## Zerodoku

For fans of all other kinds of logic puzzles

### Zerodoku

A zerodoku consists of a grid containing n x n cells. For the moment n varies from 7 to 12.
Each cell contains a number ranging from zero to 9.

The groups (or sections) consist of the rows and columns. Boxes are not involved. (Maybe I'll include the diagonals but now this is not yet the case.)
Within a group, zero can occur at most twice.
The cells containing zero are breaking the groups into pieces called segments. The cells containing zero are not part of the segments.

Within each segment:
- No digit can occur more than once
- The digits form a series - possibly permuted.

For example, if a segment would consist of 3 cells, it could contain 2|1|3 or 6|8|7, but 2|5|4 would be invalid.
Example of a completed 12x12 zerodoku:
Code: Select all
`8;7;3;2;9;5;4;1;6;0;6;07;6;4;5;0;4;3;0;5;2;3;46;5;2;3;4;0;5;6;7;0;7;89;8;0;7;5;6;0;5;8;6;4;75;0;7;8;6;5;3;4;0;4;5;30;5;4;6;1;3;2;8;7;9;0;51;0;3;1;2;4;0;9;5;8;7;60;6;5;4;3;0;7;3;4;5;6;27;5;6;0;0;4;5;7;6;3;8;96;8;2;5;4;7;3;1;0;2;0;19;7;8;0;3;5;6;2;7;1;4;08;9;0;3;5;6;4;0;8;7;5;6`

Here's a first puzzle:
Code: Select all
`0; ;1;2; ;0;3 ;1; ;0;0; ; 0; ; ; ; ;6;0 ; ;0;0;4; ; 5;0;0; ; ;7;6 ; ;8; ; ; ;0 ;0; ;7; ;0;8`
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

Code: Select all
`1;0; ;3; ;3; ; ;60; ; ;0; ;4; ; ;  ;3; ; ; ; ; ; ; 6; ;0; ; ;6; ;5;0 ;0;8;4;9; ; ;3;  ;7; ; ; ;0;6;0; 8; ;0; ;7; ;0;6; 6; ;4; ; ;0;8;0;2 ; ; ; ;0;9;0;4; `

This 9x9 zerodoku should have one unique solution, but I'm not sure about any logical solution path. I'm going to find out.
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

Hi evert & well done,

May I suggest that some clear wording describes that a "Segment" in the current form has to be in a single line. I had to look at the example & read the rules more that once to deduct that .... Well, it might be only me

I can't help but think of it at the start as islands of numbers in a sea of 0s.

Well done,

Tarek

tarek

Posts: 3761
Joined: 05 January 2006

### Re: Zerodoku

A segment is a part of a single line - or a single column - separated by zeroes (or the edge of the line/column).
Just as you say

Did you solve the puzzle from my 2nd post? I did and it wasn't that hard.
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

evert wrote:Did you solve the puzzle from my 2nd post? I did and it wasn't that hard.
No I haven't, I'm sure to give it a go when time permits. We need a savious here ... Where is a hatman when you need one ???!!!

tarek

Posts: 3761
Joined: 05 January 2006

### Re: Zerodoku

Hi, evert!
evert wrote:
Code: Select all
`1;0; ;3; ;3; ; ;60; ; ;0; ;4; ; ;  ;3; ; ; ; ; ; ; 6; ;0; ; ;6; ;5;0 ;0;8;4;9; ; ;3;  ;7; ; ; ;0;6;0; 8; ;0; ;7; ;0;6; 6; ;4; ; ;0;8;0;2 ; ; ; ;0;9;0;4; `

This 9x9 zerodoku should have one unique solution, but I'm not sure about any logical solution path. I'm going to find out.

I've manually solve this puzzle. If I was not wrong it has unique solution.
Code: Select all
`1 0 2 3 0 3 5 4 60 4 3 0 6 4 3 2 55 3 4 0 3 2 4 1 06 5 0 3 4 6 2 5 00 0 8 4 9 5 7 3 65 7 9 6 8 0 6 0 48 9 0 5 7 6 0 6 56 8 4 7 5 0 8 0 27 6 5 8 0 9 0 4 3`

It took me 1-1.5 hours to solve it (I am not experienced sudoku solver). I'd say this puzzle has moderate level. (The most complicated task was row r3 finding.)
It was interesting to solve it, but zerodoku is too similar to sudoku. Hidato puzzle (or tarek's "King's tour" modification) is more interesting to my mind.

