The New Sudoku Players' Forum

[Sojourner9]

Hi,

Just found this Sudoku forum which is nice, so I joined, so forgive me if my etiquette is weak.

I think that the accepted answer for the maximum number of Sudoku Solution Grids may be wrong.

So for those of you that have ever asked how they calculated their answer, please checkout a posting I entered on the Exchange Mathematics Stack Exchange and see of you agree with my solution. It is high school level math.

My Simple Combinatoral Method to Enumerate All Sudoku Solution Grids math.stackexchange.com/questions/2684500/my-simple-combinatorial-method-to-enumerate-all-sudoku-solution-grids#2689492

The body of this has been included below, thanks to somebody. I currently have one answer on the other site so feel free to follow there as well.

If you think I am write give it an up vote. This is not spam, I really think I have the correct answer.

And getting some assurance would be great.

Thanks, Wade Whitaker

[Sojourner9]

Hi, This is a re-post of my submission to the mathematics Stack Exchange on Friday.

Since, for what ever reason, my first attempt at introducing this topic didn't show up in the topic lists, I figured a full re-post might be better. And besides, the authors seem to be on this forum as well.

Ask yourself which is easier to believe and more useful, a complicated computed result or a simple enumeration?

How may possible Sudoku Solution Grids are there? The correct answer is: 6,670,903,752,021,072,936,960 or 6,671E21 as was proved 12 years ago! Or was it?

Their proof never really enumerated the solutions and involved a lot of mainframe time to compute, which involved a lot of reducing test cases to make the computations complete in human time.

When they said and I read that no simple combinatorial answer was possible, I immediately though, "Well I already know one."

So I ran the numbers on a calculator but did NOT get their solution and I can't see a flaw in mine.

I can not follow there solution to the end since I have to rely on their computer results. I sent this to the author but did not hear back, which is not surprising since it is not there place to prove me wrong. I have waited and looked over my method long enough that I stand by my solution.

Since this has to be expressed in the form of a question, "What is wrong with my solution?"

So, here is, what we have all been waiting for:

My Simple Combinatorial Way to Enumerate Sudoku Solution Grids

Note: The problem we are trying to solve here is for N0 which is the maximum number of correct Sudoku answer grids. So, no discussion of swapping to get equivalent solutions.

There are three Bands, rows 1-3, 4-6, and 7-9, and three Stacks columns 1-3, 4-6, and 7-9 in a Sudoku Grid, we will start with a discussion of the first Band which is the top three rows.

The three bands and stacks divide the Sudoku Grid in to 9 Blocks, labeled B1, B2, and B3 in Band1, B4, B5, and B6 in Band2, and B7, B8, and B9 in Band3, which also means B1, B4, and B7 in Stack1, B2, B5, and B8 in Stack2, and B3, B6, and B9 in Stack3.

Substituting Numbers for Symbol Positions

We will substitute the 9 numbers used in the 81 squares to leave B1 with the following Grid:

Code: Select all

    |1 2 3|
    |4 5 6|     Note: 9! cases
    |7 8 9|


With this we can now think of the numbers not as number but as symbol positions, where 9! different solutions will use the same symbol positions for all the 81 squares.

Describing Row Constraints for a Band

Say we want to fill in Band1 and we start in what I call an All-Normal Pattern for B1 Row1, as follows:

Code: Select all

   Row 1: |1 2 3|     |     |    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 2 3|     |    B3 only describe the row they are in,
   Row 3: |7 8 9|     |1 2 3|    not the column positions.


Given this as a starting point, how do we fill in the symbol positions from the B1 Row2, in B2? If we tried to put them in Row1 we have a problem when we get to B3 because we would need to use positions already occupied. So, we have to follow the same All-Normal pattern as B1 Row1, resulting in the following:

Code: Select all

   Row 1: |1 2 3|     |4 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 2 3|     |    B3 only describe the row they are in,
   Row 3: |7 8 9|4 5 6|1 2 3|    not the column positions.


Then B1 Row3 follows the All-Normal pattern to fill in the blanks.

A Normal Symbol is defined as a symbol that goes down one to the second block and down one to the third block, where if you go below the bottom you wrap to the top of the band.

Abnormal Symbol is defined as a symbol that goes down two to the second block and down two to the third block, with wrapping.

An All-Normal Pattern is where all nine symbols are normal.

Now lets look at a case with one abnormal symbol from B1 Row1, we will pick 3:

Code: Select all

   Row 1: |1 2 3|     |     |    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 2  |    3|    B3 only describe the row they are in,
   Row 3: |7 8 9|    3|1 2  |    not the column positions.


