku puzzle like that.
Once upon a time I was a teenager who was active on here 2007-2011
ocean and eleven should have paired up to make a sudoku-solving duo called Ocean's Eleven
Pat wrote:prenyast
999_Springs

Posts: 489
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: Minimum number of clues in middle Su-Doku of a Samurai?

999_Springs wrote:The subject explains it all. The intersecting boxes must be empty. The upper bound is 8 because I know that there was a (very old) Times Samurai Su-Doku puzzle like that.

I don't know, but here's one with only seven clues. None of the five component puzzles can be solved independently.
Code: Select all
` . . . . . . . . .       . . . . . . . . .  . . . . . . . . 8       . . . . . . . 2 5  . . . . 7 . 1 3 6       . . 7 6 . . 8 . .  . . 2 . . . . .         . . 4 . . 8 . . .   . . 9 . . . 6 8 7       . 8 6 . . 9 7 . 3    . . 8 . 6 7 . 5 2       . 3 . . . . 5 9 .   . 2 . . 4 3                   . 7 . . 1 .   . 1 . . . 9                   9 . 6 4 . .  7 . 4 8 . 6                   4 2   . 8 9                     7   9                   1           4                                 5   2       . . . . . .                   . . . . . .  . . . . . 9         7         8 7 . . 6 .    . 6 . . 5 4                   3 1 . . 2 4  . . . . . . . . .       . . . . . . . 3 . . . 9 . 8 2 . . 1       . 2 . 7 8 9 .   1    . . 2 . . 6 . 7 5       . . . . . . . . . . . 1 . . 7 . 6 3       . 4 9 . . . . 1 2 . 9 7 . . 1 . 4 2       . 7 . . . . . . . . . 3 4 2 .   1 7       . 6 3 9 . 5 . 7 . `

I'd be interested to know how long it takes you to solve.

Regards,

Mike Metcalf

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

Here is a Samurai with 6 clues in the center grid:

www.sudokuvault.com/images/samurai-6-centerclues.png

720000000050000009000600800100006704200300600007900000000008000040103000090005000
003001000000700840050020000906030200030000400802004060000000008000070053000000100
000000000000004000000050000000800000000001400600000000000000000000000000000000000
003040000010300000907260000072030800600090074000006030000950007000001900004080000
000000007000008032000000608500000000001000326300002000600034050000020000010005803

Solution
728549361356812479914637852135286794289374615467951238571468923642193587893725146
673841592129753846458629731946137285735286419812594367564312978291478653387965124
923187564587634291146259387271843659895761423634925718759412836462398175318576942
263148759518379462947265318172534896635892174489716235821953647756421983394687521
836219547175468932942573618527386194481957326369142785698734251753821469214695873

It is ... erm ... not easy to solve.

Ruud
Ruud

Posts: 664
Joined: 28 October 2005

With full symmetry, 8 clues in the center grid, all in a single box:

{
note by Moderator(Pat) —
new image location thanks to enxio27 }

200314000001020007080960031103000020925006300700030069000080000002601000057003000
000529008500080100420073050050000804004700312280040005000030000000804200000200940
000000000000000000000000000000521000000603000000947000000000000000000000000000000
034009000005204000000060000500010069128006500906000070060890057009050002200641000
000200750000609300000040000140090007005800123030000905670082030200030400000761008
Ruud

Posts: 664
Joined: 28 October 2005

Ruud wrote:Here is a Samurai with 6 clues in the center grid:

[snip]

It is ... erm ... not easy to solve.

Right. In fact, as far as I can tell, it needs a global technique. The desperate solver can be helped by the single hint below (not in the middle puzzle).

Regards,

Mike Metcalf

Hint:

Top Right r7,c7 = 9
Last edited by m_b_metcalf on Tue Sep 16, 2008 7:45 pm, edited 1 time in total.

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

Ruud wrote:Here is a Samurai with 6 clues in the center grid:

Ruud wrote:With full symmetry, 8 clues in the center grid, all in a single box:

And here's another, not symmetric, but with only five central clues, and all in the same box.

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

Code: Select all
`000000000000000001000123405010005007000000020008074306030048000040607000901000000000000000000100084009060200015002009007000040030004502000006000000023075000805100000000000000000000000000000000370000000006000000024000000000000000000000000000000000000000050007000089140000010006090000790030000000000001863970073024850000000200000100709000280050000000003100046007000000000089310004040053000093004002000000000`

Regards,

Mike Metcalf

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

### Re: Minimum number of clues in middle Su-Doku of a Samurai?

999_Springs wrote:The subject explains it all. The intersecting boxes must be empty. The upper bound is 8 because I know that there was a (very old) Times Samurai Su-Doku puzzle like that.

I postulate that the lower bound is at least two, and they must be on a diagonal of the central box. Reasoning:

In a limiting case, sufficient information can flow between the five sub-puzzles for the four corner ones to be solved completely. What is left is a central puzzle with its corner boxes full. We know that one other box must contain some clues, namely the central one (otherwise there are pairs of empty rows/columns), and that it must contain at least two clues positioned such that there are not two rows (or columns) intersecting it that are empty.

The lower limit is certainly higher that this, but is at least two. My attempt to go below five was unsuccessful.

Regards,

Mike Metcalf

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

So it boils down to the question:

What is the minimum number of clues for this pattern, where each X contains a clue:

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`

I tweaked my generator to search for it, but the best it could find is another 5.
Ruud

Posts: 664
Joined: 28 October 2005

Ruud wrote:So it boils down to the question:

What is the minimum number of clues for this pattern, where each X contains a clue:

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`

I tweaked my generator to search for it, but the best it could find is another 5.

