## four-box sudoku

For fans of Killer Sudoku, Samurai Sudoku and other variants

### four-box sudoku

Extracting the four boxes from Windoku (NRC Sudoku) ...

www.geocities.com/r_e_s_01/misc/four-box-sudoku.png

EDITS:

www.geocities.com/r_e_s_01/misc/four-box-sudoku-solution.png is the solution.

The following will paste the puzzle directly into <SumoCue> ...

SumoCueV1
=0J0=0J1=0J1=5J1=2J0=0J2=0J2=4J2=0J0
=0J3=1J4=0J4=0J4=0J3=8J5=0J5=0J5=3J3
=0J3=0J4=2J4=0J4=0J3=0J5=1J5=0J5=0J3
=1J3=0J4=0J4=3J4=0J3=0J5=0J5=2J5=0J3
=3J0=0J1=1J1=0J1=0J0=0J2=7J2=0J2=4J0
=0J6=3J7=0J7=0J7=0J6=4J8=0J8=0J8=8J6
=0J6=0J7=8J7=0J7=0J6=0J8=5J8=0J8=0J6
=9J6=0J7=0J7=4J7=0J6=0J8=0J8=6J8=0J6
=0J0=2J1=0J1=0J1=5J0=1J2=0J2=0J2=0J0

As Smythe Dakota points out in a posting below, this puzzle is equivalent to standard sudoku, with certain rows & columns shuffled.

[2007-04-05: Updated link addresses.]
Last edited by r.e.s. on Thu Apr 05, 2007 10:19 pm, edited 5 times in total.
r.e.s.

Posts: 337
Joined: 31 August 2005

So it's just basically a partial jigsaw sudoku or a windoku without the usual 9 "regular" 3x3 nonets... Nice one!
udosuk

Posts: 2698
Joined: 17 July 2005

Thanks -- yes it can be seen as a "partial jigsaw" sudoku (i.e. with redundant borders removed). I've added a link to the solution, with colors distinguishing the nonets.
r.e.s.

Posts: 337
Joined: 31 August 2005

Lol... My mistake... It is a "full jigsaw" sudoku because there are 9 well-defined jigsaw pieces (some of which duplicate the existing rows/columns)...

Wait a minute... Is the formal definition of "jigsaw sudoku" requires each piece to be "physically connected"?
udosuk

Posts: 2698
Joined: 17 July 2005

udosuk wrote:.... Is the formal definition of "jigsaw sudoku" requires each piece to be "physically connected"? ....

Either way, I don't see how the 4-box can be regarded as a "full jigsaw" -- at least, not if you require that there be nine total pieces, each with nine cells, and that no two pieces overlap.

In the 4-box case, you could, for example, add three "pieces" by considering rows 1, 5, 9 to each be a "piece", but what do you do with what's left? For example, if you wanted to make a "piece" out of r2c1, r3c1, r4c1, r2c5, r3c5, r4c5, r2c9, r3c9, r4c9, you would be changing the original puzzle, because there would be a new requirement that each digit 1 through 9 appear in this "piece".

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

"Jigsaw" is in scare quotes because although the puzzle has nine nonets, only four of them -- the 3x3 boxes -- are composed entirely of contiguous cells. Just as in NRC Sudoku/Windoku, the five "hidden nonets" are implied by the four boxes, and don't need to have borders drawn around them or to be colored, etc.

Some additional { broken link = www.sudoku.org.uk/cgi-bin/discus/show.cgi?tpc=1289&post=5869#POST5869 } <redundant borders could be removed> in Windoku, as well as could some in standard Sudoku. (From another POV, the five "hidden nonets" in Windoku -- and in four-box sudoku -- could be "unhidden" by adding borders and/or colors to make them easier for the player to identify.)
r.e.s.

Posts: 337
Joined: 31 August 2005

Smythe Dakota wrote:Either way, I don't see how the 4-box can be regarded as a "full jigsaw" -- at least, not if you require that there be nine total pieces, each with nine cells, and that no two pieces overlap.

In the 4-box case, you could, for example, add three "pieces" by considering rows 1, 5, 9 to each be a "piece", but what do you do with what's left? For example, if you wanted to make a "piece" out of r2c1, r3c1, r4c1, r2c5, r3c5, r4c5, r2c9, r3c9, r4c9, you would be changing the original puzzle, because there would be a new requirement that each digit 1 through 9 appear in this "piece".

I guess I might as well explain to Smythe why from logical deduction there must be 5 more disconnected "jigsaw pieces" on the 4-box configuration...

r234c123456789 contain 3 whole rows, thus these 27 cells must have 3 instances of 1 to 9 each.
r234c234 and r234c678 are both defined as "boxes" i.e. they must each contain 1 instance of 1 to 9 each.
Therefore (obviously), r234c159 must have the remaining (3rd) set of 1 to 9 each...

Similarly, r678c159, r159c234 and r159c678 are other 3 implicit "jigsaw pieces"...

Finally, since we have 4 3x3 boxes (r234c234, r234c678, r678c234, r678c678) and 4 implicit jigsaw pieces (r234c159, r678c159, r159c234, r159c678), the remaining 9 cells (r159c159) must contain the remaining (9th) set of 1 to 9 each, thus forming our 9th hidden "jigsaw piece"...

Here is the grid, with each "jigsaw piece" represented by a different letter...
Code: Select all
`igggihhhieAAAeBBBeeAAAeBBBeeAAAeBBBeigggihhhifCCCfDDDffCCCfDDDffCCCfDDDfigggihhhi`
udosuk

Posts: 2698
Joined: 17 July 2005

OK, now I get it. Thanks.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

I wrote:OK, now I get it. ....

On the other hand, this makes things totally boring. The four-box Sudoku becomes mathematically equivalent to a standard nine-box Sudoku. All you have to do is re-arrange the columns, so that columns 123456789 move to 412357896 respectively, thus pushing the four boxes to the left and right edges of the grid, and then do the same thing with the rows, pushing the four boxes to the top and bottom edges of the grid. These re-arrangements do no violence to the integrity of any row, column, or box (piece), so nothing really changes. Boo.

In fact, in a standard Sudoku, you could "weaken" the original rule to say that only four, instead of all nine, boxes each contain each digit 1 to 9. Pick any four boxes you like, as long as no three are in the same chute.

Bill Smythe
Smythe Dakota

Posts: 564
Joined: 11 February 2006

Good observation... In fact this puzzle is equivalent to the following vanilla sudoku:

Code: Select all
`1..|..3|8...2.|...|.1...3|1..|..2---+---+---..5|.2.|..4.1.|3.4|.7.2..|.5.|1..---+---+---3..|..8|4...8.|...|.5...4|9..|..6`

Which surprisingly requires a flourish of turbot fishes and forcing chains to solve...
udosuk

Posts: 2698
Joined: 17 July 2005

Smythe Dakota wrote:The four-box Sudoku becomes mathematically equivalent to a standard nine-box Sudoku. All you have to do is re-arrange the columns, so that columns 123456789 move to 412357896 respectively, thus pushing the four boxes to the left and right edges of the grid, and then do the same thing with the rows, pushing the four boxes to the top and bottom edges of the grid. These re-arrangements do no violence to the integrity of any row, column, or box (piece), so nothing really changes. Boo.

Like udosuk said, a nice observation (one I should have noticed -- sorry). I'll add a note to my original posting to point this out.
r.e.s.

Posts: 337
Joined: 31 August 2005

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