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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Wex 431 Slips by Sue, but is Caught

Finally a Weekly Extreme gets through the Sue de Coq gauntlet. Or so I thought. But after a joyful romp into XYZ-wing technique, and coloring aided by pattern analysis, I get to demonstrate instead another resource of my single alternative Sue de Coq, at the expense of Weekly Extreme 431.

wex 431basic trAfter a very reasonable basic session, . . .




it looks like déjà vu all over again with a Sue de Coq filing at the deadline.

wex 431basic sdcwex 431 verif grid

The Sue de Coq

SEr8 = 8(6+7)(4+9)

is verified when 894 forces two 7’s in c2.

But the removal is indecisive, and 431 gets beyond the grid scan to the bv map scan, conceding only a new bv.

There a regular XYZ-wing collects another bv for me.


WECers, you should know how to systematically find every available XYZ-wing, so let me demonstrate with Weekly Extreme 431:

wex 431 XYZ mapYou start with a table of all the bv. I copy one from the identical bv map used for finding XY-chains. Then you scan top to bottom and left to right for any cell containing the three digits XYZ in two nonmatching bv XY and YZ. You write in the XYZ regardless of where the bv are. Then, on the grid, you look for the unit, or ER,or forcing chain weak links for the wing. When you find both, you look for victims seeing all three Z’s.

wex 431 697-wingIn this case, I found three possible hinges, but could not find ER or forcing chain winks for 58 to 158, or for 47 to 478. But the 697 hinge already had a row wink to 67 and a box wink to 79, and 7r8c9 was caught peeping at three 7’s, one of which is true. He was expelled for that offense, and I got another bv.

Looking at the 431 X-panel was not inspiring, until I noticed the large number of slinks on the 4-panel. My eyes widened a little more at the crossing row and column slinks that hold freeforms to a small number of patterns. Looking back at the grid, I was jolted further at the number of bv sporting a 4 candidate. My regular readers know what’s coming.

wex 431 4-panelsIn the coloring on the left, two clusters cover the strong links dividing the 4-candidates into blue versus green and red vs orange. Only one 4 is left out. On the right, freeforms from the left divide the 4’s into three possible patterns, one pin and two olive. The red freeform is a pink that could not be completed.

On the left, since blue and red see each other, blue and red cannot be both true. In the logic expression I use in slide comments, I have that

!(blue&red) => (!blue | !red) => green | red. i.e. green or red is true, possibly both.

On the right, the single pink pattern has green and red 4’s. The two olive patterns have orange and green (solid) or orange and blue(dashed). One of the three patterns is true.

wex 431orange wreckSo that’s the news from the 4-panel. Back on the grid, orange has a very bad day.

The red/orange cluster expands by slinks into 6, 7 and 9 candidates. And the orange 7 and 9 candidates see 7 and 9 candidates in the same bv. One of the bv candidates is true, and that means all orange candidates are false. In turn, that means that the olive patterns are false. Red and the pink 4-pattern are true.

wex 431final trWith all of that, we still have a long follow up marking to reach the solution. That testifies to the slippery-ness of Weekly Extreme 431.





wex 431 other alternate SdCBut just when I thought that 431 had eluded Sue, the Weekly Extreme Slayer, a revisit of the indecisive Sue de Coq of SEr8 revealed another single alternative formulaton.

The contents of SEr8 can also be described as 8(7+9)(4+6)+864,

the last term applying if 7 and 9 are missing. The verification that it is not missing, shows instead that it is, and spills the solution about as fast as it can be written down. That alone might prompt WECer interest, deadlines being what they are. Here’s the breadth first trial trace that erases any doubts.

wex 431other verif trMoving right along, next post is a highlight report on the fates of Wex 432 and 433.

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Gordon Fick Tricks More Extreme 200

This post reports an alternative solution, without my pattern analysis, to Antoine Alary’s special More Extreme 200, based on a Sysudoku reader’s innovative use of a little known technique, the double common ALS pair.

Two ALS sharing two restricted common numbers are a closed set, i.e. candidates of each number they share being a toxic set. My friend Gordon Fick commented on my 3/14 post, Finishing More Extreme 200, pointing out a double common pair of ALS, and then demonstrating how to gain additional removals by comparing the two ways that the true candidates of the restricted common numbers can be shared. I’m adding this technique to my extreme repertoire. I’m calling it Fick’s trick, but that may last only until Gordon comes up with another gem.

Gordon shares my interest in human engineered Sudoku solving, and brings well grounded experience to the sysudokie reader challenges. He shares my upbringing at the hands of Andrew Stuart, but is also a student of Hoduko’s Bernhard Hobiger. He comes at extreme puzzles with an open mind and a complementary point of view.

more 200 ER ANLThis helped early on More Extreme 200. While I was dazzled by the double numbered finned 78-wing, and missed a ER assisted ANL (Almost Nice Loop). Gordon saw the two ER corners for the square AIC. I’m showing it as a grouped forcing chain, but ER makes it easier to spot.



More 200 ALS-XZAfter the double wing, while I went to pattern analysis, Gordon searched the ALS field. His attention was first drawn to this ALS toxic set with common 6 and a victim 3-candidate. It’s generally known as an ALS-XZ.  An excellent example.

