Group Therapy

Grouping is a way to uncover more slinks and create more X-chains and X-loops.  The X-candidates in a chute are grouped together.  The group is “true” if it contains a true candidate, and “false” if it does not.

Groups can function as nodes in an X-chain!  A group and an X-candidate form a slink if no other candidates are in the line.  The slink replaces a wink between any two of the candidates.  Groups in two chutes of a box are slink partners if they share no X-candidates and cover the X-candidates of the box. 

Have a look at this rather outlandish example of a grouped confirming X-loop.  Singles and groups are mixed to confirm that the NEr2 group is true.  We don’t know which candidate within that group is true, but we know of two X-candidates that see this group, and can be removed. A toxic group! Notice that the X-candidates in NE could have been grouped in column chutes instead. Then you would have a wink in r2 and a slink in c8, with no removals.

I would probably never have found this weird removal in solving by actually looking for it. Let’s imagine how we could have found our way to a now classic grouping result from Andrew Stuart’s  It is currently his Figure 2 in Grouped X-Cycles, but don’t look yet.  Imagine we are trying to connect slinks into some kind of X-chain or loop.  We’re having some success, but can go no further because slinks are at more than two winks apart.

Trying to make something of the chain we have, we see how to extend the chain from the r4 slink.  Just exploring. 

Hmm. Looks a little bit like Mr. Weird up there.  We can wink upward to singles or slink to a group, but the singles we reach will require two more consective winks to close on a slink, and the group stops the chain .  How about crossing over on r7? Know what? There’s a slink up and . . . Whoa, do you see it? There’s a wink completing a nice loop. Not exactly the one  expected, but nice. The loop, that is. 

Now look for victims. Any candidate seeing both ends of any link.  Want to tie this one down and check on the next post?  Or why not check it out on Andrew’s site?

But before you go, test your knowledge of group theory – oops – I mean Sudoku grouping.  Here’s another panel from Andrew Stewart’s . It’s Figure 4 in Grouped X-Cycles.  Andrew notes that a newer version of his solver now finds an elimination X-chain without grouping,  in preference to the more complex grouped confirmation loop. Find that if you can. And if it’s not too much trouble, how about finding me the non-grouped confirmation loop I was lacking in my earlier post. Thanks.

After this, I’m going to be looking more closely at my X-panels.  How about you?

About Sudent

I'm John Welch, a retired engineering professor, father of 3 wonderful daughters and granddad to 7 fabulous grandchildren. Sudoku analysis and illustration is a great hobby and a healthy mental challenge.
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