Here we have the symbol 3 going down 2, and down 2 with a wrap, instead of down 1 and down 1.

Now if we try to fill in B1 Row2 and Row3 in B2 and B3 we find that we have a similar problem, unless we do the same thing as Row 1 and pick one of the three symbols to be abnormal. The same thing can be said about having two abnormals from Row 1, we need to repeat this for the other two Rows. And finally we have the three abnormals case. For brevity I will leave it to the reader to validate this or they can just look at any Sudoku solution grid.

Abnormal Pattern is defined as having one abnormal symbol per row in B1.

Normal Pattern is defined as having one normal symbol per row in B1.

All-Abnormal Pattern is defined as all 27 symbols being abnormal.

The normal and abnormal patterns need further clarification. For normal pattern we need to know, for each row, which of the three positions in B1 contains the normal symbol. For abnormal pattern we need the same for the abnormal symbol. For each there are 3 positions in each of the rows for a total of 3*3*3 = 27.

So the total number of permutations of symbols in B2 and B3 for Band1 is 1 for the All-Normal 3*3*3 for the Abnormal, 3*3*3 for the Normal, and 1 for the All-Abnormal patterns. let us call this R:

Code: Select all

    R = 1 + 3*3*3 + 3*3*3 + 1 = 56 permutations of 9 symbols in 3 rows.

It should be noted that R describes all three blocks even if you were to swap them, it is a natural constraint on any block/stack and we can use it later to describe all 3 blocks and 3 stacks.

The rest is trivial, but I will highlight the important parts.

Describing Column Constraints for B2 and B3

We still need to describe the column positions for the symbols in B2 and B3, which is just the permutations of the three number in each sub-row, which is 6. Let us call this P:

Code: Select all

    P = 6*6*6 * 6*6*6 = 6*6 * 6*6 * 6*6 = 46656 permutations of 3 symbols in 6 sub-rows.

When doing the permutations only permute two symbols, A and B, through three positions:

|A B x|, |A x B|, |x A B|, |B A x|, |B x A|, |x B A|

Where A and B are:

|A B x| For each row in B1 in the All-Normal and All-Abnormal patterns.

|x A B| For each row in B1 where,
|A x B| in the Abnormal pattern, x is the position of abnormal symbol and,
|A B x| in the Normal pattern, x is the position normal symbol.

The remaining character x will find its position as the open position in its assigned row.

So for Band1 the total number of solutions using symbol positions is:

R * P = 56 * 46656 = 2612736

Note: I can use a number between 1 through 2612736 to calculate a specific permutation of these solutions or I can use a solution and use the above discussion to assign a specific number to this permutation.

The Constraining the Rest of the Bands and Stacks

If I want to describe the starting positions for B4 and B7 I can use R and P for Stack1 like I did for Band1 and know all the permutations of B4 and B7. Later I could do some renumbering for B1 when describing Stack1 to gain symmetry for the final row and column descriptions.

R can be used on Band2 and Band3 to describe the row positions for B5, B6, B8, and B9.

R can be used on Stack2 and Stack3 to describe the column positions for B5, B6, B8, and B9.


If I know the row and column positions for each symbol for B5, B6, B8, and B9 then the completion of each permutation just involves matching the row and column for each symbol for each block.

Conclusion

I can describe Band1 and the row positions in Band2 and Band3 as:

Row Contribution = R * P * R * R = 8,193,540,096

I cab describe Stack1 and the column positions in Stack2 and Stack3 as:

Column Contribution = R * P * R * R = 8,193,540,096

The total is just multiplying these two numbers and 9! for substituting numbers for symbol positions

Code: Select all

   Total = 9! * 8,193,540,096 * 8,193,540,096 = 2.436162195571x10^25

Since it is just multiplying digits I could list all the digits but my math package does not have that may significant digits.

Note, it might be to swap the P from between the two contributions since the P in row contribution fixes the column positions and vice versa.

So, now I have a total of numbers solutions and a design for a function that given a number I can derive a specific solution or given a solution I can derive its ordinal number and I know how to count through all solutions.

Humorous note: 8,193,540,096 * 8,193,540,096 = 6.7134E19

So again, where is my mistake?

[Mathimagics]

Hi Wade!