More 200 double common ALSWhile admiring that, no doubt, Gordon noticed this even more remarkable  ALS pair with two restricted commons. One gets a 7 and the other, an 8 and the union of the ALS cells must contain a true candidate of all it’s numbers. All of its numbers form toxic sets.

It’s a classic method, in this case with two removals. But Gordon goes further. More happens when you specify which gets the 7 and which, the 8, because, no matter which way it goes, both ALS become closed, and much more dangerous for onlookers.

As Gordon’s comment points out, if the green ALS gets 7 and the blue ALS, 8, 7r1c2 is confirmed as a clue, and the naked pair 36r9c17 removes 36r9c5. But if the green ALS gets 8 and the blue, 7, the closed blue ALS removes 39r1c8, leaving 8 in charge. In both cases, 8r7c7 and the 3r9c8 eliminated by the earlier ALS-XZ are removed as well. Also eliminated is the 7r8c9 hit by the earlier ER induced ANL.

More 200 trick tr 1The additional removals are enough.

More 200 trick remotes


The new bv r1c878 brings a series of naked pairs, with two remote pairs,




and an immediate collapse of More Extreme 200.

More 200 trick tr 2

With this mode of solution, I believe More Extreme 200 still ranks in the Sysudoku extreme class. It is very difficult to find ALS toxic set removals among the large numbers of ALS in the typical puzzle. Thanks, Gordon, for this effective reminder to look for them.

Next week the review continues with Weekly Extreme 431.


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Wex 430 Bewitched by Sue de Coq

After putting up a decent fight in basic, Weekly Extreme 430 joins an embarrassingly long line of Weekly Extreme Competition puzzles defeated by the single alternate Sue de Coq, one my early advanced solving discoveries. This time it’s the verification of the second SdC that marks all the way to a solution.

wex 430 bm trBasic marking of Wex 430 was a challenge. Box marking wasn’t easy and left a fairly tough line marking.

On row 8, on the last row of 5 free cells, no 6’s were found outside of the South box, bringing a boxline removal.

wex 430 lm gridwex 430 sdc 1A single alternate Sue de Coq appears on r1:

NEr1 = 5(3+7)(1+9) will remove 7r1c3, if (3+7) is not missing.


But if (3+7) is missing, the naked pair 19 in the NE box requires a 1 or 9 from r1c3, removing the same 7r1c3. So it has to go, (3+7) is missing or not. We are led to look to see if the classic Sue de Coq created by the removal, removes anything else. But it does not, because the required ALS are not present in either remainder.

wex 430 sdc 2I scan top to bottom a row of boxes at a time, looking at row chutes, then column chutes, and trying to look for APE and UR rectangles at the same time.

My next encounter is another single alternate SdC,

Wc2 = 2(7+8)(3+9)+923 ,

the extra term occurring when (7+8) is missing.


I had to work hard on the verification, but I can’t complain. Turns out, it was the solution.


The trace was folded at  (C6, C1), to get it on the screen.

wex 430 final trHopefully we’ll break Sue’s witching spell with Weekly Extreme 431.

But next week, we take a short break on the Weekly Extreme review, to return to Antoine Alary’s “most difficult” of the 200 “toughest Sudoku puzzles known to man”. The reason is to help report an alternative solution by a Sysudoku reader, who took up the challenge and independently solved More Extreme 200 using his own order of battle. His comment appears in the More Extreme 200 post

You too can use comments on the Sysudoku blog to bring out alternative solving ideas. It doesn’t have to be a current post.  You don’t have to saddle yourself with the management of a blog to do it. If you like, I’ll gladly add highlight grids and traces, while giving full credit to your contribution, in a follow up post similar to the one you’ll see next week.

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Wex 429 is Sued

In the review of a sequence of 10 Weekly Extreme Competition puzzles, Wex 429 is toppled by an easily verified Sue de Coq. The post begins with a checkpoint for the reader’s own solution of Wex 428 by single alternate Sue de Coq.

wex 428 sdc 2After line marking of Wex 428, the contents of chute SE r8 is described by

SEr8 =3(1+9)(4+7) +437

If alternate (1+9) is not missing, the verified Sue de Coq will remove 9r7c8, leaving SE5.

The wink and arrows show how SEr8 = 437 confirms two 9’s in c4, verifying the Sue de Coq.   The inference path comes directly from the trial trace

wex 428 sdc2 trThe marking following up SE5, remarkably, uncovers the same naked triple we saw in the last post,but with a different result. When it is finished, there is a field of 4789 candidates with XYZ –wings waiting to be found.


wex 428  XYZ skyscraperI looked no further when the XYZ map revealed this 189 wing. The victim “sees” the third 9 of the toxic set via a forcing chain. But hanging on the outline of the forcing chain is a 9-chain almost nice loop. The wing and the ANL are interlocked, in that the removal generated by each destroys the other.

Did you find this one? If so, are you worried that neither method works because a component candidate is lost? Believe me, it’s OK. The field of candidates left you after correct solving operations supports either removal individually. You’ve been using false candidates to help remove other candidates all along.

