Sudokuwiki Ends Nakex 63 On Its Own

This post picks up on the Sudokuwiki/Sysudoku resolution of Nakamoto Extreme 63 where it was overtaken by pattern analysis and coloring, and finishes it in two ways, one humanly practical and one not.

Sudokuwiki doesn’t maintain coloring clusters between moves, or use pattern analysis to expand them.  After the 785-wing in the solution of the previous post,  Sudokuwiki continued with a two solution paths  too instructive to pass up.

The first continues with an Andrew Stuart cell forcing chain. The idea is to remove a candidate that forces all candidates in some cell to be false. It takes a modern computer very little time to determine what a single candidate forces out, but how many such examinations fail between cell forcing chain successes? Way too many for human tolerance.

But here’s one. 3r1c9 forces 7r8c2 off via 3-chain and 7-chain. The ALS forces 1r8c1 on, 15r8c1 off. Is that a peacock or a swan?

The 3r1c9 removal enables a confirming ANL enlisting a UR slink and an ALS node. What is a UR slink? The rectangle r89c16 presents a multiple solution if both 3r8c1 and 6r9c1 are false. Therefore, If one is false, the other must be true. A strong link in the ANL.

Now, after a simple BARN,





two hidden unique rectangles, Type 2.



Next, an APE, in which the ALS excludes 15, and the bv at r4c1, 35, as placement combinations in the aligned cells.

Then an ALS AIC nice loop expands the cluster to trap 9r6c3 and bring in C8.




The further expansion traps four candidates and repeats the XY-wing of the last post.

The follow up to E5 proves green to finish Nakamoto 63 the Sudokuwiki way.






Andrew Stuart’s Sudokuwiki solver has an option to bypass specific methods. When Stuart’s cell forcing chain is bypassed, the next method after boomer 7 and the 785-wing is this ALS toxic pair. 3r9c1 is eliminated because it sees all 3’s in both ALS. Since only one ALS can have a 7, all values are locked in one of the ALS, including one of those 3’s. The toxic pair takes the place of the cell forcing chain in the Sudokuwiki solution path.

But a second elimination, of 9r6c3, Sudokuwiki  attributes to “rule 2”, saying 2,4,9 are unique to the orange ALS. We need to know what  prevents 9r6c3, and therefore blue, from being true, and leaving the ALS placements 2, 3,4, and 7?

In Stuart’s Sudokuwiki strategies, “rule 2”  is attributed to forum guru David Bird, but not explained. A clue to rule 2 is the fact that 9r6c3 forces both shared 3’s in the SW box to be true. That is because it forces 9 out, and 7 into the orange ALS, along with 3 and 2.

In general terms, this effect can be described this way:

The reason is, when an unshared value is removed, it forces the common value into its ALS,  removing the common value from the other ALS, and leaving both ALS locked. In the above example, 9r6c3 must go because single 3’s in both ALS share a box and can’t both be true.

It’s not that that the outside candidate can’t be true, removing a value set. It’s that the effect that has in this case violates Sudoku placement restrictions.

Next week, we consider what Sudokuwiki can do with Nakamoto Extreme 83. You must have the book by now. Are you using another human oriented solver? If so, tell us about it.

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Mixed Solutions on Nakamoto Extreme 63

This post reports Sudokuwiki AIC building in Nakex 63, until it is overtaken by pattern analysis and coloring. The solver’s path is continued in the following post.

Nakex 63 basic is tough, despite a productive bypass.



The first advanced move is one 3-chain ANL, extended by a NWc2 boxline, and into a longer one.









Getting to the 9-panel, grouping makes a slink in r4. The victim sees the grouped terminal in the East box.





After planting a cluster, we follow Sudokuwiki into AIC building with an easy AIC boomerang 1 from 3r4c1.

The removal expands the cluster in c1. In fact, looking at the 9-panel,


the new c1 slink brings the slinks from 9r3c1 to three, and winks from two of them bring to five the number of 9’s that are true if 9r3c1 is false.




Translated to coloring, two green patterns and one blue one are expanded, as four traps become orphans. A pink/olive result in blue and green

Sudokuwiki doesn’t see the pattern eliminations or the naked triple and hidden UR they bring, but goes ahead with the boomer parade. Next, from 8r6c5 an XY node and a reversed bv three X-chains tied together for AIC boomer 2.