I have not had time to analyse your method in depth, but there are some observations we can make:

• your figure of Band1/Stack1 completions, given fixed B1, is correct, ie 2,612,736

• so for fixed B1 there are N = 2612736 ^ 2 = 6,826,389,405,696 band 1 + stack 1 completions

• you arrive at the figure of NB = N * (56 ^4) = 67,134,099,304,759,689,216 as the total number of Sudoku grids for a fixed B1

• the accepted value for NB is 18,383,222,420,692,992, which when multiplied by 9! (the number of different B1's) gives a total Sudoku grid count of NS = 6,670,903,752,021,072,936,960

Let's assume for now that the accepted figures are in fact correct. Your NB value would then be "overcooked" by a factor of roughly 3652.

Let X(g) be the number of completions for a given g, where g is a given Band 1 + Stack 1. So X(g) is the number of ways to complete the grid, ie fill in B5, B6, B8 and B9.

Your figures suggest that X(g) is fixed and equal to 56 ^ 4 = 9,834,496. But we know that X(g) is not the same for all grids and that the average value is roughly 2,693.

It is easy with a Sudoku solver to take a random g and count the number of completions. Here are two randomly selected cases:

Code: Select all

 1 8 2 | 7 6 4 | 5 9 3       1 7 2 | 8 6 4 | 9 3 5
 4 9 3 | 5 2 8 | 1 7 6       4 9 3 | 5 1 7 | 2 8 6
 6 5 7 | 1 9 3 | 2 8 4       6 5 8 | 2 9 3 | 1 7 4
 ---------------------       ---------------------
 8 4 1 | . . . | . . .       2 4 7 | . . . | . . .
 7 6 5 | . . . | . . .       8 6 5 | . . . | . . .
 3 2 9 | . . . | . . .       3 1 9 | . . . | . . .
 ---------------------       ---------------------
 9 7 6 | . . . | . . .       5 8 6 | . . . | . . .
 2 1 4 | . . . | . . .       9 2 4 | . . . | . . .
 5 3 8 | . . . | . . .       7 3 1 | . . . | . . .


The grid on the left has 2917 completions, the other has 2738, which proves that X(g) is not the same for all grids, and is compelling evidence that the average X is far more likely to be 2693 than 9834496.

My guess is that your completion figure of X = 56^4 should be 56^2, ie 3136, and that, while your method might be sound (and I haven't really looked at it very hard), it only provides an upper bound for the number of grids, since it doesn't allow for the variance in individual (ie g-specific) completion numbers.

This was done by Jarvis & Felgenhauser originally by identifying classes of g that can be shown to be equivalent (same number of completions) and then adding up the values (NC * X), where X is # of completions per class member, NC is number of class members.

[Serg]

Hi, Wade!

My Simple Combinatoral Method to Enumerate All Sudoku Solution Gridshttps://math.stackexchange.com/question%20...%20tion-grids

This link is broken.

Serg

[Serg]

Hi, Wade!
You should describe your method in more detail (preferably with examples).

With this we can now think of the numbers not as number but as symbol positions, where 9! different solutions will use the same symbol positions for all the 81 squares.

I don't understand what do you mean.

Row 1: |1 2 3| | | Note: The symbol positions in B2 and
Row 2: |4 5 6|1 2 3| | B3 only describe the row they are in,
Row 3: |7 8 9| |1 2 3| not the column positions.

I don't understand anything. Please, show examples.

Serg

[Sojourner9]

Hello everyone,

Thank you for welcoming me to your community. The last time I was on a Sudoku Forum it was called /rec/puzzles/sudoku and I don't know how to get to the uunet newgroups anymore even if I didn't know they were shutdown years ago. Because the world required me to make a living, I didn't have time then to follow the forum then but it does not look like much has changed. I remembered a Sudoku grid called "ocean" and he seems to be a member. So am I right that this group stretches back to before the internet?

An early thanks to Mathimagics for Serg for replying back to before I had ever logged in to look at the topic once it was accepted.

A special thank you to whomever combined my two topics into one, I was sure I had a mess to clean up when I first logged in.

Please forgive me until I get up to speed on how to enter fixed font blocks, links, etc.

Serg, say to Alexis, "Simple Combinatorial Way to Enumerate Sudoku Solution Grids" and it should be at the top of the list.

It is funny/sad that this topic in this forum did not come up.

I can see that the URL got munched when it was copied. I will fix that and my ASCII art in the original post.

And I see that Serg has responded. I will be glad to expand on this, let me fix my original post up to at least be consistant with the other site.

Thanks, All

[Sojourner9]

Hi, Wade!
You should describe your method in more detail (preferably with examples).

