The New Sudoku Players' Forum

[Mathimagics]

Magic Giant (partial)

MagicGiant81-Partial.png (55.11 KiB) Viewed 863 times

The Magic Giant Sudoku is an 81 x 81 Sudoku with a twist - each of the 81 boxes (9x9 subgrids) consists of a pair of orthogonal 9x9 Sudoku's. The main grid still satsifies the 81x81 Sudoku constraints, of course - all of the 2-digit values in each of the 81 cell rows, columns, and boxes are different.

The image above just shows the top-left 9 subgrids. The complete grid image is attached below.

No puzzles yet, I've been working for some time on determining whether grids like this even exist - but indeed, they do!

There are also Magic Giant 256 x 256's, if anyone's interested ...

MagicGiant81.zip

Magic Giant Grid (complete)

(254.6 KiB) Downloaded 150 times

[tarek]

Wow well done!

[Edit: Removed part of the reply as it wasn't correct]

Tarek

[Serg]

Hi, Mathimagics!
It's unbelievable! Just to find a pair of orthogonal Sudokus is not simple thing, but to find the set of mutually compatible 81 pairs of orthogonal Sudokus is very surprising. Are all 81 pairs of orthogonal Sudokus essentially different?

Serg

[tarek]

had another look and it looks to be identical to my Sudoku^2 (which is slightly different from the classic orthogonal sudoku)

Read here http://forum.enjoysudoku.com/tarek-s-sudoku-2-puzzles-squared-power-of-2-t30338.html

The above uses the unit box of 3x3 instead of the unit box of 2x2 the and having 3x3 9x9 grids instead of 9x9 9x9 grids to make it a full Sudoku^2
With this one you will find that each long line has 27 different pairings which are displayed in the (n/9, n mod 3) format

In the classic sudoku^2 version there has to be 81 9x9 grids that will give you the essentially 81 different pairings but that would be an 81x81 grid.

I suspect that the 256 version that Mathimagics is referring to is a 4x4 16x16 which will be a 64x64 grid (64 different pairings)

tarek

[Mathimagics]

Incidentally, the smallest MG of all is an MG-16. It has the advantage of being easily presented and viewed! Here is an example:

MG-16 Example: Show

Code: Select all

++=======+=======++=======+=======++=======+=======++=======+=======++
|| 11 22 | 33 44 || 12 43 | 34 21 || 13 24 | 31 42 || 14 41 | 23 32 ||
|| 34 43 | 12 21 || 31 24 | 13 42 || 32 41 | 14 23 || 22 33 | 11 44 ||
++-------+-------++-------+-------++-------+-------++-------+-------++
|| 23 14 | 41 32 || 44 11 | 22 33 || 21 12 | 43 34 || 31 24 | 42 13 ||
|| 42 31 | 24 13 || 23 32 | 41 14 || 44 33 | 22 11 || 43 12 | 34 21 ||
++=======+=======++=======+=======++=======+=======++=======+=======++
|| 12 23 | 44 31 || 32 13 | 24 41 || 33 42 | 21 14 || 11 34 | 22 43 ||
|| 41 34 | 13 22 || 21 44 | 33 12 || 24 11 | 32 43 || 23 42 | 14 31 ||
++-------+-------++-------+-------++-------+-------++-------+-------++
|| 24 11 | 32 43 || 14 31 | 42 23 || 41 34 | 13 22 || 44 21 | 33 12 ||
|| 33 42 | 21 14 || 43 22 | 11 34 || 12 23 | 44 31 || 32 13 | 41 24 ||
++=======+=======++=======+=======++=======+=======++=======+=======++
|| 13 24 | 31 42 || 41 14 | 23 32 || 11 22 | 33 44 || 12 43 | 21 34 ||
|| 32 41 | 14 23 || 22 33 | 44 11 || 34 43 | 12 21 || 24 31 | 13 42 ||
++-------+-------++-------+-------++-------+-------++-------+-------++
|| 21 12 | 43 34 || 13 42 | 31 24 || 23 14 | 41 32 || 33 22 | 44 11 ||
|| 44 33 | 22 11 || 34 21 | 12 43 || 42 31 | 24 13 || 41 14 | 32 23 ||
++=======+=======++=======+=======++=======+=======++=======+=======++
|| 14 21 | 42 33 || 11 34 | 43 22 || 31 44 | 23 12 || 13 32 | 24 41 ||
|| 43 32 | 11 24 || 42 23 | 14 31 || 22 13 | 34 41 || 21 44 | 12 33 ||
++-------+-------++-------+-------++-------+-------++-------+-------++
|| 22 13 | 34 41 || 33 12 | 21 44 || 43 32 | 11 24 || 42 23 | 31 14 ||
|| 31 44 | 23 12 || 24 41 | 32 13 || 14 21 | 42 33 || 34 11 | 43 22 ||
++=======+=======++=======+=======++=======+=======++=======+=======++


