Bridging Coloring Clusters

With slinks left out of our Medusa coloring cluster, we can often form a significant second cluster, or maybe a third. Then the cluster colors interact through candidates of the same number that see each other or through colors of different clusters in the same cell. Let’s examine the logic with the checkpoint grid of Maestro 4.

In c8, red and blue 3’s show that

(red AND blue)

is impossible. Since at least one of these colors  must be false, then

(orange OR green )

is a true color. It could be both.

This is a bridge. The two clusters can now work together to form new toxic sets.  Any candidate seeing both orange and green in its number is false.

Now if an extension of either cluster allows orange and green candidates of a number to see each other,  the clusters merge – red becomes blue and orange becomes green.

In Maestro 4 there is more:  Red and green candidates in r9c9 show that

(red AND green )

is impossible, so

(orange OR blue)

is also true.  Since either blue or green is false, the two true statements

(orange OR green ) AND (orange OR blue)

require that orange be true.

Another way to look at it is 

not(red AND blue) and not(red AND green) => not red,

since blue or green is true.

The orange clues quickly reveal blue as true: 

Orange => (W1, E6=>SE6=> S2=>blue)

and that finishes Maestro 4.

That’s the logic of sophisticated bridging, which is not exactly merging of clusters, but is certainly a coordination of them.  A much simpler bridge occurs when two clusters share a unit and a removal leaves only the two cluster candidates in the unit. The new slink produces a direct correspondence between the colors of the two clusters, and they become one.  An example will come up later in this blog.

You may like to work on an example puzzle for the next post of June 6,  which illustrates the enhancing effect of coloring on Sue de Coq and APE patterns. It’s from Delta Sky magazine of June 2008. It was the second, harder puzzle, titled “Even More of a Challenge”, and it really was. I learned a lot from my mistakes on this puzzle.  Suppose I tell you now what to find on the way to coloring, and how to start the blue/green cluster, to match the checkpoint.  Box and line marking are routine, but it takes an XYZ wing and a successful Sue de Coq to create the slinks for coloring. I found no XY-chain or X-chain or fish removals.  Start with blue on the top left 1-candidate.

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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