Serg
Serg
2018 Supporter

Posts: 854
Joined: 01 June 2010
Location: Russia

### Re: Zerodoku

Hi Serg,

Your solution matches mine. Thanks for trying the puzzle!
I've sharpened my computer solver and now I think the following puzzles should be more interesting.
I myself will try them in train coming days.
Code: Select all
`8;3;7;2;9; ; ;4; ; ;3;  ; ;6; ;5; ; ;6; ;5; ;  ; ;5; ;2; ; ;2; ;4; ;29; ; ; ; ;3;4; ;1; ; ;  ;5;4;7;6; ;1; ; ; ; ; 3; ; ;4; ;0; ;0; ;3; ; 0;4; ;5; ; ;3; ; ; ;7;  ; ; ; ; ; ; ;9; ;6; ;8 ;0; ;4; ; ;4; ; ; ; ;  ;9; ; ;0;6; ; ; ; ;6;9 ; ;7; ; ; ; ;6; ; ; ;  ; ; ; ;0; ; ; ;0; ; ;13;5; ; ;7; ; ;3; ;3; ; 0; ; ; ; ; ; ; ; ; ;2;1 ; ; ; ;7; ; ;9; ; ; ; 4;0; ;7;9;6; ;4; ; ;6; 9; ; ; ; ; ; ; ; ;0; ;  ;7;0; ;6;8; ;7;9; ; ;8 ; ;0;2; ;9;4; ;3; ;7;  ;9; ;0; ;3;2; ; ; ; ;85; ;3; ;1;6; ; ; ; ; ;  ; ; ; ; ;4; ;0;0;6; ;  ;4; ; ; ; ;0; ; ;8; ;65; ; ; ;4; ; ;3;8; ;0;9 ; ;7;9; ;2; ; ;0; ; ;  ; ;8;0; ; ;5; ;2; ; ;  ;7; ;0; ; ;2;0;5; ; ;33;0; ; ;2; ; ; ; ; ;5;  ;0; ; ; ;2; ;7; ;0; ;  ; ;7;4;1; ; ; ; ; ; ;8 ; ; ;3; ; ;9;0;3;0;8;  ; ; ;8; ;0; ; ; ; ; ;  ;4; ; ;0; ;0; ; ; ;7; 5; ; ;2;7; ; ; ;4;3; ;4 ;5; ; ; ;5; ;4;2; ; ; 0; ; ;7;4; ;0;1; ; ;2; 1; ; ; ;2; ; ;4; ; ; ;  ;2; ; ; ; ;2; ; ;4; ;  ;1; ;7; ;1; ; ; ; ;0;6 ; ;0; ; ;2; ; ; ;0;7; 1; ; ;9; ; ;5;6;7; ; ;90; ;3;4; ; ;1;2;4; ;3;  ;7; ; ;6; ; ;5; ; ; ;  ;4;9;3; ; ; ; ;0; ;5;7 ; ;7; ;0;0; ;6; ;5; ;  ; ; ; ;5; ; ; ;3; ;8;  ;3; ; ; ;1; ; ; ;6; ;0 ;4;5; ;4; ;5;4;6; ; ;3`

Solutions: Show
Code: Select all
`8;3;7;2;9;5;6;4;1;0;3;07;5;6;0;5;4;7;6;0;5;4;36;4;5;3;2;1;0;2;3;4;0;29;0;0;6;8;3;4;5;1;2;9;70;5;4;7;6;2;1;3;9;8;0;13;2;5;4;1;0;2;0;5;3;2;40;4;0;5;4;9;3;8;2;1;7;65;3;4;0;7;8;0;9;7;6;5;86;0;5;4;3;0;4;7;6;9;8;57;9;8;6;0;6;5;0;8;7;6;99;8;7;3;1;5;2;6;4;0;4;08;7;6;5;0;4;3;5;0;2;3;13;5;6;4;7;0;2;3;0;3;4;20;6;4;5;0;7;3;5;6;4;2;13;0;5;1;7;8;4;9;2;6;3;04;0;8;7;9;6;0;4;8;5;6;79;5;7;6;8;0;3;2;4;0;5;66;7;0;3;6;8;5;7;9;4;0;82;3;0;2;5;9;4;8;3;6;7;08;9;7;0;0;3;2;6;7;5;4;85;2;3;4;1;6;7;0;5;0;6;57;8;5;3;2;4;6;0;0;6;8;70;4;6;5;3;7;0;4;7;8;5;65;6;4;0;4;5;6;3;8;7;0;96;5;7;9;8;2;4;3;0;5;4;07;6;8;0;6;0;5;4;2;6;3;75;7;6;0;4;3;2;0;5;4;2;33;0;0;9;2;6;7;8;4;3;5;14;0;4;5;3;2;6;7;1;0;1;20;2;7;4;1;5;8;9;6;3;0;86;8;5;3;7;4;9;0;3;0;8;94;7;6;8;5;0;3;2;0;7;6;52;4;3;1;0;7;0;3;5;4;7;65;6;0;2;7;8;6;5;4;3;0;43;5;4;6;0;5;7;4;2;6;3;00;3;5;7;4;6;0;1;3;5;2;41;3;4;6;2;5;0;4;0;3;2;40;2;3;0;5;8;2;1;6;4;3;72;1;0;7;6;1;4;3;2;5;0;63;4;0;6;4;2;3;7;5;0;7;81;0;2;9;3;4;5;6;7;8;0;90;5;3;4;0;6;1;2;4;7;3;54;7;8;5;6;3;0;5;3;4;6;08;4;9;3;5;7;6;0;0;6;5;79;6;7;8;0;0;3;6;4;5;7;87;0;6;1;5;2;4;7;3;0;8;95;3;4;2;6;1;7;0;5;6;4;06;4;5;0;4;0;5;4;6;7;2;3`
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