Boomer 3 in the parade, from 3r9c3. In building a boomer, after every slink you try possible winks, going with any that reach another slink, or a starting cell candidate. You can see the destination slink in every transition here.

We don’t report on the stalls, like 3r6c1 after the starting slink. There are no more stalls here after we reach 5r4c1. This is the second of four possible continuations, working clockwise. If all AIC after 5r4c1 fail, there are two possible wink/slink continuations remaining, to 3r6c4, and to 3r6c5.

Along the way, there are two possible return targets here, 2 and 7.


It takes persistence to explore all possible return paths. Many of them fail, and some after considerable effort. But there is a systematic process that is humanly accessible.

Boomer 4 looks like an ANL requiring a different spotting process. But no, it’s also a boomerang, from 3r8c8, coming back on a grouped 5-chain.




Next in the parade is the backup float. Starting with a long boomer 5 from 2r9c2, back up one slink to traverse the AIC in the opposite direction, hopping off to look back in the opposite direction for boomer 6.





Instead of the mayor in a convertible, now we get a grouped AIC ANL.



The boomer spell is decisively broken when the ANL removals enable the finned swordfish below.

The logic is: if the extra candidate spoiling the fish is seen by the victim, then were the victim to be true, its vision would remove the fin, making the fish valid and victim toast. If the victim and fin share a box or line, call it a finned fish. If the wink is an inference chain, call it a kraken fish.



Sudokuwiki can now return to the parade. It gets boomer 7 from 8r7c9, which gains a bv,

that enables a 785-wing in East.







The expansion traps 8r4c5 and 8r6c9 bring C8 and an XY wing, enabling



a simple 5-chain ANL (red), followed by XY ANL that destroys one of its nodes. N5 is confirmed as a hidden single in E or r6., whereupon

E5 => SE8 wraps blue, and the green army mops up Nakamoto Extreme 63.




Next week, we follow two Sudokuwiki paths from the 785-wing, without the pattern analysis or recorded coloring. One path is humanly practical and one isn’t.


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Nakamoto Extreme 43 Redefines Sysudoku Extreme

This post recounts another AIC building project of 15 boomerangs, Nakamoto Extreme 43. The Nakamoto Extreme collection justifies the ‘extreme’ label by possibly exhausting the time and patience of a human solver.  Let’s say, if a puzzle justifies a trial for that reason, its extreme.

As often happens a short basic trace




signals a cloudy starting grid.










Before bringing out the boomerangs, Nakex 43 throws us an easy X-chain.








And here they come. Boomer 1 from 5r4c6 and boomer 2 from 9 r9c6. Even though both are X-chains linked by XY  bv nodes, they are found in AIC building, where any means of extending a chain are employed.




Next come AIC boomers 3 from 8r1c4 and 4 from  6r1c6 => NWr1 boxline 2r3c2 and  AIC boomer 5 from 1r2c8 creates NWr2 boxline removing 9’s.






Then AIC boomer 6 from 3r1c3.










The removal allows the former boomer path into 3r1c3 for an ANL removing 3r1c4. Then extension (red) from 8r1c4 closes ANL confirming 3r1c3.






The confirmed NW3 goes nowhere, but boomer 7 from 6r1c6 finds an ALS to get home, for N6.









After two XY ANL => NW9, an XY boomer 8 from 8r6c5.







Next, AIC boomers 9 from 9r8c5 and 10 from 2r5c8. The boomer 10 removal permits


a 6-chain ANL, and the new 6-slink in r5, which brings boomer 11.









The 6-chain removal also enables AIC boomer 12 from 6r4c9, and the boxline SEr9. To the left, the grouped boomer 13.







Sudokuwiki mislabels AIC bv boomer 14, from 2r1c1, as a digit forcing chain. The interpretation does not lead to a humanly practical way to find it. Viewing it as a boomerang does precisely that.




Sudowiki finds an APE requiring three ALS, a very unlikely human solving event. But the removal is not needed until much later.

Instead, the next AIC ANL depends on boomer 10.






The AIC ANL creates the c6 3-slink for a confirming bv boomerang 15.








the removal brings a long XY chain. But coloring is past due.









The cluster makes a shortcut example of the next XY chain, as the former XY removal enables a grouped 9 chain.







