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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Super Fiendish Ends With 87 and 97

The review of Brian Challenger’s Sudoku Super Fiendish is ending quietly. SF 87 gets into beginning advanced, while SF 97 is finished in line marking. As to difficulty level, this small collection ranges widely and erratically. Is it  Fiendish? Not really. It visits advanced regularly, besting Wayne Gould’s Train Your Brain Su Doku. But not reaching AIC building, it is nothing approaching super fiendish. By contrast, the Nakamoto “Extreme” collection coming up next is much more deserving of such a title.

The Super Fiendish review table:

Super Fiendish 87 distributes basic placements over the three Basic stages:

One advanced move, a unique rectangle type 1, is enough.

Any candidate 6 seeing 6r5c2 must go, since that 6 must stay until the deadly 12 rectangle is otherwise destroyed.

As it turns out, these removals generate four new clues and the collapse of SF 87.








On Super Fiendish 97, nothing breaks loose until c1, the ninth line of line marking.

Going back to the first hidden single on r6, the line marking left 18 in r9c6, which is taken as  hidden single because the r6c6 is the only cell in r6 available for 1.

Note that r6 is a 4-fill where 1 is visible from three cells. The hidden single could have been called in the bypass at W3! And also by definition, every single or pair from a resolved 3-fill could be termed “hidden”.

Here’s the solution of Super Fiendish 97.

Next week, we embark on more ambitious review project at the expressed invitation of J.B. Nakamoto in the forward pages of his or her Sudoku Puzzle Book, Extreme Level, described on the front cover as “Diabolically Difficult Puzzles for Advanced Solvers”. The 10 puzzles pre-selected for the review are 3, 23, 43, 63, . . . , 183. Nakex 3 is below.


In this review Andrew Stuart’s SudokuWiki solver backs up the Sysudoku path through advanced methods and AIC building. As usual, the posts will provide interpretations, traces, grids and alternative paths consistent with the Sysudoku Guide.

The collaboration produces an array of examples in a completely documented context that is the best so far at illustrating the advantages of Systematic Sudoku.




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Super Fiendish Bookends 67 and 77

This post steps through the most fiendish of the 10 puzzles preselected for the review of Brian Challenger’s Super Fiendish Sudoku, SF 67. , and then adds one of the bypass victims, SF 77.

Looking at the trace, you might think basic is easy,



but the long fill strings reveal a tough line marking.

Advanced solving starts with hidden unique rectangles of type 2b. The floor to ceiling slinks project a victim clue into a reversible rectangle of four clues.

Then, after an X-chain ANL,







a regular 127-wing is extended into an inference chaining i127-wing.



The midfield action continues with a 4-BARN (Bent Almost Restricted 4-Set),


and an ungainly looking APE in which  5r4c9 cannot be combined with 1, 2 or 7 in r4c8 within sight of ALS 157 and ALS 2457.

The APE removal is telling,

leading to an XY railway of interconnected nice loops.




No removals result, but a nice loop generated cluster is expanded by a trap and wraps blue in r2 and c4, for the collapse of Super Fiendish 67.

In his comment to this post, forum member strmckr provided this quick resolution to SF 67 immediately after line marking. He terms it an ALS W-wing.

The restricted common is a forcing chain. Since one ALS will lose it’s 9’s. Any candidate seeing all of any other value in both ALS must see a true value. 1r1c9 does just that, and the removal collapses SF 67. With all the ALS in every grid, can you hope to spot such an event? Not really, but you can see the appeal of writing code that can find every instance of it.





Super Fiendish 77 might have presented similar advanced challenges, but was too vulnerable to Sysudoku Basic. It takes a while but the collapse is relentless.












Next week,  the Super Fiendish review ends with 87 and 97. For the next review, get a copy of J.B. Nakamoto’s Sudoku Puzzle Book, Extreme Level.

The 67 and 77 solutions:


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