Coloring is magic. It invokes the power of the strong link in ways that normally escape our attention. In coloring, we construct networks of slinks, called clusters. Each candidate of a cluster has one of two colors. Each of this candidate’s slink partners has the opposing color.
In Systematic Sudoku we use a form generally known as Medusa coloring. Medusa coloring includes the internal slinks of bv. Numbers change as a slink chain passes through bv.
Clusters are like chains of dominoes laid out and ready to collapse when any one of them is pushed over. Except that, when the nudge comes, only half of the candidates fall down. The other half stand as clues. It’s a rare puzzle that survives a cluster collapse.
So why do sysudokies put off coloring so long in the order of battle? Well, we don’t always. But it generally takes many of the advanced techniques of the previous posts create the slinks to grow a decisive cluster.
We become aware when a decisive cluster is becoming possible, as our actions generate more slinks. Slinks and clues are generated by box constraints. As line constraints produce more, we mark line slinks and watch them for X-wings. Bv scan removals add more slinks. On a new X-panel we often see overlooked slinks. We attempt to build alternating X-chains connecting them. Sometimes we are less than diligent in our fishing, because the slinks look promising.
When the slinks look strong enough, we willingly color, because coloring on the grid does not interfere with other pattern marking, and converts the solving effort from candidate removal to color removal. Sometimes we actually start a cluster to better analyze a potential removal pattern, not expecting it to grow immediately to a decisive size.
In this post, I will be content with explaining how to create a cluster and why coloring works. Coming later are details and examples of sysudokie coloring as a technique and in combination with other techniques. My next post has an example of decisive coloring. So in case you would like to solve the example puzzle up to the point of coloring, and anticipate the result, I’ll give you the puzzle and a trace at the end of this post. If you need it, call up the page on sysudokie tracing on the heading menu above.
Coloring is an easy application of strong links. In a cluster of connected slinks, half the candidates are true, by definition of a strong link. If a candidate in a chain of slinks is false, the next one is true, the next false, and so on. Since we don’t know which is true, this one or the next, we assign alternating colors and transform the problem into one of determining which color is true.
The cluster doesn’t grow in a single chain. Connecting slinks branch off in box, row and, and column directions. We alternate colors in all directions, forming a cluster, as in grapes. Candidates of a cluster now literally hang together. Find one false, and you can remove all of the same color and promote candidates of the alternate color to the rank of Clue. The telegraphic inference power of a large cluster is enormous. Every candidate member slinks with – and winks at – all candidates of the alternate color.
As the cluster grows, candidates inconsistent with the cluster’s inference field are found out, and removed. Any candidate seeing both colors of its number is out. So is any candidate in a cell with both colors. The cluster can destroy itself, as it forces the same color onto candidates of the same number that see each other. You don’t need to remember a set of rules. These events are obvious when you understand what a cluster implies.
Coloring is easy to do on a computer with office support. On paper, it’s easy in theory. All you need is a pack of colored pencils, or a convention of two distinctive shapes to encircle cluster candidates. In practice, it’s a problem to carry solving that far and still be able to see the color markings amid all the other detail. On a sysudoku ©PowerPoint grid, select a candidate, pick a color, and click the candidate. Or better yet, pick the color and go through the grid, clicking the candidates of that color. Box marks, line marks and bv border squares are direct visual aids to coloring.
The cautionary discipline in coloring is to be very careful to extend color only by slink and never by wink or by pattern. The end candidates of an X-chain or an XY-chain are not strongly linked. Neither is any other toxic pair. In a slink pair, only one is true. In a toxic pair, at least one is true, but both can be. If you have gone wrong, the solution will tell you where it happened. It’s often a misinterpreted link.
Light blue and green are my favorite first cluster colors. If you want to match the checkpoint diagram of the next post, start with blue 4 in r1c2. Extend the cluster as far as you can, even if it goes beyond a removal discovery. A cluster can remove multiple candidates. Enjoy!
For the second page in your coloring book, here is the box marking trace for Maestro 50:
1: Nm, Wm=>Cm. 2:SEm. 3: NWm. 4: 5: 6: NEm. 7: Wm=>SWt, NEc8np67=>NEr2np59=>N9m. 8: NW8=>SWm, Cm. 9: SWm, Cnp89=>(C2m, C4m).
Line marking starts off explosively,
3f: r2=>hs 1=>(C1=>C4=>(N4m, N3=>(NW3=>W3=>C3m=>S3m, NW4=>N4=>(N5m, N7m)), Cc5np36=>(S6m, C2=>(W2=>( NW2=>(N2=>S2m, SW5m, SW6t=>NW6m), W7=>E7m, Wnp49=>W5=>( E5m, NWc1np56=>NW1=>SW1=>(SE1m, c1np49) )), c4np57))), NW1m).
then quiets down:
3f: c2 4f: c6, c7 5f: r5, c5 6f: c8 7f: c9. Close: r4, r7, r8, r9.
For next Tuesday post, you might try the bv scan and X-panel. I found XY-chain removals, one xyt-chain removal, an SdC removal, an X-wing, and a finned swordfish. I have a kraken analysis checkpoint for you. The result is to go into your coloring book.
Also, on this coming Thursday, I will post the sysudoku box marking trace of this Maestro puzzle, number 4 in the Fall 09 issue, and a niveau 8. It illustrates a number of advanced techniques we have covered, and the merging of two clusters into one. The extra post is to give you an opportunity to work up to the point of coloring, and perhaps, discover this phenomenon, called bridging, for yourself.