This post highlights several moves from Phillip Beeby’s solver, which are alternatives to those appearing in the five earlier posts on Stefan Heine’s ultrahardcore 3. One result is a solution without the second blue green trial, left as a project for you. Just so you know, the post of June 30 has been revised to correct a significant error. Diligent reader Dov Mittelman detected it, and the modest blue/green cluster saved the day.

To Sudokuwiki’s APE of June 9, Beeby finds two alternatives. On the left, the APE  uses a bv and an ALS to eliminate combinations of 7r5c9 with candidates of r6c9. Under AICs, Beeby finds the AIC slink. If 7 is false, 9 in r5c9 is true, eliminating 7. The spotting significance is clear. Bv partner 7 does simple win-win logic for any 7 you can slink to, in E, r4 and c7.

On the right, 7r5c9 also falls victim to an ALS_XS. Note that the enlarged ALS has a single 5 and the restricted common with bv57. Its 7s align with 7r5c9,

Single candidate and aligned value sets are spotting keys in dealing with ALS.

Getting into the AIC building, the June 9 post reported the Beeby duplication Sudokuwiki’s Boomer 2. In the ALS category, Beeby finds an alternative that Phil labels an ALS–wing. It’s an ANL with links embedded in an ALS chain. Here is the first example in Gordon’s series, with Beeby description

ALS-wing of

(7=6)green – (6=2)red – (2=7)orange => -7 r1c9

I finally got the message. From now on, ALS value groups are represented as thick curves. The AIC slinks are internal slinks in the three ALS. All winks are group(restricted common) weak links. Here, one ANL terminal is the 7 value group in the green ALS. The other is the 2 to 7 internal slink in the orange ALS.

But why call it a wing? Structurally, it’s a chain, an ALS ANL of length not necessarily three.

Shortly after, on the next selection of the ALS category, Beeby recombines two of these ALS with a third one. The 7 trap is from cluster expansion, and 9r5c7 is an ALS-wing victim. Note the overlapped ALS and value groups.

The Beeby description is

ALS-wing of

(9=6)green – (6=2)red – (2=9)orange => -7 r5c7, -9 r5c7

This time, the ALS-wing is a confirming ANL. The green ALS 9 value group is confirmed. It’s a true group. One of its members is in the unique solution. Beeby’s solver misses that. The confirmation removes 9r5c7, yes, but it also removes both 9r4c7 and 9r1c9.

And Beeby’s 7r5c7 removal?  Beeby doesn’t color.

It’s justified, but the  attribution to the ALS-wing is misleading. Instead, it’s an ALS_37 defined by the overlapping green and orange ALS. 7r5c7 sees all 7s in both, so where is the restricted common? Its between the orange and green 3 value groups, also overlapped. One ALS must lose its 3s because 3r7c7 is true or false. Beeby should issue a separate Eureka for that removal.

When using Beeby, the elimination follow up, including boxlines, is done automatically and without documentation.  But some unusual follow up boxlines are recovered on your Deadly scan immediately after the move. After E9 => C9 below, invoking all Deadly obtained the boxline removing 3r7c12. The Deadly category includes, and is named after unique rectangles.

Here’s a Sysudoku project for you. Using a recommended fast browser, start Beeby with the uhc 3 grid after the red trial in the June 30 post. Load the clues, including orange. On every move, go through the Deadly, Simple ALS, Swordfish Fish, Simple AICs, Continuous AICs, Discontinuous AICs, Complex AICs sequence. Drawing curves is optional.

What I got was 3 chains catching up, to the diagram,  a hidden unique rectangle, an X-ANL, 3 AIC ANL, 2 simple discontinuous boomers, 2 complex discontinuous boomers, and 4 ALS_XZ.

That’s 13 diagrams, arriving here, where the blue green expansion traps 4r6c1, and calls for a second, red/tan cluster.

Adding the cluster, there’s bridging logic in cells:

r7c2: not (red and green) => tan or blue,    and r5c1:    not (red and blue) => tan or green

Read it either way, not red and (blue or green) =>  not red   or

tan or (blue and green) => tan,   red is wrapped and tan is confirmed.

The steep collapse confirms blue and it’s done, without a second trial.  If you’re going ultrahardcore, take up coloring. The strong link network is the human defense against seemingly endless searching.

Next time we finally get to Stefan’s ultrahardcore 47, shown at the end of the June 23 post.