The 9-chain removal expands the cluster, trapping 8 to continue the expansion and force two blue candidates in r8c4. The green army cuts through the candidate undergrowth and lays out the Nakamoto Extreme 43 solution for us.




Thanks for sticking with a a very long post. I found no logical escape hatch, but perhaps you did. If so, clue us in. I’ll publish your comment, diagram your find, from your what and where, in a following post.

Next , another two post effort is Nakamoto Extreme 63. There’s another boomerang parade, then the first post ends it with Sysudoku pattern analysis and coloring, leaving the less spot worthy, but more spectacular, Sudokuwiki ending for the second post. You could follow the parade with Sudokuwiki, looking critically at your X-panels to see if you can get that first ending before it comes out here.


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Sudokuwiki Reaches Its Digit Forcing Chain

This post illustrates the power of coloring to employ the strong link network, and again rejects Andrew Stuart’s Digit Forcing Chains?. The post bypasses the Nakamoto Extreme 23 coloring wrap of the previous post, and follows Sudokuwiki AIC building  to the “digit forcing chain” .

Leaving the original Sudokuwiki grouped AIC boomerang 9 without invoking the blue wrap extensions, Andrew Stuart’s solver finds  boomers 10 and 11 incorporating the same grouped slink.




Then comes two rare sights together, a regular XYZ-wing and boomer 12 with an ALS node.








Still counting, a grouped boomer 13 from 2r2c9.









Building on that, Sudokuwiki calls this a digit forcing chain, making 1r5c5  true whether  9r1c3 is true or false.

Picking 9r1c3 is misleading. The same attribution can be made for any other candidate along the loop.


The conclusion actually comes from the loop itself. It is a confirming almost nice loop, an ANL, asserting green.

Next week, Sysudoku Basic and Sudokuwiki advanced for Nakamoto Extreme 43.

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Cluster Maintenance Pays in Nakamoto Extreme 23

In this post, a cluster wrap interrupts the Sudokuwiki march on Nakamoto Extreme 23, with an abrupt collapse. The following post then follows the original solution through to another example of Andrew Stuart’s “digit forcing chain”, to note that human solvers can and should discard it.

The Nakex 23 line marked grid and basic trace:

After a quick scan for unique rectangles or remote pairs, a sysudokie constructs a bv map for XY chains and makes a copy for XYZ wings.

That is rewarded here with a inference chain i571-wing. Finding that a 17 bv is present to accompany the 15 wing, its not hard to find a 7-chain for the weak link. Sudokuwiki doesn’t do inference chain winks.

A coloring cluster looks promising. Green or blue removes 7r2c5.

The i571-wing removal is duplicated by an XY ANL following up the trap. Sudokuwiki misses both, but finds an APE removing 1r9c3.  ALS 157 prohibits 71 and ALS 129 bans 91. The two removals in c3 expands the cluster.




The cluster expansion traps 1r5c2 and provides a shortcut boomerang from 6r9c3, beginning a boomerang volley with boomer 2 below from 3r3c8 in black.






Boomer 3 in red branches off to come back at 6r3c8, and boomerang 4 (green) is thrown in the opposite direction from the 8-chain to another 3-chain.




The new bv 31r3c8 completes an XY ANL for NE3.








Then another boomer barrage, with boomer 5 from 2 r2c9 in black, boomer 6 from 9r1c9 in red,  boomer 7 from 1r1c2 in green, and boomer 8.







The 9r1c2 removal allows a grouped AIC ANL.





Next is Sudokuwiki’s grouped boomer with a group as the emerging slink partner.

The Sysudoku cluster reveals a remarkable coloring wrap. The red extensions create an inference chain weak link between cluster members of the same color.  If one is true, the other is false. Being the same color, both must be false, and blue is wrapped. Wow!!

From here, it still takes a naked triple, and two  1-ANL to wrap up Nakamoto Extreme 23.








Next post will forego the wrap and continue Sudokuwiki’s path to the end, for two purposes. First, to graphically reveal more beautiful boomerangs, and second, to reveal the Sudokuwiki “digit forcing chain”.  If you’re solving ahead of the blog for the feedback, here is an opportunity to find these gems on your own.





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Nakamoto Extreme 3 Exits in Full Color

This post continues the Sysudoku report and supplemented Sudokuwiki solution of Nakex 3.

