Fata Morgana Sysudokie Fried

In this post, the exocet trials are completed and a path to the monster’s cave is mapped and cleared of fog. A color wrap brings us to base camp, for the final ascent.

FM 16 nice loopThe nice loop generated by Cr5np16 is another figure 8, and its crossing cell is forced to have two loop cells of the same color. The blue candidates are wrapped.

And the blue removals in r3 leave the remaining 1’s and 3’s in the N box, forcing Nr3np16 and another deadly rectangle in 1 and 6. The trial fails, leaving np36 in Cr5. And it wasn’t that hard.

FM 36 nice loopSo now we’re ready to follow up on the third pair of ANL, a 3-chain and a 6-chain forming third fog clearing nice loop. This time there is no figure 8 and no cell crossing that wraps a color. We also note that the colors cooperate to avoid the row naked pairs that would constitute a deadly rectangle.




FM 36 clusterAs we trace out the red/orange cluster, it’s apparent that red forces two 9 clues, so it’s a more decisive choice for a coloring trial.  Monsters get no slack.







FM red trThe red trial reaches a contradiction placing two 7’s in r1. They are the N7 and NW7 effects on the bottom line.



FM 36 orangePromotion of the orange candidates produces only one additional clue, SW9, and the remaining the candidate grid resists advanced methods. But we’re almost there.

Next time we’re going for a pink olive analysis of the 1-patterns, where freeforms from the left side are significantly restricted. With a little boning up on that, you could get there first.

My pink 1-freeforms will start at r7c1. Go for it.

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Trial 13 of Fata Morgana

This post checkpoints the third “figure 8” nice loop generated by the FM exocet, and follows up on the first, the exocet trial Cnp13

FM 36 ANLAs to the homework, Yesss! It happens again. The FM exocet defines the endpoints of two X-chain ANL that link together at the ends to form a luxuriously long nice loop

Were you tempted to go ahead with the Cnp13 case?

You really could have. I wouldn’t mind.



FM 13 nice loopUnless you just joined us, it goes pretty well. First, we color the two directions of the nice loop. We have a major result right away, so obvious I missed it at first. Got it?


Yes, two orange candidates in the crossing of the loop is a color wrap. Red candidates are true, in the 13 trial, that is.


FM 13 deadlyGoing red, the departure of orange candidates leaves a naked triple c7nt457 that confirms SE6. Together they create a deadly rectangle of 1 and 3 candidates.

No, thanks. If there is a solution here, that makes it at least two. I think that our chance of discovering an undiscovered multiple in the famous Fata Morgana is too teeny weeny to bother with. I don’t play the lottery either.

So do you want to wring out the Cnp16 nice loop on your own? Be my guest, but you only have one week.

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The Fata Morgana Exocet

Here we do the Sysudoku basic solving of the celebrity monster Fata Morgana, and develop a trial plan based on its exocet. The Sysudoku interpretation of the exocet as a trial generator is reviewed and two of the three exocet trials are defined in detail. A third is left for sysudokies wishing to get close and personal with this monster.

Diligent readers might recall the relentless Sysudoku campaign of the Golden Nugget exocet. The exocet has been widely described as an elimination method, but against the GN it was a trial generator guaranteed to corral the solution. The GN had the monster trademark, an intimidating fog of candidates emerging from basic solving, blinding advanced solving methods. In GN’s case, the exocet generated a series of four closely related trials, with interchangeable forcing chains. Each trial cleared away enough fog to allow solving to a contradiction, and in one case, a solution.

Basic solving of the FM reveals another exocet. The bypass, box marking and line marking leave you in dense fog of candidates:

FM basic trYeah, that’s it, pilgrim!

FM basic gridWhen you see this field of candidates, you know you have a monster.

Having seen the exocet mentioned in discussions of Fata Morgana, I first looked for the 4 candidate base cells and two candidate target cells similar to the GN exocet, but the ones I saw were spoiled by clues. Then I realized that the r5c46 base and the r6c8and r4c2 targets qualify as a 3-candidate exocet.

