This post reveals yet another way to start a boomerang, and reconsiders the practicality of Andrew Stuart’s Cell  and Unit Forcing Chains, this time facing an ultrahardcore right page. We get to a choice between Sudokuwiki and a coloring trial enhanced by cluster bridging.

Returning to the last uhc 3 AIC building result, 4r4c2 is removed because the 4 value group in the ALS is confirmed. Sudokuwiki  has constructed a confirming ANL . The group, and at least one of its members, is true. The result isn’t a boomer, but the current issue is, can a human solver discover this ANL by scanning boomer starting cells?

The answer is  yes, but we have to be looking for a different type of boomer. We could start an AIC chain with the internal ALS slink in r4c7, anticipating a boomerang back to 4r4c2. That boomerang would have a starting line, not a starting cell. Do we still call it a boomerang?  It would be up to us to know, as we build the chain, that a wink into any other value set would close an ANL confirming the starting value set.

The new bv from the 4r4c2 removal enables a shortcut  XY-chain. Yeah, it’s a show off shortcut.

Speaking of shortcuts, think of the host of blue/green slinks a coloring cluster adds to the boomer starting pool. Sudokuwiki happily supplies one below, with a group filling out r4 for the second slink in boomer 6.

Sudokuwiki didn’t use coloring to find this, but the example suggests we put clusters in and carry them along as strong link network markers in very hard puzzles.

This cluster may be decisive in a trial, but let’s pass it up now, and. reconsider it later as an alternative to the forcing chain methods Sudokuwiki goes into next

In the cell fc method, find a cell such that any of its candidates being true forces a candidate elsewhere to be false. In this case, forcing chains from 2, 4 and 6 of r7c7 force 2r9c1 out. Cell r7c7 must have a placement, so 2r9c1 is indeed out.

Is this humanly practical? Even if you limit candidates to 3, how many cells do you have to examine to find one whose 3 candidates see the same victim? Solvers like Sudokuwiki don’t need to care how many.

Let’s color code the forcing chains to suggest a feasible order. Here I’d start with cells having a colored candidate, and a matching candidate in another unit seeing possible victims. That includes the black arrow. Then I would look for a forcing chain from another candidate in the starting cell to one those victims, the red chain.  Then with a cell in hand with two fc’s I’d go all out for the third candidate, the green chain requiring the ALS node.

Sudokuwiki’s next elimination is equally or more impractical, Andrew Stuart’s  Unit Forcing Chain.

In the Unit Forcing Chain, all candidates of a value in a row or column force that value in a target cell to be false. You  scan unit by unit,  looking for more than one of, let’s say, 3 forcing chains  of the same value to target the same victim. That event then triggers an intense search for another from the unit. Sudokuwiki does try for as many as four candidates. Such a case is called a quad forcing chain.

In this case, r9 has three 4-candidates targeting a single 5-candidate. The set of 5’s also qualifies for a unit forcing chain. How many such sets of three on this grid?

As a new cluster is added, netting two traps, a bridge between the two slink networks is possible. A scan for  common values finds 7 !  Red and blue 7 in the c8 make the  Sudoku logic assertion:

Not(red and blue), so logically,

(not red) or (not blue).

Whoa. Orange or green is true!! Maybe both.

If a candidate sees orange and green, it gets removed. As to trials,  if we try orange and it fails, the red and green armies are both true. I’m so tempted to look there.

So what do you want to do? The orange trial, or do we turn back to Sudokuwiki? Experience is that cell fc and unit fc are generally followed by more of the same. It may be different this time. The general rule is to hold the trial in reserve, while we go as far as we can without it. It depends on your tolerance for search frustration, but I’m bypassing that with Sudokuwiki, so here goes. It will tell us something.