With this we can now think of the numbers not as number but as symbol positions, where 9! different solutions will use the same symbol positions for all the 81 squares.

I don't understand what do you mean.

Row 1: |1 2 3| | | Note: The symbol positions in B2 and
Row 2: |4 5 6|1 2 3| | B3 only describe the row they are in,
Row 3: |7 8 9| |1 2 3| not the column positions.

I don't understand anything. Please, show examples.

Serg

Serg,

For the first part let me say that I am doing nothing here that is different than anyone else just normalizing the numbers in B1. I started by reading "Enumerating possible Sudoku grids" and "Mathematics of Sudoku I" where they were doing some swapping of B1 with B2 and B3 which I found confusing. So here I am just stating that this can't be done, no swapping of numbers.

For the second part, I have reformatted the text to use a fixed column width and it reads better now. I would like you to try to fill in 4,5, and 6 in the grid in several ways and see if you can see what I am trying to explain. If you can't figure it out, let me know and I will elaborate.

[Sojourner9]

@Mathimagics: I don't think the @ is needed here but I got dinged on the other site.

I am tired for today so I will address you question tomorrow, but for everyone let me quote from Frazer Jarvis, et al.

56 x (3!)^6 = 2612736

And paraphrase by saying there are 56 ways to assign the row positions and (3!)^6 to assign the column positions for each band and stack. Now if I don't need to assign the column positions...

Now I want you all to assume the Lotus Position and chant over and over:

There are 56 ways to assign the row positions, there are 56 ways to assign the column positions.

and wait for Enlightenment.

We'll talk tomorrow.

[Mathimagics]

.
Another piece of compelling, if not completely overwhelming, evidence that Jarvis & Felgenhauser got it absolutely right is provided by the next paper, "Mathematics of Sudoku II", by Jarvis & Russell.

This concerns the counting of the number of essentially different Sudoku grids, ie those that can't be obtained from any other grid by geometric permutations and/or relabelling.

If they got the original count wrong, then the Burnside's Lemma calculations that they performed would fail horribly, and this would be self-evident (Serg will know just what I mean here!).

This latter process requires computation, but is backed by established Group Theory, and so it is quite a rigorous demonstration that the number of Sudoku grids they obtained originally is correct. Additionally, that result was confirmed independently by gsf who gave a full accounting of different grids by automorphism group, and arrived at exactly same answer!

But you don't need Group Theory to prove that your numbers are suspect, just try the simple grid completion count test that I suggested above.

[Serg]

Hi, Wade!
I found your post and discussion at Stack Exchange forum. You reformatted your description in this thread, and now it looks better, but the main problem is textual description. I don't understand your considerations (maybe I am stupid). But nevertheless I am trying to crosscheck your intermediate result.

So for Band1 the total number of solutions using symbol positions is:

R * P = 56 * 46656 = 2612736

Do you mean that there are 2612736 variants of filling Band1 (B1+B2+B3), provided that B1 contains digits in the right order?

Code: Select all

  B1

1 2 3
4 5 6
7 8 9

Serg

[blue]

There are 56 ways to assign the row positions, there are 56 ways to assign the column positions.

So again, where is my mistake?

Take Mathimagics' first Band1 + Stack1 example, from above.

Code: Select all

 1 8 2 | 7 6 4 | 5 9 3
 4 9 3 | 5 2 8 | 1 7 6
 6 5 7 | 1 9 3 | 2 8 4
 ------+-------+------
 8 4 1 | . . . | . . .
 7 6 5 | . . . | . . .
 3 2 9 | . . . | . . .
 ------+-------+------
 9 7 6 | . . . | . . .
 2 1 4 | . . . | . . .
 5 3 8 | . . . | . . .

Try to fill box 5, in a way where band 2 and stack 2, each have the "All-Normal" type.
You'll have trouble at r4c5.
[ If you relabel the grid to have "123,456,789" in box 1, the problem will persist. ]

Enlightenment

[Mathimagics]

Enlightenment

The first step on the golden road to unlimited perception ...

[Sojourner9]

blue,

Ok, I see my mistake. 4 and 8 would need to occupy the same space.

I am wrong, and thanks to all for my new Enlightenment.

Something may work going both ways only because it is already true and does not mean that all cases will work going both ways.

Thanks, now I don't have that stuck in my brain.

I will enumerate the 56 cases in the near future as they are still useful.

The next question might be what is it about the input that limits which of the 56 will work and can they be predicted?

[Sojourner9]

As promised, here is a list of all 56 R values.