Perhaps. One can imagine that, with only two initial clues, Y, an intermediate solution is reached:
Code: Select all
`X X X|. . .|X X XX X X|. X .|X X XX X X|. . .|X X X-----+-----+-----. . .|. . Y|. . .. . .|. . .|. X .. . .|Y . .|. . .-----+-----+-----X X X|. . .|X X XX X X|. X .|X X XX X X|. . .|X X X`

where the additional three clues needed for a solution in the central puzzle are provided by the corners. To be investigated.

Regards,

Mike Metcalf

P.S. Did you know that, in your symmetric 8-clue puzzle above, clue '2' is redundant?

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

m_b_metcalf wrote:To be investigated.

After some further attempts to go below five, I'm giving up. I'm prepared to believe that four is possible, but am unwilling to search further for it.

Regards,

Mike Metcalf

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

After some further attempts to go below five, I'm giving up.

I'll have my generator try for one more night before I give up.

Did you know that, in your symmetric 8-clue puzzle above, clue '2' is redundant?

Yes, but removing it would degrade the symmetry.
Ruud

Posts: 664
Joined: 28 October 2005

### Re: Minimum number of clues in middle Su-Doku of a Samurai?

It turns out that the 5-clue central box samurai that I posted above is not minimal (something I was not actually checking for back then). Here's a minimal version:

Regards,

Mike Metcalf

Code: Select all
`.................1...1234.5.1...5..7.......2...8.743.6.3..48....4.6.7...9.1......  TL............1...84..9...2....5..2..9..7....4..3...45.2.....6.......23.7....8.51..  TR..............................37.........6.......24..............................  M..........5...7.....914.....1...6.9....79..3............1863....73..485.......2..  BL...1..7.9...28..5.........31...46..7..........8931...4.4..53....93..4..2.........  BR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . 1 . . . 8 4 . . . 1 2 3 4 . 5 . . . . . 9 . . . 2 . . . 1 . . . 5 . . 7 . . . . . 5 . . 2 . . 9 . . . . . . . 2 . . . . . . 7 . . . . 4 . . . 8 . 7 4 3 . 6 . . . . 3 . . . 4 5 . 2 . 3 . . 4 8 . . . . . . . . . . . 6 . . . . 4 . 6 . 7 . . . . . . . . . . 2 3 . 7 . 9 . 1 . . . . . . . . . . . . 8 . 5 1 . . . . . . . . . . . 3 7 . . . . . . . . . . . . . . . . . . . . . 6 . . . . . . . . . . . . . . . . . . . 2 4 . . . . . . . . . . . . . . . . . . . . . . . . 1 . . 7 . 9 . 5 . . . 7 . . . . . . . . . 2 8 . . 5 . . . 9 1 4 . . . . . . . . . . . . . . . 3 . 1 . . . 6 . 9 . . . . 1 . . . 4 6 . . 7 . . . 7 9 . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 8 9 3 1 . . . 4 . . 1 8 6 3 . . . . . . . 4 . . 5 3 . . . . 7 3 . . 4 8 5 . . . . . 9 3 . . 4 . . 2 . . . . . . 2 . . . . . . . . . . . . . .   5 clues in central puzzle, minimal`

m_b_metcalf
2017 Supporter

Posts: 10762
Joined: 15 May 2006
Location: Berlin

### Re: Minimum number of clues in middle Su-Doku of a Samurai?

.
To establish whether 4 clues in the central box can produce a valid puzzle (together with all 4 corner boxes), I am searching the ED grid catalog (by band) prove or disprove this notion once and for all.

Testing of sample bands for the 5 clue case shows 1% to 2% of grids have this property.

Testing of the 4 clue case is underway, every indication so far suggests it can't be done. Completion will take a few days … only one core is available.

Mathimagics
2017 Supporter

Posts: 1485
Joined: 27 May 2015
Location: Canberra

### Re: Minimum number of clues in middle Su-Doku of a Samurai?

Testing this in real life is easy … I did mention a similar way in the Kazaguruma empty middle grid thread.

On a Samurai:
1. Obtain a database of Samurai solution grids
2. Strip all clues from central grid except for central box for each solution grid.
3. Solvable Stripped down grids are kept & the rest discarded.
4. For each solvable stripped down grid: remove all central box clues Except 2 clues (Many permutations) & solve
5. Step 4 can be done keeping 3, 4 or 5 clues

On a Vanilla sudoku:
1. Obtaina database of slution grids
2. strip clues from Box 2,4,6,8 (so boxes 1,3,5,7,9 are full)
3. Solvable Stripped down grids are kept & the rest discarded.
4. For each solvable stripped down grid: remove all central box clues Except 2 clues (Many permutations) & solve
5. Step 4 can be done keeping 3, 4 or 5 clues

Tarek

tarek

Posts: 3531
Joined: 05 January 2006

### Re: Minimum number of clues in middle Su-Doku of a Samurai?

About one-crossing-free pattern, previous discussions have been done here.
Any subset of any pattern listed shall not make a valid puzzle. Number below shows the corresponding pattern
Code: Select all
`o  oo oo  o   ooo  o  oo  oo o  oo  o    o   o  28 28 28  40  8`

For patterns with givens in multiple boxes, we have to check one-by-one, but it is not to hard to prove that 4 is impossible(if so).
qiuyanzhe

Posts: 86
Joined: 21 August 2017
Location: China

Next