If you don’t mind a little math, this FN exocet is even better than the GN’s. In the GN exocet, there were four candidate numbers in the two base cells. That comes to 6 combinations of two numbers, or possibly 12 grids to be tested.  In the FN exocet the three candidates can form only three combinations and only six possible solutions. The critical issue is the effect that each combination has on the candidate fog.

For the GN I prepped for the 6 exocet trials by working hard on pattern analysis, and had some restricted pattern sets to help out, but it was still a lot for Sysudoku readers to wade through. But the effort convinced me, at least, that a logical solution to such a monster is humanly possible, and worthy of the attempt.

Let’s start the FM with a base solution of 1 and 3. We can hope, as in the Golden Nugget, that we can meet the exocet condition without having to designate which cell has which. Remembering the GN trials, we look for 1-chains and 3-chains between the target cells. X-chains are not as easy as XY-chains, but they glow infra-red in the monster fog.

FM 13 ANLThe X-panels reveal a long-1-chain ANL c1. This prompts a look for the similar 3-chain ANL, which is just a bit harder, requiring some grouping.

But that’s OK. Chain construction, as opposed to searching, is what sysudokies do.

Remarkably, the ends of the ANL stitch together to form an AIC nice loop, eliminating 246r4c2 and 6r6c8. The target solutions are guaranteed to be 1 and 3 by the ANL forcing chains, meeting the exocet condition for base solution 13, without specifying cell locations for 1 and 3 candidates.

There are more immediate consequences of the nice loop for you to discover, and I will catch up to you in the next post, but let’s continue here with the other exocet combinations.

FM 16 ANLIn the Cr5np16 trial, the same 1-ANL is joined by a 6-chain ANL and the two form a nice loop removing 234r4c2 and 3r6c8. Can you believe it?

Again we leave the nice loop follow up for later, and move on to the 36 combination.

Have you been paying attention? If so, you will have no difficulty in finding a nice loop for the Cnp36 trial, to be displayed next time.

Covering the three possible exocet solution pairs will guarantee the elimination of candidates 2, 4 and 6 from the exocet target cells. That is the stated objective of the exocet elimination method. But it will also guarantee that one of the three trials can end only in a solution of Fata Morgana. That is the objective of an exocet trial generator.

Experts, please compare the logical transparency and human accessibility of this mode of solution with the others published. That is the objective of Sysudoku.

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Closing the Weekly Extreme Review

This is the final post of the Sysudoku WEC collection review. It opens with a checkpoint on the finish of a pink/olive pattern analysis and with it, Weekly Extreme Competition 435. A review table shows the features found in each of ten consecutive competition puzzles, and their points of collapse. My rating has the WEC collection somewhat less than extreme.

wex 435 pink setupMy grid setup for the pink patterns trial doesn’t include removals of the orphan and other candidates bypassed by the pink freeforms, because each of these candidates “see” a pink shaded candidate belonging to all pink patterns. They will be eliminated in the normal trial marking.

The arrows show how pink quickly confirms blue.


wex 435 pink trIn a trace remarkably similar to the jump ball trace earlier, marking stalls.

The extension of the red/orange cluster also stalls leaving the grid below, with one trap.

wex 435 pink clusterThe red trial fails to provide a 1-clue for c6, and the orange marking easily reaches the solution.

I’m happy with Wex 435 as an example of the dogged effort you may have to make to truly solve a competition puzzle. At least no arbitrary guess was made. There was a logical game plan and logically constructed trials in effect the whole way.


Special thanks to the Weekly Extreme Competitors who followed these posts, and to the Akron competitor who favored me with a copy of Wex 426 at Akron’s 2014 tournament. I hope that many WECers have found something interesting to follow up on in this review.

The conventions for the review table are on the review page.

wex review table

The Sysudoku blog has evolved four primary ranking categories for reviewed puzzles: basic, advanced, extreme and monster. Advanced methods are those requiring the full set of candidates. Except for Sue de Coq verification, I defer color and pattern trials until I come to believe the puzzle won’t be solved by advanced methods. Therefore such a trial means the puzzle has at least a provisional ranking of extreme. Also I consider pattern analysis, ALS enumeration and nice loop coloring as extreme methods.