Hi, evert!
I solved your first 12x12 zerodoku puzzle. (My solution coincides with your solution.) Thanks, I enjoyed by solving it. Overall strategy is obvious - try to determine first as much zeroes as possible, then analize digits in "pensilmark" style, for example. I consider this puzzle has moderate complexity level, because it can be solved by rather obvious logical considerations and small amount of "Try and Error" method used locally (to 4-5 neighbour cells only). Is it possible to construct really "tough" or even "diabolic" zerodoku?

The next open question - optimal puzzle's size. I don't feel any principal difference beetween 9x9 and 12x12 puzzles you published. But I solved 12x12 puzzle approx. 4 times longer than 9x9 puzzle (though it contains 2 times more cells only).

In any way you invented interesting puzzle. Congratulations!

Serg
Serg
2018 Supporter

Posts: 854
Joined: 01 June 2010
Location: Russia

### Re: Zerodoku

Serg wrote:Is it possible to construct really "tough" or even "diabolic" zerodoku?

I can't answer that on short terms

What I found quite tough was:
Code: Select all
` ; ; ;4; ;0;0; ; ; ; ;  ;6;0; ;8;5; ;6;7;9; ;  ; ; ;1; ;0; ; ;4; ;5;  ;8; ; ; ; ;1; ; ; ;0;5 ;7; ;2; ; ;5; ; ;9; ;  ;6;4; ; ; ; ; ;7; ; ;00;3; ; ;6; ; ; ;0;5; ;76; ; ;8; ; ; ;4; ;8;3; 0; ; ;6; ; ;0;3; ; ;5; 3; ; ;3; ;1; ;0;8; ; ;  ; ; ;4; ; ;1; ;4; ; ;  ;2;0; ; ;7; ;2; ;8; ;7`

After some obious steps - and some fish :

Code: Select all
` ;5; ;4;7;0;0; ; ; ;4; 7;6;0;0;8;5;2;6;7;9;3;44;0;3;1;2;0; ; ;4; ;5; 2;8; ;3; ; ;1;7; ;0;0;5 ;7; ;2; ; ;5; ;0;9;7;85;6;4;0; ; ; ; ;7;4;8;00;3; ;7;6; ;4; ;0;5;6;76;5;7;8;0;6;7;4;5;8;3;00;4;2;6; ; ;0;3;7;6;5;43;0;5;3;4;1;2;0;8;7;9;62;3;1;4;0;2;1;3;4;0;4;51;2;0;5;4;7;3;2;6;8;0;7`

Really couldn't tell how to proceed from here.
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

Hi, evert!
This puzzle is really tough. I came to your final position, but don't know the further way yet. I'll try to go forward coming days.

Serg
Serg
2018 Supporter

Posts: 854
Joined: 01 June 2010
Location: Russia

### Re: Zerodoku

Code: Select all
` ; ; ; ;0;3; ;3 ; ;9;0;0;5; ; 8;2; ; ; ;4; ;  ; ; ;5; ; ; ; 9;7; ; ; ;6;7; 7;6; ; ; ;7;9;6 ; ;0; ; ; ; ;08; ;3; ;8;4; ; `

Also tough...
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

Hi, evert!
evert wrote:What I found quite tough was:
Code: Select all
` ; ; ;4; ;0;0; ; ; ; ;  ;6;0; ;8;5; ;6;7;9; ;  ; ; ;1; ;0; ; ;4; ;5;  ;8; ; ; ; ;1; ; ; ;0;5 ;7; ;2; ; ;5; ; ;9; ;  ;6;4; ; ; ; ; ;7; ; ;00;3; ; ;6; ; ; ;0;5; ;76; ; ;8; ; ; ;4; ;8;3; 0; ; ;6; ; ;0;3; ; ;5; 3; ; ;3; ;1; ;0;8; ; ;  ; ; ;4; ; ;1; ;4; ; ;  ;2;0; ; ;7; ;2; ;8; ;7`

