How Not to Find a Multiple on Moito’s Road

After a jaw dropping irregular XYZ-wing, a marking mistake reveals an unavoidable UR and two solutions of Very Hard IV-6, then the correction leads to an even more remarkable, but no less unavoidable uniqueness violation. Coloring reveals the logical path to all four solutions, no errors required.

In Sysudoku Basic, a very stingy bypass and a routine box marking lead to a demanding line marking. It was tough, but our reward is an irregular XYZ for the trophy case.



The hinge of the 345-wing is r3c7. The 45 regular wing is attached in the . . . , well, regular way. The 35 weak link is a grouped 3-chain. The 5r6c7 removal is regular, its winks to the square marked toxic set, being unit weak links. The 5r6c7 sees the irregularly attached Z through a grouped 5-chain.

Have to ask how such a crazy wing is ever spotted?  It becomes possible with the XYZ Map, a derivative of the bv map. The map is where you can view all the XYZ-wing ingredients and examine the grid for the enabling AIC.

What happens next is embarrassing to admit, but is one of my more instructive mistakes.  I follow the two effects of the irregular removal, the 3r3c3 removal by the disrobed pair c7np39, and the boxline Wbxl5m removal of 5r6c2. But somehow I miss 5r5c2 and take the removal as a boxline, removing two 5’s to produce NW47.

A quick collapse came to a very strange end, when I reached a deadly rectangle on the very last removal. No chance to identify a preventing action. 

Then a check with Moito’s IV-6 solution matched neither of these two. 




Going back to the boxline removal above, and scratching my head over its mistaken effect, I had a much better alternative, a double Type 1 UR. Sorry, extra candidates are in one corner, and it must be reserved for one of them. Rectangle candidates are asked to leave quietly.

The 9r5c2 removal is an effect of the r5np39 naked pair created by the 5r5c7 removal. The 5r2c2 removal comes from NE3, and its c1np25 effect.

OK, back on track, and knowing we have a multiple, we naturally go to our customary multiple tracking bloodhound,  the coloring trials.







The green trial gives us a 6-node remote pair loop. Like the earlier deadly rectangle, it arrives too late to be avoided at the puzzle’s expense. Moito’s solution is the orange side.

Like too many composers, Moito doesn’t run its solver long enough to avoid multiples.

I was a little surprised to find that blue is just as decisive, and leads to the deadly rectangle I encountered first. That means there are exactly four logically derivable solutions. Does your favorite solver agree? If not, I’d like to know.

This is the first time I’ve seen a UR multiple solution formations turn up with all other placements decided. The occurrence doesn’t affect the dispute  I have with the UR detractors who say that use of UR is an additional assumption of uniqueness that we should not make. It is not that. It is nothing more than a belief that the puzzle was published with adequate processing and rudimentary proof reading.

Suppose the remote pair loop or the UR had occurred earlier, and we took avoiding action, then became stuck. That would have proved nothing, because, what can you expect from a known multiple? It has already given up claim to the rules of Sudoku.

Next time, the review continues with a walk through of Moito Very Hard III-15, shown here. It’s another example of coloring/ X-panel cooperation.

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Moito’s I-4 Inspires a Trace Refinement

This post introduces an improved trace convention for 3-fills, in response to a complex of 3-fills in the bypass solution of Very Hard I-4 from Moito’s Sudoku Road to Mastery, the puzzle Moito rates as highest possible difficulty. The bypass solution demonstrates a tracing refinement for the newest addition to Sysudoku Basic, the 3-fill. Then we watch X-panel methods cooperate with a coloring cluster with a decisive wrap generated by finned fish.

Here is a I-4 bypass trace, as you might have done it by following the examples since  3-fills were introduced last April. Spotting the naked pair SW28, this trace goes into a collapse before the customary number list of first causes begins.

The Traces page and the Basic pages explain the trace notation, in which 3-fills are represented implicitly as effect lists. Now with some experience with the complications of 3-fill interaction, I’m going with a more explicit and simpler tracking of 3-fills in traces. Instead of resolving some 3-fills immediately and leaving unresolved ones unaccounted for, I will list every 3-fill in an unresolved bracketed list form as an effect, and use square brackets to mark its 3-fill resolution list, wherever that occurs, as an effect of whatever cause resolves it.

With this treatment, I can consistently list the four 3-fills in the starting grid of Very Hard I-4, just as I listed the SW28 subset, before starting the numbered cause list.