And here is the same grid labelled as a normal Sudoku 16 x 16:

MG-16 as Sudoku 16x16: Show

Code: Select all

 0 5 A F | 1 E B 4 | 2 7 8 D | 3 C 6 9
 B E 1 4 | 8 7 2 D | 9 C 3 6 | 5 A 0 F
 6 3 C 9 | F 0 5 A | 4 1 E B | 8 7 D 2
 D 8 7 2 | 6 9 C 3 | F A 5 0 | E 1 B 4
 -------------------------------------
 1 6 F 8 | 9 2 7 C | A D 4 3 | 0 B 5 E
 C B 2 5 | 4 F A 1 | 7 0 9 E | 6 D 3 8
 7 0 9 E | 3 8 D 6 | C B 2 5 | F 4 A 1
 A D 4 3 | E 5 0 B | 1 6 F 8 | 9 2 C 7
 -------------------------------------
 2 7 8 D | C 3 6 9 | 0 5 A F | 1 E 4 B
 9 C 3 6 | 5 A F 0 | B E 1 4 | 7 8 2 D
 4 1 E B | 2 D 8 7 | 6 3 C 9 | A 5 F 0
 F A 5 0 | B 4 1 E | D 8 7 2 | C 3 9 6
 -------------------------------------
 3 4 D A | 0 B E 5 | 8 F 6 1 | 2 9 7 C
 E 9 0 7 | D 6 3 8 | 5 2 B C | 4 F 1 A
 5 2 B C | A 1 4 F | E 9 0 7 | D 6 8 3
 8 F 6 1 | 7 C 9 2 | 3 4 D A | B 0 E 5


MG's have subgrid box sizes 2x2, 3x3, 4x4, etc. The grid size is 2^4 = 16, 3^4 = 81, 4^4 = 256 etc..

[Mathimagics]

had another look and it looks to be identical to my Sudoku^2 (which is slightly different from the classic orthogonal sudoku)

The above uses the unit box of 3x3 instead of the unit box of 2x2 the and having 3x3 9x9 grids instead of 9x9 9x9 grids to make it a full Sudoku^2

Tarek, my friend, have a third look! The image is just a partial one ... the full grid is too big to attach directly, so I attached the complete 81 x 81 image in a zip file.

I suspect that the 256 version that Mathimagics is referring to is a 4x4 16x16 which will be a 64x64 grid (64 different pairings)

I would have called that an MG-64, but 64 doesn't have a 4th root, so there's no MG-64 ... (see above)

Cheers
MM

[Mathimagics]

It's unbelievable! Just to find a pair of orthogonal Sudokus is not simple thing, but to find the set of mutually compatible 81 pairs of orthogonal Sudokus is very surprising. Are all 81 pairs of orthogonal Sudokus essentially different?

Hello Serg!

Nice to hear from you!

Finding pairs of orthogonal 9x9 Sudoku's is actually quite easy. It took only 2 days (using 8 cores) to count the orthogonal pairs for all of the ED grids. It worked out at about 8500 grids/second for each core.

You can find my results, recently posted in Orthogonal Sudoku.

As for the problem of finding a set of 81 pairs of mutually compatible orthogonal Sudoku's, well, I spent months looking for them the first time around (back in 2018), with no success. Now I have revisited the problem, and encountered the same difficulties ...

... but yesterday an idea ( ) suddenly occurred to me, and that led to a simple and direct construction method.

I will reveal this algorithm in due course, but meanwhile I wonder whether you might be able to intuitively work it out, if I give you a big clue - the answer to your question is "No". We can build an MG-81 directly using just a single pair of orthogonal 9x9 grids.

Cheers
MM

[tarek]

had another look and it looks to be identical to my Sudoku^2 (which is slightly different from the classic orthogonal sudoku)

The above uses the unit box of 3x3 instead of the unit box of 2x2 the and having 3x3 9x9 grids instead of 9x9 9x9 grids to make it a full Sudoku^2

Tarek, my friend, have a third look! The image is just a partial one ... the full grid is too big to attach directly, so I attached the complete 81 x 81 image in a zip file.

