The New Sudoku Players' Forum

[BryanL]

I don't have time to create a pair of puzzles from them but here are the two grids I created with both way 9 digit revealing groups...

Code: Select all

123|789|456
456|123|789
789|456|123
---+---+---
312|978|645
645|312|978
978|645|312
---+---+---
231|897|564
564|231|897
897|564|231

123|897|645
456|231|978
789|564|312
---+---+---
645|123|897
978|456|231
312|789|564
---+---+---
897|645|123
231|978|456
564|312|789



Maybe you could create a puzzle or two from them?

[dyitto]

I don't have time to create a pair of puzzles from them but here are the two grids I created with both way 9 digit revealing groups...

Code: Select all

123|789|456
456|123|789
789|456|123
---+---+---
312|978|645
645|312|978
978|645|312
---+---+---
231|897|564
564|231|897
897|564|231

123|897|645
456|231|978
789|564|312
---+---+---
645|123|897
978|456|231
312|789|564
---+---+---
897|645|123
231|978|456
564|312|789



Maybe you could create a puzzle or two from them?

At first sight I would never believe someone could do this with pen and paper. Amazing!
Could you explain your method for finding the complete grid in some more detail?

A puzzle would be:

Code: Select all

1..|...|...
.5.|.2.|...
...|.5.|..3
---+---+---
...|...|.45
.4.|...|...
..8|...|..2
---+---+---
...|8..|...
.64|...|.9.
...|...|...

.2.|...|6..
4..|.31|...
.8.|...|312
---+---+---
6..|.2.|...
97.|.56|...
...|7..|...
---+---+---
...|6..|1..
..1|...|45.
...|3..|..9


[dyitto]

Code: Select all

..3|...|8..
51.|..8|..4
...|.5.|7..
---+---+---
...|...|6.9
...|..9|..5
...|..1|...
---+---+---
4.7|8.6|...
8..|4..|...
...|.95|...

...|...|...
.73|5..|9..
.5.|...|8..
---+---+---
...|8..|..5
...|.69|7..
...|...|.64
---+---+---
7..|...|..3
.1.|...|..6
3..|..8|...


There's an error with those two...
A couple of moves reveal an 8 in r5c9 in pzl2, but there is already a 5 in both r5c9 and r9c6 in pzl1

This one had only one digit each direction:
1 in grid 1 reveals a group in grid 2 & 9 in grid 2 reveals a group in grid 1

[dyitto]

Just another point on 9 digits revealing groups in both directions. I suspect (prove me wrong?) that for any grid 1, there will be one and only one grid 2 with the correct property - except perhaps with symmetric relabeling.

The first random sudoku grid I checked hasn't.

Do you mean it hasn't got another 2nd grid with the correct property or it hasn't got only one???

It has zero twin grids with the correct property.

But I did find a second twin for the first grid of your 9-9:

Code: Select all

123|789|456
456|123|789
789|456|123
---+---+---
312|978|645
645|312|978
978|645|312
---+---+---
231|897|564
564|231|897
897|564|231

123|897|645
479|536|128
586|124|397
---+---+---
241|973|856
798|465|231
365|281|974
---+---+---
812|749|563
937|658|412
654|312|789


[dyitto]

Another one with few just a little more advanced steps.

Code: Select all

8..|7..|...
1.3|9..|27.
.5.|.1.|...
---+---+---
.3.|...|...
...|35.|...
.2.|4..|...
---+---+---
4..|5..|3..
...|...|.84
..2|.3.|...

...|...|29.
4..|...|..8
..7|6..|..5
---+---+---
2.3|...|.4.
1..|...|3..
.7.|3..|...
---+---+---
...|..1|...
7.9|.4.|...
.21|.65|..3

1 in grid 1 reveals a group in grid 2
9 in grid 2 reveals a group in grid 1


[BryanL]

At first sight I would never believe someone could do this with pen and paper. Amazing!
Could you explain your method for finding the complete grid in some more detail?

As I mentioned yesterday I was playing around with some symmetries (rotations, reflections, transpose either way) in the 2nd grid from the 1st grid. I had started with the MC grid but it was too hard to follow so then I created the 1st grid with repeating mini-rows (digits 1-3, 4-6, 7-9 always appear together in a mini-row). This made it easy to see where digits needed to be in the 2nd grid. I had been seeing the odd pattern with symmetries, and with the new 1st grid, it suddenly became clear what was required for the digits in the 2nd grid to map back to the 1st. Note that both grids have repeating mini-rows and that the ordering of the digits in each mini-row repeats in a pattern and also the order of the mini-rows in each box.
If you open each grid in a gui solver like simple sudoku where you can filter digits, select all the 1s in 1st grid and observe the matching digits in the 2nd, repeat for the 2s and 3s etc, you will see a pattern. Then if you do that again by selecting the 1s in the 2nd grid and observe the matches in the 1st grid, you will see similar patterns. The pattern is in how the digits move around in their mini-rows. I had seen this in braid analysis.

Bryan

p.s. I have been getting a sense of deju vu with revealing twins for a while. I have done a quick search here and on the Programmers Forum but didn't find anything... Yet i still feel (especially with the 9 way revealing twins) that I have seen it before...