The total effects of these causes are seen here. Then, as the trace continues above,  the hidden dublex finishes the puzzle before numbered causes even begin.

That makes the super tough I-4 look super easy on the order of battle scale.

Maybe it’s how Moito approaches basic.

Now I’d like to comment on the interesting solution path of Very Hard I-35. Here is the fully line marked grid, with the scars of the  boxline event of column 1 still in place.

The removals leave a 3-slink in NW,

but we call it a boxline because it’s the removal of the last 3 in the NWr3 box remainder that creates that slink.

Box marking is easy, making line marking hard.

The seldom seen regular XYZ-wing is first, but a search for irregular wings on the new 236-wing hinge calls attention to the slink network around these numbers, and I get out the crayons for a look.





A nice cluster develops, but no results, so I go to the X-panel and find 6-chain ANL, expanding the cluster and the bv field. One of the removals traps a 3 in c6.

Coloring early pays dividends, and the cluster coloring has no adverse effect on chain construction. And its easy to do.


The same 6-panel delivers a  finned swordfish on rows 1,2 and 8. If the fin 6r1c9 is false, the perfect swordfish gobbles up the victim.  But if true, the fin does the gobbling. Seeing the fin is fatal.

Of course, I look at the other potential victim, 6r7c2. Is there any way it can see the fin, and become a kraken fish victim? That is a fine reason to look for an AIC, but it doesn’t work.

Then, on the  8-panel, a finned 8-wing on c2  and c3 delivers NW9 and green.  The cluster wrap finishes I-35.

Next, we take up a very unusual multiple solution puzzle, Moito Road to Mastery Very Hard IV-6. This one plays like a unique solution Sudoku, until it doesn’t, so you may prefer to avoid it.  In this review, are you a player, or a spectator?

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Moito’s Road to Sudoku Mastery

My first Sysudoku review of 2018 is one of a host of new collection books with the same grid format, some with an unexplained rating system, some with listed authors A.D. Ardson and Rebecca Bean. Moito and Bean’s  Extremely Hard Sudoku Volume 10 seem to share common difficulty rating system. The puzzle ratings are two decimal places, from a presumed zero, up to 1.00.

Sudoku: Road to Mastery is a collection of 400 puzzles without listed author, described on the cover as “Hard to Devilishly Hard”. It is divided into a “Hard Sudoku” section of 200 puzzles, ranging in difficulty from 0.60 to 0.74, and a “Very Hard Sudoku” section, ranging from 0.75 to 1.00.  Each section is divided into four sets, I through IV,  of 50 puzzles. Rather than my dart board selection of a start and a fixed size step to cover the collection, I tested the rating system as well by preselecting the 10 most difficult puzzles, according to the difficulty rating,  from the “Very Hard” section. The result is the following review table:

The Moito difficulty rating is very inconsistent with the review results. The most highly rated puzzle fell in the bypass, and three more of the highest rated collapsed in box marking. The third above, III – 47, is the only contender for an extreme rating. I finished with a coloring trial, but perhaps I missed something you can find.  

The 0.91 rated IV-6 has multiple solutions of a peculiar nature. Missing a boxline removal after a spectacularly irregular XYZ-wing, I reached a double solution on unique rectangle too late to reject it.  The Moito solution was different, so I retraced, and found the omission. This time it was a  different double solution on an extended unique rectangle formation of six bv, and this time, including the Moito solution. Related to earlier 2017  revelations, anything can happen with multiple solutions. They follow no inference rules. Details to come.

I happened upon two more glaring errors on the Road to Mastery.  Puzzles III-5 and III-6 are destroyed by a sloppy paste-in of puzzles III-7 and III-8. Raising a more serious issue of trust, your homework III-47 above is identical to I-28.

Now to checkpoint your III-47. First a bypass typical of the Moito collection, reading:

C29, W4, W5, SE5.           Yes, that’s it.

ANL with grouped ER victim in Moito III-47Box and line marking are routine, and the bv scan brings only this XY ANL with a grouped ER victim, or if you prefer, a grouped AIC ANL.






Next, fishing is bad on the X-panel, but an extraordinary set of 9-slinks can be assembled into an X9-chain ANL to trigger a SW/c2  boxline 9m removal to create a naked triple.





No fixes on the X-panel, but coloring around the 9-slinks produces three coloring clusters that do not bridge.







