## Extended Sudoku Board

Advanced methods and approaches for solving Sudoku puzzles

### Extended Sudoku Board

In my recent book "The Hidden Logic of Sudoku", I have analysed in full detail and formalised all the logical symmetries of Sudoku. Not only have I fully explicited the relationships between Naked subsets, Hidden subsets and Fishy patterns, which seem to have been partly known for some time (see the post by Arcilla), I have also introduced a universal representation that allows to deal uniformly with these symmetries, the rn- and cn- spaces.

As my approach is totally player oriented (instead of programmer), I have devised an Extended Sudoku Board that allows to use these representations in practice: these new "spaces" can be appended to the standard Sudoku board. This extended board cannot easily be displayed in the format of this forum, but you can download it from my web page dedicated to this book (http://www.carva.org/denis.berthier/HLS). On the same page, you will also find instructions on how to build and how to use this extended board, together with a substantial excerpt of the book. Feel free to use it and tell me your opinion about it. As any new tool, you may need some time to get used to it, but I think it is worth making the effort.

The general idea is that you can play as usual on the sandard part of this extended board and you can apply the rules of Latin Squares (Sudoku without block constraints) on the other two sub-boards. You maintain the consistency between the three sub-boards by asserting a value or eliminating a candidate on the three sub-boards at the same time.

What are the advantages of this extended board?

First:
- hidden subsets appear as naked subsets,
- fishy patterns (X-wing, swordfish and jellyfish) also appear as naked subsets; for this reason, I call them the super-hidden subsets.
But if these were the only advantages, you could probably forget all about this: although it may ease the spotting of the fishy patterns, no new pattern in obtained at this stage.

More interestingly, the above symmetries can be pushed further to define new rules.
I have thus introduced hidden xy-chains (or hxy-chains). In each of the two new sub-boards, they appear exactly as xy-chains would in the standard board and I have proved that, in their sub-board, they allow the same eliminations as the usual xy-chains do in the standard board. The only difference is that the cells in these two new sub-boards cannot be linked by blocks, but only by rows and columns (in this sense, hxy-chains are even simpler than xy-chains).
Hidden xy-chains are a very powerful tool. They may correspond to very complex AICs in the ordinary board.

I have also defined a natural (from a logical standpoint) and simple extension of xy-chains, the xyt-chains. Of course, the xyt-chains have counterparts in the new sub-boards, the hxyt-chains.

Consider the following rules:
- elementary constraints along rows, columns and blocks
- subset rules : naked, hidden and super-hidden
- interaction rules (between blocks and rows or columns)
- xy-wing, wyz-wing
They are generally consider as the basics for a medium level Sudoku player.

Apart from these rules, some complex or very complex techniques have been devised by and for the expert players.
What my approach allows is not only solving a small part of the as yet unsolved top level puzzles, but making these very complex techniques unnecessary for more puzzles than usual. In practice, this means that you will be able to solve more puzzles than you could solve before.

xyt-chains and hxy-chains have already been discussed in the Eureka section of the http://www.sudoku.org.uk forum.
denis_berthier
2010 Supporter

Posts: 1261
Joined: 19 June 2007
Location: Paris

I am sorry to say that "re'born" and "Mauricio" do not
support me in general.

I think that they will buy it.
wintder

Posts: 297
Joined: 24 April 2007

I think I've forgotten to signal it here, but I've put online a new (still free)version of this extended grid, with notations (R1 C1 N1 instead of A a 1) fully consistent with those in my book.
denis_berthier
2010 Supporter

Posts: 1261
Joined: 19 June 2007
Location: Paris