[BryanL]

Just another point on 9 digits revealing groups in both directions. I suspect (prove me wrong?) that for any grid 1, there will be one and only one grid 2 with the correct property - except perhaps with symmetric relabeling.
later

But I did find a second twin for the first grid of your 9-9:

Well that blows that idea out of the water.

Interestingly, which I guess sorta proves it, I was checking whether the grids are isomorphs or not and found that my grid 1 and 2 are isomorphs, but your grid 2 for my grid 1 9-9 is essentially different - i.e. not an isomorph - leaving the door open for any number of 9-9 2nd grids pairing with a common 1st grid.

[simon_blow_snow]

Excellent stuffs guys!

It seems it is not very hard at all to create 9-9 revealing twins.

From the attempts to create the setdoku grids, I came across this very special sudoku variant:

Constraints:

• Each row
• Each column
• Each 3x3 nonet (i.e. blocks)
• Each 3x7 nonet (i.e. corresponding mini-columns within horizontal bands)
• Each 7x3 nonet (i.e. corresponding mini-rows within vertical stacks)
• Each 7x7 nonet (i.e. spots or disjoint groups)
• Each diagonal
• The asterisk (i.e. R258C5+R37C37+R5C28)

Also, each mini-row or mini-column sums to the same total.

All mini-rows are cycles of 1-5-9, 2-6-7, 3-4-8.

All mini-columns are cycles of 1-6-8, 2-4-9, 3-5-7.

The central 3x3 nonet (block) is a full magic square.

All (horizontal/vertical) adjacent cells are in different range (123/456/789) and different modulo of 3 (147/258/369).



Turns out, there are only 2 valid grids for this variant. And they form a pair of 9-9 revealing twins!

[dyitto]

Very interesting!

Could you post the grids?

What's the minimal definition of your variant?
(Which properties are derived from the others? The same-total property is a consequence of the cycles I guess.)

[simon_blow_snow]

Very interesting!

Could you post the grids?

What's the minimal definition of your variant?
(Which properties are derived from the others? The same-total property is a consequence of the cycles I guess.)

I will describe how I developed the concept.

First of all, I defined 2 attributes for each cell value: range (small={123},medium={456},large={789}) and modulo of 3 (1={147}, 2={258}, 0/3={369}).

The reason I did that was to create an elegant graphical representation of Sudoku, which you can find in the "One for the X'mas" thread in this forum.

So I wanted to see with this concept if it was possible to establish a very concise and elegant variant rule. Immediately one came up:

Neighbours have different attributes.

With this fundamental rule the solution space is much refined already. Note that with this constraint, all 3x3 nonets automatically become semi-magic squares (or Euler squares if you consider the 2 attributes). Then once the disjoint group constraints are applied, it is further "restrained" (as HATMAN, one of the experts in Sudoku variants, love to say). I believe only 2 essentially different solutions exist, if we can freely swap the cell values based on their attribute structures.

So what's remaining is to fix the contextual and positional aspect. I do this by adding 2 extra constraints:

3. The central 3x3 nonet must be a full magic square (essentially saying R5C5=5).

4. The mini-rows must be cycles of 1-5-9/2-6-7/3-4-8 (left to right). The mini-columns must be cycles of 1-6-8/2-4-9/3-5-7 (top to bottom). This fix the reflection/rotation unstability.

(Alternatively, one can bypass constraints 3 and 4 by specifying the cell values of N5 as the [834159672] full magic square.)

With these 4 constraints, there are only 2 valid solutions and as I said above, they form a pair of 9-9 revealing twins!

I'm sure using a simple solver software (e.g. JSudoku) anyone can find the 2 grids within seconds. But if people still want it, I will post the grids next time.

[simon_blow_snow]

To clarify a bit more on the minimal requirements of the constraints:

With the "neighbours have different attributes" rule and the "disjoint groups", it takes only the "full magic square in N5" and "mini-row/mini-column cycles in N5" to make it a 2-solutional puzzle.

If the "mini-row/mini-column cycles" are applied on the whole grid, then "disjoint groups" are not necessary. Instead only the 2 "diagonals" (together with the "neighbours have different attributes" and "full magic square in N5") are enough to make it 2-solutional.

Also, here is a formal specification of the "mini-row/mini-column cycles" constraint without quoting the specific numbers:

All mini-rows and mini-columns consist of cycles of six particular triplets. Also, in their numerically minimal form, each of them must be strictly increasing sequences (left-to-right, top-to-bottom). Finally, the smallest numerically minimal mini-row is smaller than the smallest numerically minimal mini-column.

You can see why it is much easier to specify this constraint quoting the actual numbers.

[dyitto]

What I could feed to my solver - without essentially programming new stuff - is:

Each row
Each column
Each 3x3 nonet (i.e. blocks)
Each diagonal
Each 3x7 nonet (i.e. corresponding mini-columns within horizontal bands)
Each 7x3 nonet (i.e. corresponding mini-rows within vertical stacks)
Each 7x7 nonet (i.e. spots or disjoint groups)
The asterisk (i.e. R258C5+R37C37+R5C28)
Also, each mini-row or mini-column sums to the same total (which is in that case 15).
The central 3x3 nonet (block) is a full magic square (meaning that its mini-diagonals also sum to 15).