Anyway, the collapse of 428 is immediate. Weekly Extreme vs single alternate Sue de Coq is a rout so far.

Wait, wasn’t this post about Wex 429?

wex 429 bypass gridA generous slink marking bypass leaves very little box marking:

The bypass, which I came to only recently, is almost free of pencil marks. The exception is the marking of subsets. Free cells are clearly marked. It is my favorite phase of basic solving, and I try to go as far as possible with it, before box marking, which adds strong links (slinks) defined by units.

Are you still finding all candidates first? If so, stop. Life is too short.

wex 429 bypass trThe bypass trace follows the same sequencing as box marking, but is not separated into numbered lists. It is box marking, with slinks followed to their conclusions. Writing them in pencil marks is bypassed, except for naked subsets.

wex 429 bm trBox marking of Wex 429 is very short. Numbers are omitted when all 9 clues are in place. No marks were found for 2, 3 and 6.

wex 429 lm swordHere is line marking in progress, showing a swordfish found on rows that have been marked. The “fish” are left in place to prevent 5-candidates being added to prohibited columns as the remaining rows are marked.

The lists to the right and along the bottom are fill strings, giving the numbers eligible to fill that line. The line marking process is much more efficient than going over every cell scanning through all numbers to identify fill candidates.

Lines are processed, fewest free cells first. After all rows or all columns are marked, the remaining lines in the other direction are “closed” by marking line slinks and X-wings. Line slink marking makes it easy to spot x-wings in line, but I don’t try for higher order regular fish. They’re much easier found on the X-panel.

wex 429 sdcThe first Sysudoku action upon completing the candidate field is to exploit the grid’s bi-value cells. In sysudokie speak, it’s the bv scan.

The grid is inspected, chute for chute, for Sue de Coq, Aligned Pair Elimination (APE), and unique rectangles. APE generally require more bv, so SdC are more frequent. This SdC is hit near the end of the grid scan. The chute contains

SWc3 = 7(3+5)(2+6) +537.

The SdC is verified by the diagram below, and has two victims.

wex 429 verifiedThe resulting removals trigger an immediate collapse of 429, traced below for the record.

wex 429 final tr

I don’t think Sue is perturbed at WEC for any reason, but the Sysudoku single alternate Sue de Coq  is bringing the “Extreme” label into question here.

Maybe the come back will be strong, and will start with 430. We’ll find out next week.

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Sue Defeats the Slippery Wex 428

wex 428 basic trThe third Weekly Extremes review puzzle is another rewrite, in which an extreme level pattern analysis is made unnecessary by an advanced technique overlooked at first reading.

Taxes in? On the review, Wex 428 was easy on us through basic solving. A very generous slink bypass gave us quick box marking and very moderate line marking, with a naked triple to keep us awake.

Again, a potential Sue de Coq on the line marked grid appears, taunting me for missing it before. I kinda’ hope it doesn’t erase the coloring cluster extension via pattern analysis I was planning on.

wex 428 basic grid sdcYou have the verification dance by now. The chute is

SWr9 = 5(1+9)(7+8),

or 5 plus naked pair 78.

If (1+9) is an alternate, so is (7+8). But if it’s missing, the two cells in the remainder of r9 can supply the 1 and 9. But if it’s not missing, we can discard the possiblility of a naked pair 78. Then bv19 has to supply 1 or 9 needed in the chute, and 9r9c5 will never be a clue.

wex 428 trial tr 1So I scratch out a trial trace of SWnp78. It’s a long and tedious one, with a consolation prize in the middle, the naked triple(nt).  Whdx9m,  stands for “hidden dublex=>W9m”. My hidden dublex is a form of double line exclusion. See Sysudokie Speak.

wex 428 trial gridThe trial would have bogged down without the naked triple.

The SWnp78 is found guilty to a charge of forcing two 7’s into c6.

wex 428 trial tr 2Regular readers may be scratching their heads, but a trial trace can be divided like that because it is breath first and is read line by line.

wex 428 XYZ remote pairNow 428 comes apart, as a direct consequence of that 9r9c5 removal. A grouped forcing chain (or ER wink) creates a remote 798-wing, but there are no victims. However, a parallel 8 forcing chain completes a remote pair pair, a.k.a conjugate pair, removing 7r8c6 and confirming 7r3c6 as N7. I wasn’t even aware that could happen!

Listen up, WECer friends: Such revelations happen when you work advanced techniques at the fundamental, sysudokie level. Don’t bypass these catches  by guessing! Well, not after your WEC solution is submitted, anyway. Do you have that grouped forcing chain thing down?

wex 428 kraken swordWex 428 has a pretty final act. My 9-panel now reveals a kraken swordfish. The victim sees the fin via an XY forcing chain.

The collapse follows immediately. I’m showing the trace to give you another example of a regular 2-D trace. Unlike the trial trace, all consequences of a cause is explored before the next cause its effect list is explored. Its depth first.

wex 428 final trBoth traces are left to right, but in the regular trace, unused causes sprout out of a descending leg to start their own descending leg.

The review continues next Tuesday with Wex 429.

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