Hypothesis -- any Sudoku matrix can be transformed into any other Sudoku matrixes through swapping digits.

Verification: I have verified the hypothesis in 4*4 Sudoku matrix, however, I am not able to verify it in the normal 9*9 Sudoku matrix due to its complexity.

Swapping digits: after swapping some digits, a valid Sudoku matrix will still remain a valid Sudoku matrix.

Swapping digits happens when: 1. rotation (all digits swap with the digits against the centre point); 2. Swap columns/rows of boxes(such swapping the top three blocks with the middle three blocks); 3. Reflection (similar to No. 2 swap except the border between boxes acts as a mirror); 4. swapping rows or columns; 5. Renumbering (swapping all digits of any two like swapping all '1' and '2' digits); 6. Swapping digits along an open chain (discussed in detail below).

Some of the above contents come from Condor's:

http://forum.enjoysudoku.com/viewtopic.php?t=44&postdays=0&postorder=asc&start=210

Relationship between these swappings: 1. Rotation can be achieved by several boxes swapping; 2. reflection can be achieved by boxes swapping plus row/column swapping; 3. boxes swapping can be achieved by renumbering plus row/column swapping (still need to be proved strictly); 4. Renumbering and row/column swapping can be achieved by swapping along open chains.

Summary of relationship between swappings: there are three main kinds of swaps, they are boxes swapping, rows/columns swapping and swapping through opening chains. Because boxes swapping and rows/columns swapping are special cases of swapping though open chains, swappings can be divided into only two kinds of swapping: swapping through global open chain and swapping through local open chain -- shorten as open chain (will be discussed in details below).

Swapping through (local) open chains: when two different digits in two grids are swapped, by swapping these two digits in other grids, it will remain a valid Sudoku matrix. It forms a kind of chain and that is where its name comes from. More than two digits chain: if an opening chain (must be local chain) involves more than two digits, it can be divided into several two digits chain. I have found a few examples of this, however, I am not 100% sure because it stil needs a strict proof.

Take Condor's Sudoku Queen which is recently discussed in the forum as an example below:

453|671|928

921|853|674

876|249|153

-----------

594|182|367

162|397|485

387|465|291

-----------

715|924|836

238|716|549

649|538|712

If we look at block 1 at the left upper corner. If we swap digit '1' and '2' there, it will take 3 swaps, 8 digits involved (length=4) to remain a valid Sudoku matrix which is shown below. The rest of '1' and '2' form another local chain. However, if we swap r2c2 = 2 and r4c4 = 1, all digits of '1' and '2' must swap in order to remain valid because they are taken from different chains.

453|671|928

912|853|674

876|249|153

-----------

594|182|367

261|397|485

387|465|291

-----------

725|914|836

138|726|549

649|538|712

If you try all digits in block 1, you will find that most of the digits need 18 digits involved (8 swaps). The length of the chain varies from 2 to 9. If you find a length=x, the rest of the digits will form another chain with length = 9-x. Because swapping won't destroy any chains in either global or local level, so according to the hypothesis, all Sudoku matrixes will have same number and same kinds of local chains. I don't have time to verify this yet, it is very time consuming by hand. If someone is able to prove a few examples by programming, that will be great. If not true, then my hypothesis about Sudoku matrix must be wrong.

Some possible conclusion of the hypothesis:

Uniqueness = when no open chains exist = only have close chains (which needs to involve the swap with hints)

If a Sudoku matrix can be transformed so that it coincides with the hints of another Sudoku matrix, the transformed Sudoku must be the solution to the hints, i.e. equal to the Sudoku needs to be found out.

There are still many questions remained. And I hope the above will help to explore more about Sudoku matrix.