We pick up from last week with matching  8 pairs in E and SE.

The starting slink has been available since Sudokwiki entered AIC building, but new bv now enable the bv boomer placing SE8.

After a follow up of




we get this example of coloring support in AIC building. Instead of the grouped return of the boomerang found by Sudokuwiki, the cluster provides the more direct and more easily spotted shortcut slink in red. There is a slink of blue 4 with every green candidate on the grid. If blue 4r1c7 is false, all blue candidates are false and all green candidates are true.


Adding a modest red/orange cluster, there is immediate evidence in r1 and NE that red and orange are not both true. By logic,

not (orange and green)

=> red or blue.

9r2c8 sees red and blue 9’s, a bridge removal, which colors 9r3c7 orange, trapping 9r3c2.

Unlike Sysudoku grids, Sudokuwiki does not remember clusters.

While you’re contemplating that, here’s a shortcut to Sudowiki’s green wrap on the next move. Here we start AIC building from 4r3c1, that would do it. All we need is a force of any green candidate, because blue 4 will also force that candidate for an ANL.

Tell’m where you got it.



It’s a testament to Nakamoto Extreme toughness that any more is needed, but . . .

red 5r3c7 forces removals in c3.







And these wrap red in c7, for the immediate collapse of Nakamoto Extreme 3, in orange and blue.








Next week a Sudokuwiki path is followed through Nakamoto Extreme 23, until a Sudowiki boomerang is modified for a coloring wrap. A second post will continue  the longer Sudokuwiki path to re-interpret one of its successful moves for better human spotting. Get the book, but if you’re not ready for that, prepare for patience by doing what you can.


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Reviewing a Truly Extreme Collection

This post and the next initiate a review of Sudoku Puzzle Book, Extreme Level by J. B. Nakamoto, with Nakex 3. The review combines moves by Andrew Stuart’s Sudokuwiki solver with Sysudoku alternatives, while reporting and interpreting moves in accordance with the Sysudoku Guide to  systematic human solving.

Nakex 3 Sysudoku Basic portends an extreme level, with a total of four clues and six fully claimed cells.

The line marked grid offers two hidden unique rectangles of type 1, in which one UR value is slinked along both lines from the corner opposite the single extra free cell. The other UR candidate in the opposite cell must be false, because otherwise, the slinks force a reversible solution on the rectangle corners.


The hidden UR removal 1r5c1 enables a BARN, which Sudokuwiki labels a WXYZ wing. The Bent Almost Restricted n-Set applies to other values of n. Here, and most commonly, n = 4. The single value set 3 that is not restricted to a single unit must contain a true value, and is therefore toxic to any outside candidate seeing the whole set.  This one brings W3 and SW3 clues.

Moving on to the X-panel methods, two grouped X-chain ANL follow:


When a coloring cluster in the South stalls, Sudokuwiki steps into AIC building with the  AIC boomerang below. On the diagram are the slinks that you would examine as boomerang starters. These you would scan, looking for slink terminals that see another slink candidate. In this case, r6c4 is the first successful starting cell, and it takes two 8 nodes, a cell wink, a 7 node, and an XY node to reach a 9 seeing into the starter cell.


It is a good idea to leave AIC building results in place, even if unsuccessful, because they are often extended by later removals and clues.






In this case, the 9r6c4 removal leaves an XY node extending the 5r2c6 boomer 2 to the 8-chain.



The new XY node also enables the 5r3c7 boomer 3. This time the next node is the ALS 289 internal slink between 8 and 9 value sets to reach an XY node back into starting cell r3c7.




The boomer’s removal 1r3c7 expands the cluster and allows the AIC grouped ANL, with follow up









with coloring traps in r1, a naked quad and naked pair in r4.




The naked pair 12 in r4 sets up a Type 4 unique rectangle. The slink across the floor guarantees that one slink partner is true, which guarantees that both UR partner 1-candidates must be false. The 5r4c3 removal also permits the grouped 5 slink of the almost nice loop.

It’s good to look at new aligned pairs for UR, and line removals for new grouped slinks.

The story is getting long, and perhaps you’d like to take full advantage of added coloring to shorten the Sudokuwiki solution path, which continues in AIC Building mode.  Those results are reported next week.



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