Hear me out

Everything about Sudoku that doesn't fit in one of the other sections

Hear me out

This:

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

is equivalent to this:

9 6 8 | 3 1 7 | 5 4 2
7 5 1 | 2 9 4 | 8 3 6
4 2 3 | 6 8 5 | 1 7 9
-------+-------+------
1 7 5 | 4 2 9 | 6 8 3
3 4 2 | 5 6 8 | 9 1 7
8 9 6 | 7 3 1 | 2 5 4
-------+-------+------
2 3 4 | 8 5 6 | 7 9 1
6 8 9 | 1 7 3 | 4 2 5
5 1 7 | 9 4 2 | 3 6 8
sciguy47

Posts: 9
Joined: 17 July 2005

what?
skilly91

Posts: 3
Joined: 17 July 2005

Of course that's true!

Rotate the 2nd grid by 90 degrees clockwise, then do some row and column exchanges, follow by some digit subsititution... you should get the 1st grid somehow. I'm not going into details there.
udosuk

Posts: 2698
Joined: 17 July 2005

sciguy, if what you say is true, then all sudokus are equivalent, are they not
MCC

Posts: 1275
Joined: 08 June 2005

These two are connected in a very special way.

They acually represent the same tesseract in 4D, but have been "sliced" and laid out in different ways.
sciguy47

Posts: 9
Joined: 17 July 2005

sciguy47 wrote:These two are connected in a very special way.

They acually represent the same tesseract in 4D, but have been "sliced" and laid out in different ways.

Aaahh, I get it. Your grids can be addressed by coordinates (R,C,r,c) where R,C specify the 3x3 box and r,c specify the cell within that box. Write out the first grid as 81 5-tuples (R,C,r,c,d) where d is the digit at position (R,C,r,c). For example, the leading diagonal is:
(1,1,1,1,9)
(1,1,2,2,4)
(1,1,3,3,7)
(2,2,1,1,5)
(2,2,2,2,6)
(2,2,3,3,2)
(3,3,1,1,3)
(3,3,2,2,1)
(3,3,3,3,8)
The second grid just takes the (R,C,r,c,d) list from the first and reads it back as (r,c,C,R,d).

You can get away with this because you've insisted on the condition, not normally found in sudoku, that no digit appears in the same position in more than one box. There are 201105135151764480 grids that satisfy that criterion, or about 0.003% of the full set of 6.67x10^21 unrestricted grids.

Since you've gone further and put 1-9 down the two major diagonals as well, it might interest you to know (or it might not!) that there appear to be only 48534220224000 grids with that property in addition to the property mentioned previously. Any of these will preserve the diagonal feature under the particular transformation (R,C,r,c,d) -> (r,c,C,R,d) that you used.
Red Ed

Posts: 633
Joined: 06 June 2005

Actually, the reason I can get away with cutting it up differently and have it still remain a valid Sudoku grid is that every plane on the tesseract has the numbers 1-9 only once. Each type of plane on the tesseract can represented by the two co-ordinates in which it varies. For example, The Rr plane is repesented by a set of numbers where the R or r co-ordinates are different for each of the numbers and C and c are all the same. To be able to get away with swiching the co-ordinates like that you have to have all 6 types of planes (Rr,Rc,RC,Cr,Cc,rc) have numbers 1-9 only once. The diagonal thing was a happy accident.
sciguy47

Posts: 9
Joined: 17 July 2005

sciguy47 wrote:To be able to get away with swiching the co-ordinates like that you have to have all 6 types of planes (Rr,Rc,RC,Cr,Cc,rc) have numbers 1-9 only once. The diagonal thing was a happy accident.

Oops, yes, I missed the Rc and Cr planes. Duh. Well in that case I think we have only 9! x 938 tesseract sudokus, of which, viewed in some pre-assigned way, 9! x 104 have nice long diagonals.
Red Ed

Posts: 633
Joined: 06 June 2005

Red Ed wrote:
sciguy47 wrote:To be able to get away with swiching the co-ordinates like that you have to have all 6 types of planes (Rr,Rc,RC,Cr,Cc,rc) have numbers 1-9 only once. The diagonal thing was a happy accident.

Oops, yes, I missed the Rc and Cr planes. Duh. Well in that case I think we have only 9! x 938 tesseract sudokus, of which, viewed in some pre-assigned way, 9! x 104 have nice long diagonals.

I get 9!*104 tesseract sudokus of size 3*3*3*3.
I didn't bother about the diagonals.

Now we have a 3*3*3* tesseract, but we put numbers 1..9 into it,
which is not so nice. Let's take numbers 0..8 and write them
in base 3 ! That gives two orthogonal 3*3*3*3 tesseracts
filled with numbers 0,1,2 . Can we get them both to be
latin tesseracts ? Sadly it seems that this is not possible :-(
Is it possible for 4*4*4*4 ? Seems it's not. Othen n : also doubtful.
Maybe it can be proved that it's impossible for all n ?

Define a 9^d sudoku in d dimensions, as a latin hypercube with the additional property that any 9*9-plane is a normal sudoku-grid.
We had such hyper-sudokus for d=2 and d=3. Do they exist
for any d ?
dukuso

Posts: 479
Joined: 25 June 2005