With this necessarily limited sample, I have to rank the Weekly Extreme Competition review puzzles at around 40% extreme. That’s lower than I expected, but high enough to say that consistent success in the Competition, without arbitrary guessing, is a very worthy achievement in Sudoku solving.

The primary reason for the low ranking is the consistent appearance and success of the single alternative form of Sue de Coq. It may just be a statistical fluke. On the other hand, it could come from an unintended bias in puzzle composition. Whatever the reason, my Sysudoku single alternative Sue de Coq may keep you on the WEC honors list with a minimum of T & E.

I’m happy to acknowledge that the toughest puzzle of the review, Wex 435, shoved me forcibly into the a new type of trial, the “jump ball” trial between two ALS.

The review table can be compared with those of other collections reviewed on this blog. Your rating could depend on how extreme you consider the single alternative Sue de Coq. Based on theory required and difficulty of implementation, I rate the SASdC as advanced, but not extreme.

WECers visiting my blog for this review, please note also how competitors might enjoy comparing their solving experiences after solutions are published, by exchanging emails of grids and traces. The puzzles certainly deserve this type of attention.

If my ranking explanation above left you wondering what a monster is in sysudokie land, it’s a puzzle that requires some extreme measure, usually a trial, to clear a dense fog of candidates before advanced and extreme methods can even get started.

fata morganaYou can get to my previous experiences with monsters on the Find It page, but also just stay tuned . For 2015, I’m pinning a sysudokie target on the infamous Fata Morgana, and will begin the campaign next post with the often mentioned, but rarely detailed, exocet skirmish. Sign on by doing the basic solving on Fata Morgana and reviewing the post on Golden Nugget’s Exocet.

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The ALS Jump Ball

The last review puzzle from the Weekly Extreme Competition, Wex 435, evades the Sue de Coq to give us: a precaution about grouped forcing chains in XYZ-wings, a genuinely extreme patterns trial, and an obvious form of ALS trial to invoke when the WEC deadline looms. With the Cavaliers in the NBA playoff finals this week, what else can an Akron guy name the new form of trial?

wex 434 collapse 3 trBefore getting into it with Wex 435, here is the checkpoint trace finishing the collapse of Wex 434. The last run is shifted left to continue the trace.

If you are just came in, this is a continuation of the previous post and the end of a long basic trace of a Weekly Extreme Competition puzzle solved in Sysudoku box marking.

wex 435 basic gridNo such result for Wex 435. The basic trace follows below.

WECers, are you on to slink marking now?  Don’t you agree that this pencil marked grid is more informative than any you’ve seen elsewhere?


wex 435 basic trIf your version of the basic grid didn’t get that far, you have something to learn from these traces of bypass, box marking and line marking:


True to form, the first advanced solving event is a single alternative Sue de Coq:


wex 435 sdcThe contents of Wr5, described as 1(5+7)(4+8), excludes 814, as shown by the weak links (winks) of the diagram. The verified SdC gets only another bv, and a new strong link (slink) in c9.

Time out for ALS story! Since covering the toxic sets created by a pair of ALS with a restricted common, I have made very little use of them. Now I’m on the lookout for them in the bv scan, either in the conventional ALS-XZ manner, or with a mini trial, ala Gordon Fink.

wex 432 ALS-XZFollowing my review of Wex 432, Gordon let me in on a beautiful example of the classic ALS-XZ that I overlooked. It comes right after line marking, where I tried a 123-wing(gold) and used the single alternate Sue de Coq (purple chute). The blue and green ALS have 2 in common, leaving their 9 candidates toxic. The new clue 7r4c7 collapses Wex 432 forthwith.

wex 435 ALS jump ballBack on Wex 435, I encountered this victimless ALS toxic set . Gordon’s forcing strategy on the Alary More Extreme 200 made me do it. I’m using this ALS pair in a mini trial. For reference I’m calling this form of trial an ALS jump ball.

