What I say here I have already (more or less) said in the Sudoku.org.uk forum (Eureka thread), where it has generated much discussion. But I can see there are different participants here. And it may be good to have a short summary of what I said previously.

Some symmetries (in the general mathematical sense) of Sudoku are obvious and well known:

- you can permute rows and columns,

- you can permute rows (or columns) in the same group of 3,

- you can permute whole groups of 3 rows (or columns)

…

I'm not going to speak of these obvious symmetries.

The topic of this post is other generalised symmetries, that had never been explicited and which I have called supersymmetries in my book "The Hidden Logic of Sudoku".

Roughly speaking, they consist of permuting rows and numbers or colums and numbers. But when you say things that way, it is incorrect, because these supersymmetries are not valid in general, they are subject to some precise restrictions. And nobody can claim to know or understand these supersymmetries if he doesn't also know or understand these restrictions.

I have proven the following theorems:

- a block-free formula (or rule if you prefer), i.e. a formula that does not mention blocks/boxes, is valid for Sudoku if and only if it is valid for Latin Squares (i.e. without the constraint on blocks);

- for any valid block-free resolution rule (and only for such rules), the rules obtained by applying to it the above supersymmetries are valid.

These theorems cannot be extended to non-block-free resolution rules - so that saying (as I've sometimes seen written) that this is a severe limitation can only be a proof of misunderstanding. (Said otherwise, maths are not ruled by our desires.)

Simplest examples:

- from Naked-Subsets rules, one gets Hidden-Subsets rules

- from Naked-Subsets rules, one also gets Super-Hidden-Subsets rules, commonly named X-Wing, Swordfish and Jellyfish.

Part (and only part) of the factual symmetry relationships between these rules have been locally known for some time, although the above general transposition theorems were not formulated.

Having formalised the axioms for Sudoku Resolution Theories and proved the above general theorems allows to go much further than the above mentioned local remarks on relationships between Subsets.

Indeed, to any block-free rule in rc-space there corresponds a rule in each of two new spaces, the rn- and cn- spaces.

As a consequence, you can take almost any (in fact any block-positive) resolution rule R and restrict it to cases which do not make use of blocks. You get a specialisation of R, call it R'. R' is not interesting in itself, but it can now be transposed to both of the rn- and cn- spaces, using the above general theorem. Now you get something new. And anything you used to do with R in the standard representation, you can also do, as easily, in the two rn- and cn- representations. You can even do it more easily, because, in these spaces links between cells are restricted to what is displayed as rows and columns.

In order to ease the application of the super symmetries in the general case, the rn- and cn- representations of the puzzle can been grouped into an Extended Sudoku Grid (a new version of which I have now made available on my web pages, with notations consistent with those used in the book) and I have written a systematic procedure for building them (so that writing a program or a spreadshhet for managing the coherency of the 3 subgrids would be easy) together with recommandations on how to use them.

Let us now see if the above transposition theorems bring anything new.

Firstly, to xy-chains, there correspond hxy-chains in rn- and cn- spaces. Here, "h" stands for "hidden", as in hidden-subsets. hxy-chains appear in rn- or cn- spaces as xy-chains would in the usual rc-space.

hxy-chains do not allow to solve new puzzles if you already know the complex generalised AICs, because they can be shown to be particular cases of them. But they allow to make some (not all) complex AICs appear as much simpler xy-chains in the proper rn- or cn- space. So that, if you didn't know AICs or if you find it difficult to spot them, you can anyway solve puzzles you couldn't solve before. hxy-chains make some complex chains appear simpler.

Secondly, I have also introduced new kinds of chains, the xyt-chains and xyzt-chains, that are of interest in themselves, because they are very powerful, although very simple, generalisations of xy-chains and of respectively XY-Wing and XYZ-Wing. xyt-chains and xyzt-chains cannot be reduced to any known resolution rule.

Moreover, to these new chains there correspond new chains in rn- and cn- spaces, the hxyt-chains and hxyzt-chains. And the rules for the new hxyt-chains and hxyzt-chains cannot be reduced to any known rule.

Finally, what I consider noticeable is that with only these types of chains (plus of course the elementary rules for Subsets and Interactions) you can solve 97% of the randomly generated puzzles - without resorting to any of the much more complex rules that have been devised. It does not mean that these complex rules are no longer justified. But it means that there are more puzzles you will be able to solve without them.