Ok then it is exactly a Sudoku^2. Your 2x2 unit box and the example is identical to what I presented there. I didn't manage to see to the bigger picture so I thout that it was a smaller grid.

3x3 unit is something I've been trying to do for a while so well done for doing that.

the 4x4 unit box your (256x256) can be done as a Sudoku^3 with a unit 2x2 within it … That is something I suspect you of being very close to achieve and well done if that has been done.

I'll be interested to see if you've produced some puzzles based on these for me to attempt solving. creint has adapted his solver to solve them so his solver can be used to independently verify them.

I hate renaming things so I may stick to using the name I used for mine unless with time I'm convinced that the terms you use to describe them have a better ring to them

This was puzzle 01:

Code: Select all

.. .. .. ..  .. .. .. ..  .. .. .. ..  .. .. .. .. 
.. .. .. 21  .. 44 .. ..  .. .. 14 ..  31 .. .. .. 
.. .. .. ..  .. .. 42 ..  .. 12 .. ..  .. .. .. .. 
.. 31 .. ..  .. .. .. ..  .. .. .. ..  .. .. 21 .. 

.. .. .. ..  .. .. .. ..  .. .. .. ..  .. .. .. .. 
.. 44 .. ..  .. .. .. 21  31 .. .. ..  .. .. 14 .. 
.. .. 42 ..  .. .. .. ..  .. .. .. ..  .. 12 .. .. 
.. .. .. ..  .. 31 .. ..  .. .. 21 ..  .. .. .. .. 

.. .. .. ..  .. 23 .. ..  .. .. 33 ..  .. .. .. .. 
.. .. 14 ..  .. .. .. ..  .. .. .. ..  .. 44 .. .. 
.. 12 .. ..  .. .. .. 33  23 .. .. ..  .. .. 42 .. 
.. .. .. ..  .. .. .. ..  .. .. .. ..  .. .. .. .. 

.. 23 .. ..  .. .. .. ..  .. .. .. ..  .. .. 33 .. 
.. .. .. ..  .. .. 14 ..  .. 44 .. ..  .. .. .. .. 
.. .. .. 33  .. 12 .. ..  .. .. 42 ..  23 .. .. .. 
.. .. .. ..  .. .. .. ..  .. .. .. ..  .. .. .. ..

and puzzle 15

Code: Select all

3.   ..   ..   ..   ..   ..   ..   1.   2.   ..   ..   ..   ..   ..   ..   1.
..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..
..   ..   34   ..   ..   31   ..   ..   ..   ..   11   ..   ..   23   ..   ..
..   ..   ..   4.   1.   ..   ..   ..   ..   ..   ..   2.   1.   ..   ..   ..
..   ..   ..   3.   4.   ..   ..   ..   ..   ..   ..   2.   3.   ..   ..   ..
..   ..   21   ..   ..   23   ..   ..   ..   ..   13   ..   ..   22   ..   ..
..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..
3.   ..   ..   ..   ..   ..   ..   1.   1.   ..   ..   ..   ..   ..   ..   1.
3.   ..   ..   ..   ..   ..   ..   1.   3.   ..   ..   ..   ..   ..   ..   1.
..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..
..   ..   12   ..   ..   13   ..   ..   ..   ..   14   ..   ..   11   ..   ..
..   ..   ..   4.   3.   ..   ..   ..   ..   ..   ..   2.   4.   ..   ..   ..
..   ..   ..   2.   3.   ..   ..   ..   ..   ..   ..   1.   3.   ..   ..   ..
..   ..   11   ..   ..   12   ..   ..   ..   ..   23   ..   ..   42   ..   ..
..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..   ..
3.   ..   ..   ..   ..   ..   ..   4.   1.   ..   ..   ..   ..   ..   ..   3.

tarek

[Mathimagics]

I hate renaming things so I may stick to using the name I used for mine unless with time I'm convinced that the terms you use to describe them have a better ring to them

Ok, "Magic Giant" kind of rings my bell, but at least I can see now that your original Sudoku^2 puzzles are indeed synonymous with my MG's (MG-16), so hats off to you for the original concept!

I hope to be to deliver some MG-81 puzzles over the next few days for you and creint to try.