This list was made specifically for enumerating the max solution count and so it has redundancy, based upon the context, like all the abnormal cases become normal if I just swap B1 and B2, if their order does not matter, in which case there would be only 28 values in groups, "All Same" and "One Off". The R value could be used to determine the row values of B1 only, ignoring B2. Also, the P number was listed as being the permutations of the column positions in B1 and B2 but the permutations could be created just for B1 and combined with the R values for a different number that describes B2 in terms of B1. Then if we had a way of describing B4 in terms of B1 and B2 we could maybe come up with a method of describing the set of solutions that would work for B5 and extend it to all the rest. Has someone tried to do this before?

AN All-Normal

Code: Select all

   Row 1: |1 2 3|7 8 9|4 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 2 3|7 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 5 6|1 2 3|    not the column positions.


A111 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 8 9|7 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 2 3|1 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 5 6|4 2 3|    not the column positions.


A211 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 8 9|7 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 1 3|2 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 5 6|4 1 3|    not the column positions.


A311 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 8 9|7 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 1 2|3 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 5 6|4 1 2|    not the column positions.


A121 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 8 9|7 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 2 3|1 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 4 6|5 2 3|    not the column positions.


A221 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 8 9|7 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 1 3|2 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 4 6|5 1 3|    not the column positions.


A321 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 8 9|7 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 1 2|3 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 4 6|5 1 2|    not the column positions.


A131 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 8 9|7 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 2 3|1 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 4 5|6 2 3|    not the column positions.


A231 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 8 9|7 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 1 3|2 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 4 5|6 1 3|    not the column positions.


A331 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 8 9|7 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 1 2|3 8 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 4 5|6 1 2|    not the column positions.


A112 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 7 9|8 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 2 3|1 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 5 6|4 2 3|    not the column positions.


A212 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 7 9|8 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 1 3|2 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 5 6|4 1 3|    not the column positions.


A312 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 7 9|8 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 1 2|3 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 5 6|4 1 2|    not the column positions.


A122 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 7 9|8 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 2 3|1 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 4 6|5 2 3|    not the column positions.


A222 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 7 9|8 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 1 3|2 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 4 6|5 1 3|    not the column positions.


A322 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 7 9|8 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 1 2|3 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 4 6|5 1 2|    not the column positions.


A132 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 7 9|8 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 2 3|1 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 4 5|6 2 3|    not the column positions.


A232 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 7 9|8 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 1 3|2 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 4 5|6 1 3|    not the column positions.


A332 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 7 9|8 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|8 1 2|3 7 9|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 4 5|6 1 2|    not the column positions.


A113 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 7 8|9 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 2 3|1 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 5 6|4 2 3|    not the column positions.


A213 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 7 8|9 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 1 3|2 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 5 6|4 1 3|    not the column positions.


A313 - Abnormal

Code: Select all

   Row 1: |1 2 3|4 7 8|9 5 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 1 2|3 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 5 6|4 1 2|    not the column positions.


A123 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 7 8|9 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 2 3|1 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 4 6|5 2 3|    not the column positions.


A223 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 7 8|9 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 1 3|2 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 4 6|5 1 3|    not the column positions.


A323 - Abnormal

Code: Select all

   Row 1: |1 2 3|5 7 8|9 4 6|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 1 2|3 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 4 6|5 1 2|    not the column positions.


A133 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 7 8|9 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 2 3|1 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 4 5|6 2 3|    not the column positions.


A233 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 7 8|9 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 1 3|2 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|2 4 5|6 1 3|    not the column positions.


A333 - Abnormal

Code: Select all

   Row 1: |1 2 3|6 7 8|9 4 5|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|9 1 2|3 7 8|    B3 only describe the row they are in,
   Row 3: |7 8 9|3 4 5|6 1 2|    not the column positions.


AA All-Abnormal

Code: Select all

   Row 1: |1 2 3|4 5 6|7 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|7 8 9|1 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|1 2 3|4 5 6|    not the column positions.


N111 - Normal

Code: Select all

   Row 1: |1 2 3|7 5 6|4 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 8 9|7 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 2 3|1 5 6|    not the column positions.


N211 - Normal

Code: Select all

   Row 1: |1 2 3|7 5 6|4 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 8 9|7 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 1 3|2 5 6|    not the column positions.


N311 - Normal

Code: Select all

   Row 1: |1 2 3|7 5 6|4 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 8 9|7 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 1 2|3 5 6|    not the column positions.