The ALS in r9 and c1 in c1 are competing for the rights to the true 5 of the SW box. One contains the true 5 and the other doesn’t. The 1’s and 3’s of the two ALS are toxic sets, but it’s useless to look for a candidate seeing all candidates in either set. However, assignment of the restricted common value to one of the sets can produce a decisive trial set.

wex 435 jump trThe trial assignment of 5 to r9c2 is a good example. The trial trace reveals a promising response. But there it stalls, but I have looked long and hard and the Competition deadline looms, so I press on, adding two coloring clusters. Can you identify with that?

wex 435 3 clustersI started with red/orange and added tan/aqua. The clusters show that Wex 435 is extremely slippery, coming close to multiple solutions.



Ready for some bridging logic? We can look at what the clusters forced into r3c8, and say that orange is true.

Or if you want to explain it to a programmer, use “!” for not, “&” for and and “+” for “inclusive or”, and say

!(red & aqua) & !( red & tan) = (orange or tan) & (orange or aqua)

= orange &(tan or aqua) = orange

Then r1 reveals that orange => tan.

WECers, you know about bridging logic, don’t you?

wex 435 ALS and ANLThe resulting collapse forces all 1’s from c5 and all candidates from r1c5. The r1 ALS loses and 5 goes to the c1 ALS. This doesn’t produce much, only (SW5, r9np3) in trace talk, but hey, progress has been made.

The next elimination comes on the XY railroad, with the XY chain ANL in black, or it may come on the amended 1-panel, with the 1-chain ANL in red. One more bv, but checking back on the bv maps, no fireworks.

wex 435 diconnected clustersBeyond that, I’m bogging down. But with all those bv around, I have to color it up again. It’s the most effective way to enlist the strong link network. This time, the two clusters are not speaking to each other via common numbers. Also, I’ve not found AIC extensions or successful chains, so I’m going with pattern analysis to build stronger trial set.


Make it decisive and make that Competition deadline in good logical form!

The 1-panel looks promising, containing slinks in both directions. I start the freeforms from the bottom, which allows the smallest number of starting points for freeforms. Note that placing that 5 over the 1 in rpc1 makes a big difference in that pattern.

But there’s more, The naked pair SWnp13 earned by that hard fought trial brings the 3 patterns into line with the 1 patterns in a combined pink/olive pattern analysis:

wex 435 freeforms

As intended, the freeforms divide the patterns into two distinct sets. One of these sets contains the true pattern; the other doesn’t. In each set, the shaded cells are in every pattern of that set. In a trial of a pattern set, the candidates not crossed by a freeform of that color can be removed. For either outcome of the trial, we will have clues and removals. Probably a decisive number of them.

There is one removal without trial. Both sets of 1 freeforms bypass r2c2. This candidate belongs to no pattern. It’s an orphan.

Did you know about the humanly practical use of patterns before this review? If so, where did you learn about them? I had to figure it out for myself, in an attempt to confirm Andrew Stuart ‘s evaluation of patterns as beyond human comprehension in his The Logic of Sudoku. Look on the Find It page to locate the posts of my experiments. At first I called it limited pattern overlay, or LPO.

Anyway, I chose the pink patterns, which will have more shaded cells, and almost as many removals, in the trial. But why should I have all the fun? Yes, WECers, there’s homework. Your assignment is to go back to the full grid, and looking at the pink freeform panels, formulate the trial and carry out as much of it as you care to. (I bet you can’t put it aside.) If you hit a stall, color your way out. I’ll have the finishing checkpoint, plus a review table and conclusions on this Weekly Extreme review series in next Tuesday’s post.

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A Tracing Exercise with Wex 434

A careful marking session finishes Wex 434 in box marking! This surprising event demonstrates how 2-D depth first tracing fully exploits removals and confirmations.

My first time through to a solution I considered Wex 434 among the most deserving of the review puzzles for the Extreme ranking. I had found a naked triple in line marking, and very subtle hidden boxline on line marking close. Then there was the verification trial of another Sue de Coq, with a removal that 434 shook off easily. Then the XY railroad delivered a 13 link XY-chain removing two candidates, allowing it to curl into a nice loop removing 3 more. Finally, a coloring cluster was extended for trial by pattern analysis.

