Here we complete the report on the puzzles selected in *The Hidden Logic of Sudoku* to show that hidden XY chains are an “inescapable tool for the advanced player”, and therefore require the generation and maintenance of symmetry grids. None of the selected puzzles actually require this burdensome effort on the part of a human solver.

Picking up where we left off in THLS Chapter XV on hXY chains, the elaboration candidates of Royle 17-4167 are restricted to 3,6 and 9, and admit a decisive 6 cell remote pair. Since Berthier reports it to have a 5-cell XY chain in crn space, he can claim a solution of lower logical complexity. The solutions have the very same basis, the hidden XY removing 3 r5c5 and the remote pair removing 69r4c5 by means of five slinks.

But give me a break.

Remote pairs are so easy and require nothing extra.

Next, we have a 6-cell hXY chain in crn space, another pushover in good old nrc space. It’s Royle 17-5546. There’s not as many bv, but the 8-panel has the slinks for a decisive 8-chain.

I included the line marking fill strings to show what an easy line marking it was.

It’s true that the XY railway comes before the X-panels in SOB, but where would you put hidden chains? I take them as very extreme, and would do symmetry transformations only on a weeded grid.

In the collapse, you wouldn’t want to miss this little four cell remote pair.

But now, watch out! We’re down to the heavy hitters. Coming up in Sudogen-9617, you’ll need an XYZ wing, a 6-cell hXY chain in crn space, then a 7-cell hXY chain in rcn space, or alternatively a 5-cell xyzt-chain (details later), followed by the above hXY chains. Are you sure you want to go on?

No? OK, just assume the puzzle composer would not embarrass herself with an obvious rectangular multiple placement in the solution (post 12/13/16).

The candidate 3r6c2 has to be true or the composer is embarrassed. Adding this guardian triggers an immediate collapse.

Making this UR assumption has no effect on the risk of a less obvious multiple solution, even if it happens to occur as a part of one.

Finally we get to the end of this series on hidden XY chains, and your homework. Royle 17-1020 was unreasonably stingy in basic. The trace looks simple but how long did it take you? How about number scanning it first and then eliminating the extra candidates, to see how long that takes?

For those not participating, here’s the line marked grid.

The results aren’t pretty either. All three bv in the same unit. Let me know if you did better. But you know this doesn’t look that bad to those who actually do number scan hard puzzles. They wade though this kind of swamp regularly.

OK, where we go next is the X-panel. X-chains can cut through the candidate fog.

The 2- and 3-panels yield three indecisive clues. It’s not unusual for my X-panels to recover some missed line slinks when I do them. The 3-chain removal yields a box slink and box/line clue.

Of course, the x-panel is also my fishing hole, and I pulled out a creature that didn’t make it into THLS, a finned fish. In the left panel below, you can see that my 2-chain removal actually demoted a jellyfish to a kraken swordfish.

The right panel illustrates another frequent contribution of X-panels. The 2-panel suggests limited patterns. When a coloring cluster is defined, the solution pattern must be consistent with one color or the other. With freeforms left to right, the red freeform starts on green at r5 but cannot cross c5. It must leave the S box, has used r3, and cannot include a blue cell. It is the only chance for 2r3c3 to have a pattern. An orphan is a candidate belonging to no pattern and is removed.

Confirming the orphan is not enough, and we move on to AIC hinges, where I know there are counterparts in standard nrc space to the hidden XY chains in the symmetry spaces. Pattern analysis being rather extreme, I’m leaving the orphan unremoved, but I’m not forgetting it.

Thanks to the reader whose comment linked below prompted me to correct the right panel above, and illustrate coloring restrictions in Sysudoku Limited Pattern Overlay. Pattern analysis, as explained in Andrew Stuart’s *The Logic of Sudoku*, was known to Berthier. We know that because of his criticism of Andrew’s near BUG example. But pattern analysis also, is omitted from THLS.

An AIC hinge is a cell holding two slink partner candidates. Being in the same cell puts a wink (weak link) between them, so there is a alternative inference chain segment of three links around the hinge. If you can complete a nice loop through a hinge, it removes the other candidates from the hinge. In this case, I found that my chain of slinks and hinges could be completed as an AIC almost nice loop(ANL), eliminating the single candidate between winks. The collapse is immediate.

My Sudoku accomplice Gordon Fick, armed only with THLS, and not knowing that I was selecting this puzzle for the review, sent me two more ways to exploit the same two lines r4 and r5 for the same critical removal 2r5c1. One is an AIC almost nice loop that does not depend on my removal of the kraken swordfish victim. Can you find it? Go back to line marked grid and map the AIC hinges. Checkpoint next time.

The other way is an amazing ALS-XZ. The Almost Locked Sets r4c347 (blue) and r5c34789 (green) have a grouped restricted common 4 in the E box. That means candidates of every other number in the two sets contain a true one. 2r5c1 sees all 2’s in the two sets. Do you spot things like that?

I wish I could.

As to hidden XY chains, the first 2017 post showed there may be no Sudoku grids solvable **only** by hidden logic XY chains. Each one seems to convert to a reverse-XY AIC in standard nrc space.

These next two posts have demonstrated humanly accessible alternatives to hidden XY chains. These are supported by efficient, clutter avoiding basic solving.

SudoRules doesn’t do finned fish, much less krakens, or ALS, much less ALS-XZ, or AIC, much less reverse-XY AIC ANL, or patterns, much less orphans, or slinks, much less coloring. Its reliance on hidden logic is understandable. It runs on modern computers, where any number of definable extra grids is acceptable.

Next we look at c-chains, as defined in The Hidden Logic of Sudoku. C-chains are the Berthier counterpart to X-chains, only more complex and less capable.

Why is (3,3) an orphan? What’s wrong with (5,1),(3,3),(9,4),(2,5),(7,8),(4,9)? I see that (5,1) is green and (2,5) blue, but the free-form seems to not fail until it’s past that so I’m guessing the colors are only applied after the fact.

You’re right. Thanks for checking. I seem to have forgotten column 4 on this freeform as well. In the correction, r3c3 did not get a pattern. Its freeform encountered a color barrier.

Usually, colors are applied after all freeforms, to identify significant candidate/pattern memberships. It can extend coloring. Your comment prompted me to apply Medusa coloring first. The cluster isn’t large, but it causes the removal. A lesson for both of us.