N121 - Normal

Code: Select all

   Row 1: |1 2 3|7 4 6|5 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 8 9|7 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 2 3|1 4 6|    not the column positions.


N221 - Normal

Code: Select all

   Row 1: |1 2 3|7 4 6|5 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 8 9|7 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 1 3|2 4 6|    not the column positions.


N321 - Normal

Code: Select all

   Row 1: |1 2 3|7 4 6|5 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 8 9|7 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 1 2|3 4 6|    not the column positions.


N131 - Normal

Code: Select all

   Row 1: |1 2 3|7 4 5|6 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 8 9|7 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 2 3|1 4 5|    not the column positions.


N231 - Normal

Code: Select all

   Row 1: |1 2 3|7 4 5|6 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 8 9|7 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 1 3|2 4 5|    not the column positions.


N331 - Normal

Code: Select all

   Row 1: |1 2 3|7 4 5|6 8 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 8 9|7 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 1 2|3 4 5|    not the column positions.


N112 - Normal

Code: Select all

   Row 1: |1 2 3|8 5 6|4 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 7 9|8 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 2 3|1 5 6|    not the column positions.


N212 - Normal

Code: Select all

   Row 1: |1 2 3|8 5 6|4 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 7 9|8 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 1 3|2 5 6|    not the column positions.


N312 - Normal

Code: Select all

   Row 1: |1 2 3|8 5 6|4 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 7 9|8 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 1 2|3 5 6|    not the column positions.


N122 - Normal

Code: Select all

   Row 1: |1 2 3|8 4 6|5 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 7 9|8 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 2 3|1 4 6|    not the column positions.


N222 - Normal

Code: Select all

   Row 1: |1 2 3|8 4 6|5 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 7 9|8 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 1 3|2 4 6|    not the column positions.


N322 - Normal

Code: Select all

   Row 1: |1 2 3|8 4 6|5 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 7 9|8 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 1 2|3 4 6|    not the column positions.


N132 - Normal

Code: Select all

   Row 1: |1 2 3|8 4 5|6 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 7 9|8 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 2 3|1 4 5|    not the column positions.


N232 - Normal

Code: Select all

   Row 1: |1 2 3|8 4 5|6 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 7 9|8 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 1 3|2 4 5|    not the column positions.


N332 - Normal

Code: Select all

   Row 1: |1 2 3|8 4 5|6 7 9|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 7 9|8 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 1 2|3 4 5|    not the column positions.


N113 - Normal

Code: Select all

   Row 1: |1 2 3|9 5 6|4 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 7 8|9 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 2 3|1 5 6|    not the column positions.


N213 - Normal

Code: Select all

   Row 1: |1 2 3|9 5 6|4 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 7 8|9 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 1 3|2 5 6|    not the column positions.


N313 - Normal

Code: Select all

   Row 1: |1 2 3|9 5 6|4 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 7 8|9 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|4 1 2|3 5 6|    not the column positions.


N123 - Normal

Code: Select all

   Row 1: |1 2 3|9 4 6|5 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 7 8|9 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 2 3|1 4 6|    not the column positions.


N223 - Normal

Code: Select all

   Row 1: |1 2 3|9 4 6|5 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 7 8|9 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 1 3|2 4 6|    not the column positions.


N323 - Normal

Code: Select all

   Row 1: |1 2 3|9 4 6|5 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 7 8|9 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|5 1 2|3 4 6|    not the column positions.


N133 - Normal

Code: Select all

   Row 1: |1 2 3|9 4 5|6 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|1 7 8|9 2 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 2 3|1 4 5|    not the column positions.


N233 - Normal

Code: Select all

   Row 1: |1 2 3|9 4 5|6 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|2 7 8|9 1 3|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 1 3|2 4 5|    not the column positions.


N333 - Normal

Code: Select all

   Row 1: |1 2 3|9 4 5|6 7 8|    Note: The symbol positions in B2 and
   Row 2: |4 5 6|3 7 8|9 1 2|    B3 only describe the row they are in,
   Row 3: |7 8 9|6 1 2|3 4 5|    not the column positions.


[StrmCkr]

http://forum.enjoysudoku.com/post262717.html#p262717

here's an idea/ method for generating all unique essentially different grids,
alternative methods have been done as well which takes around 2-3 weeks to generate in full IE all
5,472,730,538 essentially different grids

unique grids x 9! {digit swaps}, 3!^8{TRANSFORMATIONS} = total number of grids.
5472730538 * 36280 * 3359232 = 6671248172291458990080

which also verified the total number of all Sudoku grids with a valid single solution.