## Orthogonal sudoku

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

### Orthogonal sudoku

Here comes one of Paul Vaderlind's orthogonal sudokus (orthogonal in the same meaning as for latin squares).
Fill each cell with a two-digits number from 11 to 99, without zero digit, in such a way that
a) The first digits represent one regular sudoku grid.
b) The second digits represent one regular sudoku grid as well.
c) (orthogonality) Each number from 11 to 99 (without the digit 0) occurs exactly once in the grid.

59 .. .. 18 .. .. 84 .. ..
.. .2 .. .. .. 94 .. 43 ..
1. .. 74 .. 63 .. .. .. 98
.6 9. .. .5 7. .2 6. .. .1
6. .. .2 2. .. 5. .9 .. 1.
.4 .. 1. .6 3. .3 .. 9. .2
93 .. .. .. 15 .. 31 .. 6.
.. 85 .. 91 .. .. .. .9 ..
.. .. 47 .. .. 69 .. .. 25
giant

Posts: 18
Joined: 09 August 2005

now the question comes, how many orthogonal pairs of
sudokugrids are there ?
How many of the 5e9 grids have an ortogonal mate ?

You could see this as a constraint programming problem with
81 variables, all different, with domains {11..99} and with
sudoku-constraints on both digits.
dukuso

Posts: 479
Joined: 25 June 2005

I would say that a better question is how many PARWISE orthogonal sudoku grids one can construct. It is known that one can constuct eight parwise orthogonal latin squares of order 9. How many parwise orthogonal sudoku grids? Can you imagine how nice puzzles you could then make?
giant

Posts: 18
Joined: 09 August 2005

### Re: Orthogonal sudoku

A typical sudoku puzzle is 9x9 or 3^2x3^2. The maximum size of a complete family of mutually orthogonal sudoku puzzles where n=k^2 (for instance when n=9 or 3^2) is k^2 - k. So the upper bound of MOSP is 6 when n=9.
ncoursey

Posts: 1
Joined: 12 August 2011

### Re: Orthogonal sudoku

The same variant has been mentioned in this thread: "Graeco-Latin Soduku" challenge

tarek

tarek

Posts: 2647
Joined: 05 January 2006

### Re: Orthogonal sudoku

tarek wrote: .... ncoursey, your reply comes nearly 6 years after the last post ....

-- And so does my attempt to solve it.

It took a few hours, just using elementary techniques. Every so often I had to ask myself a question like, "Which box could contain a 47?" Not exactly a systematic solving method, to be sure.

Bill Smythe
Smythe Dakota

Posts: 543
Joined: 11 February 2006