Conflicts in c3, c7 and r3c9 give bridging combinations:

not(green&red) => blue or orange,  not(red&tan) => orange or purple, and

not(blue&purple) => green or tan.  Bridge toxic sets don’t materialize.

With the X-panel showing little prospect for pattern restrictions, I’m ready for a color trial. From the bridge combinations, blue => tan. I go for a blue trial.

The trial trace reveals how blue inferences coordinate to force two 2’s into SE. Green => orange, but the marking stalls, but an XY chain from my updated railway wraps purple, with an immediate collapse to green, orange and tan.


Next post, I checkpoint your version of the I-4 bypass, and show several highlights reported in the review table above.

Also, four of the five Guide pages are up. They explain and demonstrate Sysudoku Basic via worthy examples.




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Only Extreme 390 – a Fitting Finale

In this post, the last preselected Sudoku of the review, Only Extreme 390, is solved by only advanced methods, but by an extremely thin margin.  The first crack in the armor is an irregular XYZ-wing. At first I missed an XY-chain ANL on a very extensive XY railway, that expands into a decisive nice loop. The railway is included so you can trace around it to verify the ANL and see if I missed any others.  I had solved 390 with a coloring trial, which would rate it as Sysudoku Extreme. After the ANL, however,  the expanded cluster merges with a second cluster for a follow-up wrapA review table and general conclusions on Only Extreme Sudoku end the post. 

As for Only Extreme 390, here is a basic trace that carries you to the first grid:




In the  irregular 893-wing, the 83 wing winks at the hinge via a forcing chain. The almost nice XY loop (black) removes one 8, but then an 8-slink  in c1 winks in to the chain terminals to form an AIC nice loop, duplicating that removal and adding three more.



To practice the Sysudoku search for ANL along the XY railway superimposed on the bv map, here is the 390 example.  Find all of the starting terminals, and on each unvisited terminal follow the chain. On each incoming  digit, test each repeated outgoing digit along the railway but not in the same unit, for an almost nice loop victim. It’s faster than it sounds. Some segments are one-way. That’s why it’s called a railway.

Powerful as it is, the railway exploits only one feature of the bv field. A more powerful , but still only advanced feature is the Medusa coloring network that the bv field supports. After giving single value X-chains and cross unit fishing restrictions their due diligence, as we did here, the Sysudoku Order of Battle returns to coloring. The blue/green cluster traps three candidates.

A second cluster brings a merge:

Not(blue & orange) and not green & red makes red function as blue and orange function as green.






Nothing changes when we merge the two clusters, but the traps of the bridge are exposed.

The marking confirms green, and 390 is done.




Overall, Manuel Castillo’s Only Extreme Sudoku shares with Krazy Dad Insane the honor of being the hardest reviewed here, outside of acknowledged monsters. As to the “Extreme” label , the Sysudoku “extreme” rating is reserved for necessary trials.  Among the 10 selected, Only Extreme had 5 extremes, and 5 only advanced. That’s equal to KD’s Insane’s. That collection t forced me into the effective freeform approach to pattern analysis. Only Extreme Sudoku forced me into series of trials extending a coloring  over AIC slinks.

The Weekly  Extreme Competition series I selected for review had 7 of 10 extremes, but there, in consideration of the by week deadline of the competition, I took the first trial opportunity presented. Advanced alternatives were probably available. Outside of that, my position is to defer, and often discard, trials. Thanks, Gordon.


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A Coloring Trial for Only Extreme 347

Here a Manuel Castillo Sudoku sends me directly to last resort city.  Although the bv field eludes productive XY-chains or AIC, coloring is decisive enough to solve 347 in a single trial.

Your first view of the grid comes with a blue green cluster, and six AIC hinges I marked when I came up with nothing from the bv scan and X-panel.

If you’d like to work it from the beginning and didn’t  get Castillo’s Only Extreme Sudoku yet, just load the clues and go for it. The grid is in the line marked state.


You can check your basic against my Sysudoku basic trace.

My inner engineer asked me why I continued to put the naked pair “np” and other subset symbols in the trace. He’s right. It’s clear and consistent with the clue marks to just list the numbers in any subset, after the box name.

When I had nothing better than a coloring trial, I did look at several X-panels with slink corner formations that might add candidates to the trial sets. The 8-panel illustrates the idea.  In the cluster on the corner, here’s only one pattern that includes 8r1c2 and 8r4c7. There are four green patterns, reaching every other 8-candidate.  I pick blue.