This is already so restrained that for example the following grid has exactly 1 solution:

Code: Select all

24.|...|...
...|...|...
...|...|...
---+---+---
...|...|...
...|...|...
...|...|...
---+---+---
...|...|...
...|...|...
...|...|...


The cycles would be essentially new in my solver. Is there an existing free-ware solver that supports them?


By the way,

Looking at Setoku with 4 attributes.

Suppose I'd have a mapping from the digits 1-9 to two attributes.
And also a mapping from the digits 1-9 to the other two attributes.

I want both mappings to result into a vanilla sudoku.

Then I end up with a representation of a 9-9 revealing twin.

[simon_blow_snow]

What I could feed to my solver - without essentially programming new stuff - is:

Each row
Each column
Each 3x3 nonet (i.e. blocks)
Each diagonal
Each 3x7 nonet (i.e. corresponding mini-columns within horizontal bands)
Each 7x3 nonet (i.e. corresponding mini-rows within vertical stacks)
Each 7x7 nonet (i.e. spots or disjoint groups)
The asterisk (i.e. R258C5+R37C37+R5C28)
Also, each mini-row or mini-column sums to the same total (which is in that case 15).
The central 3x3 nonet (block) is a full magic square (meaning that its mini-diagonals also sum to 15).

You can add two more constraints:

9x9 nonet (i.e. R159C159)
Central 5x5 nonet (i.e. R357C357)

HATMAN once said these 2 are equivalent diagonals for the disjoint groups.


It would be really interesting if you can find the implicit relationship between setdoku and 9-9 revealing twins, the only 2 ongoing variant topics in this forum.

[dyitto]

Once again the exact set of constraints as suggested by simon_blow_snow:

Each row
Each column
Each 3x3 nonet (i.e. blocks)
Each diagonal
Each 3x7 nonet (i.e. corresponding mini-columns within horizontal bands)
Each 7x3 nonet (i.e. corresponding mini-rows within vertical stacks)
Each 7x7 nonet (i.e. spots or disjoint groups)
The asterisk (i.e. R258C5+R37C37+R5C28)
Also, each mini-row or mini-column sums to the same total (which is in that case 15).
The central 3x3 nonet (block) is a full magic square (meaning that its mini-diagonals also sum to 15).
9x9 nonet (i.e. R159C159) (1st diagonal equivalent disjoint groups)
Central 5x5 nonet (i.e. R357C357) (2nd diagonal equivalent disjoint groups)

I'v found 16 solutions:

Code: Select all

249573816735168492681924357168492735924357681573816249357681924816249573492735168
267591834915348672483726159348672915726159483591834267159483726834267591672915348
276195384438627519951843762519438627762951843384276195843762951195384276627519438
294375186618429537753861942537618429942753861186294375861942753375186294429537618
429537618375186294861942753186294375942753861537618429753861942618429537294375186
438195762276843519951627384519276843384951627762438195627384951195762438843519276
483591672915726834267348159726834915348159267591672483159267348672483591834915726
492735168816249573357681924573816249924357681168492735681924357735168492249573816
618375942294861537753429186537294861186753429942618375429186753375942618861537294
627519438195384276843762951384276195762951843519438627951843762438627519276195384
672915348834267591159483726591834267726159483348672915483726159915348672267591834
681573492735924816249168357924816735168357249573492681357249168492681573816735924
816735924492681573357249168573492681168357249924816735249168357735924816681573492
834915726672483591159267348591672483348159267726834915267348159915726834483591672
843519276195762438627384951762438195384951627519276843951627384276843519438195762
861537294375942618429186753942618375186753429537294861753429186294861537618375942


The 1st of these solutions forms a 9-9 revealing twin with 8 of the other solutions.
The 3d of these solutions forms a 9-9 revealing twin with the other 8 solutions.

(I suspect there might be two groups of 8 solutions, where each of one group goes with each of the other group. I didn't check this.)

This means that the above combination of constraints turns out to exclusively generate 9-9 revealing twins.

Questions arise here, like:
Is there a less neat set of constraints for 9-9-exclusive?
Are all 9-9-revealing twins subject to similar constraints?

[simon_blow_snow]

Notice that your 16 solutions can be divided into 2 groups of 8, within each group, all grids are mirror/rotational image of each other. So there are only essentially 2 different solutions.

That they are all isomorphs of the Most Canonical grid is also an important fact.

Also I think you have shown yourself that 2 grids which are not isomorphs of each other can be 9-9 revealing twins. So I think there is a large space of 9-9 revealing twins, some of them are tidy like MC grids isomorphs while some of them are just random, arbitrary grids.


BTW, if you have some spare time, I would like your (and other programming experts') help to my "Secret Communicator Matrix" problem posted in the "Inventors' studio" forum. That problem starts relatively trivially but gets more complicated real quick as the size grows bigger. I think a program will help a lot but I am not that good in programming. So please have a look, thanks!