Sudoku solutions...: Clarification

[hv at crypt.org]

zak seidov <zakseidov at yahoo.com> wrote:
:
:Dear seqfans,
:let me be more clear:
:
:any sudoku solution may be transformed such that
:the numbers in first row, block and column
:are as below:

I don't believe this is true.

:>  1 2 3 | 4 5 6 | 7 8 9
:>  4 5 6 | ? ? ? | ? ? ?
:>  7 8 9 | ? ? ? | ? ? ?
:>  ------+-------+----
:>  2 ? ? | ? ? ? | ? ? ?
:>  3 ? ? | ? ? ? | ? ? ?
:>  5 ? ? | ? ? ? | ? ? ?
:>  ------+-------+------
:>  6 ? ? | ? ? ? | ? ? ? 
:>  8 ? ? | ? ? ? | ? ? ?
:>  9 ? ? | ? ? ? | ? ? ? 
:
:To do it, 
:first we simply "rename" numbers such that
:first block looks like this:
:1 2 3 | 
:4 5 6 | 
:7 8 9 |
:------+ 
:then we rearrange the 4-9 columns such that
:first row looks like this:
:1 2 3 | 4 5 6 | 7 8 9

If at this point the top looks like:

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

.. it seems to me your suggested rearrangement is not possible.
While it is safe to swap two rows or columns within a block, or a row
or column of blocks with another, you cannot swap a row or column
from one block to another.

The best simple approach I know to find a lexically early equivalent is:
- rename the numbers so that top row is 1-2-3-4-5-6-7-8-9
- reorder rows 2-3, 4-6 and 7-9 so that the first digits ascend
- swap block of rows 4-6 with 7-9 if needed.

However it may be possible to get lexically earlier results by
doing some reordering of rows, columns or blocks before renumbering
the top line; I don't know how to go about guaranteeing a canonical
lexically earliest result without trying all the possibilities.

Hugo van der Sanden