This post explains lite coloring as I understand it from comments and emails from its apparent inventor, Dov Mittelman. I only know Dov as a reader of this blog. He (or she) does his analysis in his head and on his cell phone, and is not given to definitions or proofs, but is a very talented Sudoku solver. All this doesn’t help much in making things clear for ordinary solvers like me. On the other hand, I’ve gone much further than most Sudoku writers with the interpretation of strong link networks by coloring, so I can understand why, seeing this, Dov had the patience and fortitude to explain it to me.
Although I can tell you how lite coloring works, I hadn’t been blessed with good examples until another reader friend Gordon Fick put me onto the “Red Russel special”. Here’s why examples are scarce: A lot of hard review puzzles finish with coloring, but it’s pick, pick , pick, candidate by candidate until a cluster or two can be started, and then it ends quickly. Lite coloring is needed most when the clusters are there but aren’t working.
The puzzle I name here the Red Russel Special is a Sudoku presented long ago on the EnjoySudoku forum by contributor “Red” Russell. Its givens and prospects for Sysudoku solvers appeared at the end of the previous post, the last one of 2020.
The basic trace shows where the four X-wings and hidden quad occur. Yes, nada on the bypass. Beginners, you can learn all about traces in The Guide, linked above.
Now as we stop on that line marking 4-wing, the grid shows why, as we mark the r8 slink in its southwest cell corners, it’s so easy to spot the matching slink in r3. Of course we need to leave the fish icons on to pick up more 4-wing victims as we line mark more rows.
Pardon me for dwelling so long on Basic Sysudoku, but the Red special illustrates line marking X-wing spotting so well. If you don’t actually do it this way, take the time to follow the line marking as line fill strings get longer and longer. We line mark the easiest first. If it’s confusing, then read the line marking page in the Guide along with the grid pictures.
I missed one X-wing at first, but here is all four of them together, the first two on rows. and the second two on columns.
Yes, Virginia, X-wings are fish. For what makes a fish a fish, go through Advanced to fish in the Guide.
Let’s leave the hidden quad for later, and recognize that each X-wing is a coloring cluster that could grow. Coloring? Better see
The clusters are small and bear no apparent relation to each other. This “Red “ Special shows how lite coloring connects the clusters, extending their capability to trap and remove outside candidates.
Here is a definition of lite coloring:
The definition itself explains how lite color extensions create new traps. If an outside candidate sees two opposing cluster colors, lite or regular, it is seeing a true candidate, because all candidates of a true color are true.
The definition doesn’t tell us how to identify the lite color candidates. They come in a sequence. With any color candidate, full or lite, identify slink partners seen by that candidate. They are false when the starting candidate is true. And in each case, the slink partner is color lite in the starting color. The color lite procedure is, generate a branching tree of lite candidates of the starting color. Naturally, the root of the tree is a full color candidate. The sequence does not include candidates of the opposing color, which generate branching sequences of lite candidates of the opposing color.
Let’s walk through the marking of a tree of lite candidates of the Red Special clusters.
Starting with blue 4r3c1. If true it erases 5, 6, and 7, and no other 4’s, except the green ones. Only 6 is a slink partner, and its slink partner is blue lite. It is blue when blue is known to be true. The cell corner slink marking directly assists lite marking.
In Sysudoku, we draw arrows to mark lite candidates. The arrow stands for a wink to a candidate erased by blue and slink to the lite blue candidate. You can use either the arrow or the wink/slink sequence for marking. It doesn’t matter which blue candidate the arrow comes from. Any candidate on any arrow path from a blue candidate is blue lite. What matters is blue becoming true.
Lite coloring inventor Mittelman uses lighter shades of the same color for lite candidates, but I use arrows. For one reason, a candidate can be lite in more than one color. Also, in PowerPoint, my colors are already light shaded to make values visible, so it’s hard to make them lighter.
Continuing the lite sequence from blue 4r3c1, it’s OK for blue lite 8 to claim yellow r5c2 is also blue lite. But that doesn’t mean yellow is blue. It becomes blue only when blue is confirmed to be true.
The other blue candidate 4r8c5 claims red 6r7c9 as blue lite! A red candidate can act as blue lite in a trap. It does that, making 4r9c9 blue lite. Since 1r9c1 is green lite, 4r9c3 which sees blue lite and green lite in r9, is false. Red 6r7c9 is red, but blue lite 4r9c9 is not necessarily blue, but for sure, blue or green is true, so blue lite 4r9c3 or green lite 4r9c3 is true. The trap works.
But there is something else. Blue makes 3r7c5 true and that forces orange candidate to be false when blue becomes true. But orange and red are opposing cluster colors. If any orange candidate is false, they all are, and all red candidates are true. So now, red is true when blue is true. How about that? Red is blue. A bridge between clusters.
We don’t see it here, but a lite coloring wrap occurs when lite coloring chains from opposing cluster colors meet on a candidate of another full color. The full color is true, and its opposite is false. No waiting for a chain color to become confirmed.
Let’s merge the red/orange cluster with the blue/green cluster and re-mark the red/orange lite sequences as new blue/green lite sequences.
1r3c6 is blue lite and green lite. It wins either way .
6r1c7 sees green in the NE box and lite blue 6r1c1.
3r6c7 sees green lite 3r1c7 in c7 and blue lite in the East box.
It’s a short follow up to blue 3r7c5 and a green wrap.
Now green lite sequences disappear, and blue lites become blue clues. Including yellow 7r5c2. Yellow 7’s are confirmed as blue, and the purple 7’s they see are removed.
The investment in lite coloring pays off with the additional blue clues, and collapse is immediate. Just be careful picking through the clutter.
Here’s the lite coloring solution with the surviving blue color.
Way back on the Basic trace, the hidden quad appears as X-wings and removals are removed, on the closing scan of c5. The subset candidates are encircled in light blue and the blue diamonds are the removals.
One more line marking fact: although all the candidates of the quad are identified when r5 is marked, the columns have not been examined for subsets. Rows or columns, whatever set finishes first, the unmarked “other” lines are examined for slinks and subsets, i.e. they are “closed”.
The quad removals bring a sharp collapse. Here is the Sysudoku trace, down roughly to the green wrap above.
The Red Russel Special gained attention for several reasons. It’s posted here, and in the Guide, because it illustrates so well the technique and possible results of Dove Mittelman’s lite coloring.
Next week, we correct an error in the December 8, 2020 post and make another attempt at a trial free solution to ultrahardcore 399 with lite coloring. The 7 in r2c8 belongs in r3c8. I’ll leave it uncorrected for now. Aw go ahead and DIY it, you’ll see where the error applies.