After some obious steps - and some fish :

Code: Select all
` ;5; ;4;7;0;0; ; ; ;4; 7;6;0;0;8;5;2;6;7;9;3;44;0;3;1;2;0; ; ;4; ;5; 2;8; ;3; ; ;1;7; ;0;0;5 ;7; ;2; ; ;5; ;0;9;7;85;6;4;0; ; ; ; ;7;4;8;00;3; ;7;6; ;4; ;0;5;6;76;5;7;8;0;6;7;4;5;8;3;00;4;2;6; ; ;0;3;7;6;5;43;0;5;3;4;1;2;0;8;7;9;62;3;1;4;0;2;1;3;4;0;4;51;2;0;5;4;7;3;2;6;8;0;7`

Really couldn't tell how to proceed from here.

I used "Try & Error" method to go forward.
Cell r5c1 can contain "3" or "6" only. If we set "6" to this cell, we'll get solution:
Code: Select all
`3 5 6 4 7 0 0 5 6 8 4 77 6 0 0 8 5 2 6 7 9 3 44 0 3 1 2 0 3 8 4 7 5 62 8 6 3 4 9 1 7 5 0 0 56 7 8 2 3 4 5 0 0 9 7 85 6 4 0 5 3 6 2 7 4 8 00 3 9 7 6 8 4 5 0 5 6 76 5 7 8 0 6 7 4 5 8 3 00 4 2 6 3 5 0 3 7 6 5 43 0 5 3 4 1 2 0 8 7 9 62 3 1 4 0 2 1 3 4 0 4 51 2 0 5 4 7 3 2 6 8 0 7`

If we set "3" to r5c1 cell, we'll come to contradiction. So, solution found is unique.
I'll try to localize cells' chain providing solution.

Serg
Serg
2018 Supporter

Posts: 854
Joined: 01 June 2010
Location: Russia

### Re: Zerodoku

Hi Evert,

ZeroDoku seems to be an interesting game.
It is sufficiently different from Sudoku to bring new experience, although one can still have (Naked, Hidden and Super-Hidden) Subsets, whips, braids, g-whips, g-braids and all my fauna.

The grid structure is the same as in Kakuro (cells with 0 correspond to black cells), but the game avoids all the tedious sum checking work required by Kakuro. What I appreciate here is, the resolution state is visible directly in the grid, as in Sudoku.

Two remarks (with the background idea that large size puzzles often involve lots of obvious boring eliminations).

1) I think you should forget your restriction about a maximum of 2 zeros per row and column. It has no impact on the solving process and it excludes the possibility for the puzzle creator to generate some interesting cases.
One interesting point in Kakuro (and probably in this modified game) is, one can create hard puzzles of small size by a good placement of the black cells.

2) Did you investigate puzzles with no clues but the 0's ? I think it'd be another way of creating hard puzzles of small size.

As for the notation, could you adopt something closer to the usual one in Sudoku, e.g. a dot for an empty cell and no semi-colons ?
denis_berthier
2010 Supporter

Posts: 3899
Joined: 19 June 2007
Location: Paris

### Re: Zerodoku

denis_berthier wrote:The grid structure is the same as in Kakuro (cells with 0 correspond to black cells), but the game avoids all the tedious sum checking work required by Kakuro.

One difference is that in Zerodoku not all zeroes are necessarily given as clue. They must be located during the solving process.

denis_berthier wrote:Did you investigate puzzles with no clues but the 0's ?

That would be impossible. For if in the final solution you replace 1 by 9, 2 by 8, 3 by 7 etc, that would give you another valid solution.

denis_berthier wrote:I think you should forget your restriction about a maximum of 2 zeros per row and column.

That would be impossible too. The puzzler could fill in as many zeroes as he/she likes and the final solution would then be valid.
evert

Posts: 187
Joined: 26 August 2005

### Re: Zerodoku

evert wrote:
denis_berthier wrote:I think you should forget your restriction about a maximum of 2 zeros per row and column.

That would be impossible too. The puzzler could fill in as many zeroes as he/she likes and the final solution would then be valid.

I must correct myself.
Code: Select all
`...|...|......|...|......|...|...---+---+---28*|***|*.....|...|......|...|...---+---+---...|...|......|...|......|...|...`

A segment containing 2 and 8 requires at least 7 positions where thus no zero is possible.
evert

Posts: 187
Joined: 26 August 2005

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