## Utrahardcore 3 Lifts Off

The review of Sudoku ultrahardcore 1 right pages, a separate collection from (and tougher than) its left pages, begins with number 3, shown here post before last.  In this post, we follow Sudoku Basic and the Andrew Stuart’s Sudokuwiki solver along 3’s boomerang parade, noting its challenge to the human solver.  The solver, and possibly, comments from readers, back me up on humanly practical methods. Thanks again to reader Alex (review page) for pointing out this collection. And of course, Stefan Heine for creating it.

On 3, bypass is short and box marking is shorter, leaving a hard line marking and a monster level line marked grid.

A small network of strong links stand out in a relatively clear area, and coloring produces a trap.Nothing else stands out until we get to the last X-panel.

First, spot the slinks on the 9-panel. There are three. Now find winks connecting slinks.

There are two ANL on the 9 chain. All of this goes beyond the bv scan, because the few bv are connected only within r8.

Turning to ALS and bent regions, an APE turns up.

It’s an aligned pair of cells whose candidates are seen by pairs of ALS value sets. No combination of values across the cells are possible when seen by a pair of value sets, because such a combination would remove both values from the ALS.  Combinations 74 and 76 are contained in ALS 3467 and 75 in the bv r4c8, so 7r5c9 must go.

Finding nothing else in ALS territory, it’s time for AIC building, a stage dominated by the boomerang. It’s search intensive, because boomer starting candidates are numerous, and a human solver must try every type of AIC node to extend branching chains to unpredictable destinations. There are the 18 strong links on the grid, representing 36 possible starting points for boomerangs.

Scanning box by box in our normal order, we start with 1r2c2 looking for a wink from 1r3c3 to a slink. Possibles are one of the two slinks as a wink or an ALS containing 1 as a value set. Then we try the reverse with starting cell r3c3, looking for a wink from 1r2c2. Then it’s starting with 1r3c3 and a wink form 1r4c4. The search ends with a wink back to the starting cell.

The starting cell is critical in forming a boomerang. A candidates can be on a other boomerangs and fail to be a starting candidate for its cell. Here is Sudokuwiki’s first boomerang. We know it doesn’t scan the boxes in our preferred order, because the first one starts in East, on 9r5c9, and

the second one starts in NE, from 5r1c9. If we start a boomer from r3c8 and move along the path of boomers 1 or 2 , it fails on r5c9 (1) or r1c9 (2).

The boomer 2  7r1c9 removal turns 7r7c9 blue. The cluster expansion traps 7r5c7, as it sees green by cell wink.

Sudowiki’s boomer 3 is from 2r5c7. The trap shows up on it’s selfie.

In boomer 4, grouping creates a starting slink, showing that our list of 36 possible boomer starting candidates is incomplete.

The AIC is shortened to illustrate the shortcut that is possible because of the slink between every pair of opposite colors.

The removal brings an event of interest only, a dead BARN. The bent value is 4, and there is no victim. To get reported by Sudokuwiki, you need a victim.

Continuing in AIC building, the solver finds this AIC nice loop, built around an ALS. Would it be found on the AIC building boomer scan? The answer is yes, given some flexibility in the solver’s goal. Reaching 5r3c8, you would realize that the boomer wink into r1c9 not there.  It’s OK, though if you have in mind that a slink into the starting cell closes a nice loop.

Of course, you automatically examine every link in a nice loop for a victim seeing both ends. That nets 5r6c8. And you have to check every elimination for a boxline. This one is SEc9, from the 3 removal.

The nice loop structure is now  re-used in  boomer 5 from 9r4c4.

This week, we’ve passed up a trial, and our reward has been an exploration of the searching task awaiting the unaided human solver in a right page ultrahardcore  Sudoku.

Next week  with this picture, ultrahardcore 3 brings news that an AIC building search plan based on starting cells is not quite complete. Why is 4r4c3 removed, and is this a boomerang?

Next week also,Ultra 3 brings  Andrew Stuart’s Cell and Unit Forcing chain methods, and the searching task they present for the human solver.  According to Stefan Heine predictions, there will be trials.

## 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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