Then, getting ready to write all this up, I went over the anemic bypass again, in case I missed something. Indeed I had, but as is often the case, a better bypass led to an even better box marking. So much better, that it came out with the solution.

wex 434 bm 1-8 trFirst off, let’s take a look at where we are after the bypass, and the first 8 lists of box marking:


wex 434 bm 1-8This grid is the third slide of my PowerPoint file. The first was the puzzle, with givens grouped onto the template. The second was the bypass, allowing me to check it and go a little further. Now there’s usually one slide for the box marking grid. The traces go in the slide comments window. Would you have uses for such a record of your solving experiences?


The collapse comes next, on box marking list 9. It demonstrates the value of tracing as a solving aid. The trace enforces a systematic unfolding of the collapse, saving every unfinished inference, and presenting it at the right time for you to record the effect. Nothing is overlooked.

Tracing also helps you communicate with other solvers by exchanging slides and files. Writer and reader follow the same order of events. And, as writer and reader, you can determine exactly where you went wrong when a contradiction arises. I use that advantage a lot.

Now let’s look at the stacking action as the collapse begins. Only the first cause on a list gets immediate follow up action. The remaining causes are left on a list. The lists are effectively stacked as we move down the trace, and retrieved for continuation last in , first out, order.

wex 434 collapse 1 trwex 434 collapse 1The grid at this point:

The north boxes are filled, and the only reason that Center and South boxes are not, is that effects of stacked causes are not marked yet.

We have reached the end of the depth probe, and next we move back up the trace to pull out the stacked lists as we encounter them.

We replace the NE4 pencil mark with its full size digit. On the trace, wex 434 collapse 2 trwe shove its symbol ‘NE4’ right, and list its effect E4. When all the effects generated by this list is finished, C2 is pulled to the right. As all of its effects are worked out C4 waits its turn as backup, if the bottom list runs out of steam.

wex 434 collapse 2 gridI often “assign” readers a follow up on a newly introduced topic, and in this case, that topic is the collapse of Weekly Extreme 434 in sysudokie box marking. Read the trace to fill the grid, and to become familiar with the Sysudoku 2-D trace conventions. Then complete the puzzle from here, with tracing. You can compare your trace with mine on the next post.


Next week, we begin the last puzzle of this review, Weekly Extreme 435.


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Two More Wex Join Sue’s Parade

wex 432 basis trIn Weekly Extreme 432, a Sue de Coq shows some promising removals, but we never get there, because the verification trial solves the puzzle.

After moderate basic solving in all three Sysudoku stages,



wex 432 SdCWex 432 presents the Sue de Coq    Cr5 = 8(2+3)(1+9) + 189,

with tempting removals and two clues in r5. But first, it is necessary to verify that alternative (2+3) is not missing from cells Cr5, leaving 189.

The trial reveals that Cr5 actually is 189, and the removals would not have helped us meet the WEC deadline.


wex 433 basic gridA similar fate awaits Wex 433 whose line marking ended in a rather spectacular set of removals from a naked pair.

Again, the wannabe Sue de Coq NEr3 = 2(1+6)(3+4) , does not remove 4r3c5 as anticipated, because alternative (3+4) turns up missing, and the naked pair NEnp16 leads to a collapse.

Like me, you may be getting weary of these SdC trials, but here’s something worth your time.

wex 433 AICextensionSuppose we missed the Sue de Coq and colored the bv remaining from the naked pair, extending it as far as possible. Now it happens that 3-slink of c5 can be combined with a 9-slink in r5 in an AIC that is a slink between 3r4c5 and 9r5c9, extending the cluster to the 9 and to its 1 bv partner. This traps the other 1 in r5, confirming 2r5c6.

The AIC strong link can fit into any methods where a slink is used.

Two more Weekly Extremes in this review. Let’s see what they bring.

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