Only Extreme 347 is useful as an illustration of the trial trace and corresponding graphic demonstration. In the trial trace, instead of finding all effects of each cause, we list only the first level effects, of each item on the list, returning on the next line to pick up another level of effects.

Here is the trial trace for the blue trial, arriving at a contradiction when two 2’s are forced into r4.

In a graphic overlay, we trace down from cause to cause on a direct path to the contradiction, ignoring effects that have no bearing on it. 

Going back to the trial setup grid, we use clue cause-to-effect arrows and supporting weak links to display the shortest, most direct path to the contradiction.

Here blue candidates see two candidates in r9c1, starting a chain of inferences that place 2r4c4, and 2r4c8.




Now knowing that green candidates are all true, we can use normal depth first marking to reach an immediate solution.



Next, the final post of 2017 takes up the final Sudoku of the Only Extreme review, 390. Another review will follow, but in 2018,a series of pages will be added to consolidate and illustrate Systematic Sudoku developments in these seven years of the Sysudoku blog. 

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Dragging in Only Extreme 304

This post continues with three more trials, two to reach a solution, and one to guard against multiple solutions, in Castillo’s Only Extreme 304. This is getting reeeaallly extreme.

Last week, we managed to extend and XY-chain across the grid and over a the bv field of a coloring cluster, and by three not-both trials, extended coloring over this chain. The cluster expansion trapped a 7-candidate, but little progress had been made against the brambles above and below the cluster.

The last not-both trial was close to a blue trial on the completed coloring of the XY chain, so between blue and green, I choose blue for a coloring trial. As before, an 3-chain covers one 8 in c6, and 4r2c7 is confirmed to help confirm 8r7c7 to eliminate the other 8 in c6. Blue goes down.

When green bogs down,  another trial possibility emerges.There is a Single Alternate Sue de Coq:

SEr7 = 3(1+8)(4+9) + 934.

One chute cell is (1+8) or that alternative is missing from the chute.   In the latter case, the solution is exactly 934.

It’s not too surprising that (SE9, SE4)  marks immediately into the solution.  



The difficulties to this point argue for a final trial to guard against a multiple solution.  

We should see if the alternative chute description

SEr7 = 3(1+ 8)(4+9)

leads to other solutions.  The  good news is that the resulting clue 7r7c6 forces 1’s in r7c2 and r8c4, allowing no 1 in r9. Knowing this is impossible without a trial trace.

Next week, it’s on to Only Extreme 347.

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Definitely Extreme 304

This post encounters what seems to be a monster in Castillo’s Only Extreme 304. A stingy basic leaves a near monster cloud of candidates resistant to the ordinary advanced repertoire. Following a theme of this review, a small coloring cluster is extended across the grid by not-both trials. It takes two more trials in the next post to defeat 304.  Or perhaps you can find a less extreme path.

The ordinary looking basic trace

leaves this formidable grid to navigate:

There is a Type 1 UR and a resulting box/line to start, but then I find nothing else.








In the swath of bv cells across the middle, I latch onto a tight little XY-chain intersecting a very modest coloring.

The potential for  coloring the XY-chain is enhanced by an AIC with numbers common to both the cluster and the xy chain.

If blue 8r4c2 and 8r3c3 are not both true, then 8r3c3 is green, and working back 7r3c3 is blue, 7r2c2 is green, and 2r2c2 is blue.

Testing terminal 8’s on the subchain from r4c2 is a start. The 8’s are not both true, along with blue,  because this forces both 3 and 8 in r1c7.






The cluster expansion removes four candidates. Clearly, a further expansion to 3r3c1 does further damage.

Again the not-both trial, testing blue + 3r3c1, is required for the extension.






If both terminals of the X-chain are true, c6 gets no  3’s.  Not acceptable.








The expansion turns 8r3c1 blue, generating three traps that expand the XY-chain further.








The final not-both trial to extend coloring over the XY-chain determines that 3r3c7 and 4r6c7 are not both true. If they are, 1,3, and 7 are removed from r2c5, and 1 and 3 are removed from r9c4, leaving two 4’s in c5.

Unless you have found a less extreme path, let’s see how you would continue with the fully colored XY-chain.  My finish of Only Extreme 304 comes next week.






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