An interesting fact about MG-16's. There is really only one ED pair of orthogonal 4x4 Latin Squares. Fortunately they are also Sudoku squares, as our 16 x 16 puzzles would not be possible otherwise! But it was this property that made me think about a direct approach to the MG-81 grid construction problem.

Now, about "Sudoku^3" ... I think I've worked out what you mean.

Here is a 4 x 4 box that might be part of a 16 x 16 orthog pair grid:

Code: Select all

14 B9 AB 63
A3 6B 19 B4
BB 13 64 A9
69 A4 B3 1B


This pairs 4 symbols on the left {1, 6, A, B} with 4 on the right {3, 4, 9, B} and does so such that it works as a 4x4 orthogonal Sudoku subgrid.

I assume that this is what you had in mind? It's a pretty severe restriction but it might just be possible ...

[tarek]

Now, about "Sudoku^3" ... I think I've worked out what you mean.

Here is a 4 x 4 box that might be part of a 16 x 16 orthog pair grid:

Code: Select all

14 B9 AB 63
A3 6B 19 B4
BB 13 64 A9
69 A4 B3 1B


So your MG-256 has 2 x 16 MG-16s . You will have 4 symbols in each cell. each 2 combine to form 1 of possible 16 symbols. All 4 symbols combine to form 1 of 256 possible symbols (playing card suits are suitable symbols too)

You can also restrict or expand further by limiting/unlimiting which 2 symbols in a cell can pair to form 1 of possible 16 symbols. In the most restricted version you may have 3 different possible pairing of symbols in a cell. You may then have 6x16 possible MG-16s in in the most restricted version of MG-256

tarek

[Serg]

Hi, Mathimagics!
I missed your "Orthogonal Sudoku" thread. Good work!
About the method to produce Giant Sudoku 81 x 81 cells.
It seems, simple method of cyclically shifting up rows/bands and cyclically shifting left columns/stacks (in MC grid style) of the given orthogonal pair works well.

1. We have an Orthogonal Sudoku grids pair (9 x 9 cells).
2. To construct 9 x 81cells row we should add to given 9 x 9 cells orthogonal pair on the right the same 9 x 9 cells pair with cyclically shifted up rows in the same bands. We can construct 9 x 27 cells fragment in this manner.
3. Now we should shift up cyclically bands and continue shifting rows up to build another 9 x 27 cells fragment. At the finish we'll get 9 x 81 cells fragment ("giant row").
4. To build another "giant row", we should use the first "giant row" (9 x 81 cells), but with columns cyclically shifted left. In such manner we can build 27 x 81 cells fragment ("giant band").
5. To construct other "giant bands" we should shift stacks cyclically left in the every 9 x 9 cells fragments.

It seems to me, this strategy avoids cells duplication in a row/column and provides non-repeating cell values in a row/column.

Serg

[Mathimagics]

Hi Serg!

Your method works fine, nice one!

So, in the space of a few days, we have each found a different method for direct construction of Magic (Sudoku^2) grids from a single pair (A, B) of orthogonal subgrids.

My method is remarkably simple. Here A and B are the orthog pair, and GA(1 to 9, 1 to 9) and GB(1 to 9, 1 to 9) are the "giant box" arrays (each element is a 9x9 grid).

• for all x,y set GA(x,y) = A

• for x = 1 to 9
  
  • for y = 1 to 9
    
    • set GB(x,y) = B
    • increment B (modulu 9)
  • increment B (modulu 9)

The "increment B" operation is just a re-labelling of B, with a cyclic shift of the cell values.

Thus we construct a complete MG81 = GA|GB from just 9 different labellings of grid B.

[Serg]

< Withdrown >

[tarek]

When I decided to do this variant I was using DLX. There were extra columns for N/9 and N%9 … Then I had to change it again & double the columns to allow orthogonal clue separation … tedious but fun!!

tarek

[Mathimagics]

I think Magic Giant Sudoku is just another representation of "traditional" N^4 x N^4 Sudoku.

I did say that from the very start:

The Magic Giant Sudoku is an 81 x 81 Sudoku ...

The "magic" (or Sudoku^2) grid has all the properties of a normal 81 x 81 Sudoku (ie with box size 9 x 9), plus the additional "magic" (Sudoku^2) property, namely that when each cell value (1 ... 81), is assigned a mapped pair [a,b] (a, b both in {1 ... 9}), then each 9 x 9 box corresponds to a pair of orthogonal 9 x 9 Sudoku's.