## Revealing twins

For fans of Killer Sudoku, Samurai Sudoku and other variants

### Re: Revealing twins

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|456456|123|789789|456|123---+---+---312|978|645645|312|978978|645|312---+---+---231|897|564564|231|897897|564|231123|897|645456|231|978789|564|312---+---+---645|123|897978|456|231312|789|564---+---+---897|645|123231|978|456564|312|789`

Maybe you could create a puzzle or two from them?
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Revealing twins

BryanL wrote: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|456456|123|789789|456|123---+---+---312|978|645645|312|978978|645|312---+---+---231|897|564564|231|897897|564|231123|897|645456|231|978789|564|312---+---+---645|123|897978|456|231312|789|564---+---+---897|645|123231|978|456564|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`
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

BryanL wrote:
dyitto wrote:
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.|...|..63..|..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
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

BryanL wrote:
dyitto wrote:
BryanL wrote: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|456456|123|789789|456|123---+---+---312|978|645645|312|978978|645|312---+---+---231|897|564564|231|897897|564|231123|897|645479|536|128586|124|397---+---+---241|973|856798|465|231365|281|974---+---+---812|749|563937|658|412654|312|789`
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

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|..31 in grid 1 reveals a group in grid 29 in grid 2 reveals a group in grid 1`
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

dyitto wrote: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

Posts: 247
Joined: 28 September 2010

### Re: Revealing twins

BryanL wrote: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

dyitto wrote: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.
BryanL

Posts: 247
Joined: 28 September 2010

### Re: Revealing twins

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!

simon_blow_snow

Posts: 85
Joined: 26 December 2010

### Re: Revealing twins

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.)
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

dyitto wrote: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

Posts: 85
Joined: 26 December 2010

### Re: Revealing twins

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.

simon_blow_snow

Posts: 85
Joined: 26 December 2010

### Re: Revealing twins

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.
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

dyitto wrote: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.

simon_blow_snow

Posts: 85
Joined: 26 December 2010

### Re: Revealing twins

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
`249573816735168492681924357168492735924357681573816249357681924816249573492735168267591834915348672483726159348672915726159483591834267159483726834267591672915348276195384438627519951843762519438627762951843384276195843762951195384276627519438294375186618429537753861942537618429942753861186294375861942753375186294429537618429537618375186294861942753186294375942753861537618429753861942618429537294375186438195762276843519951627384519276843384951627762438195627384951195762438843519276483591672915726834267348159726834915348159267591672483159267348672483591834915726492735168816249573357681924573816249924357681168492735681924357735168492249573816618375942294861537753429186537294861186753429942618375429186753375942618861537294627519438195384276843762951384276195762951843519438627951843762438627519276195384672915348834267591159483726591834267726159483348672915483726159915348672267591834681573492735924816249168357924816735168357249573492681357249168492681573816735924816735924492681573357249168573492681168357249924816735249168357735924816681573492834915726672483591159267348591672483348159267726834915267348159915726834483591672843519276195762438627384951762438195384951627519276843951627384276843519438195762861537294375942618429186753942618375186753429537294861753429186294861537618375942`

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?
evert on the crashed forum

dyitto

Posts: 118
Joined: 22 May 2010
Location: Amsterdam

### Re: Revealing twins

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!

simon_blow_snow

Posts: 85
Joined: 26 December 2010

PreviousNext