Ardson Very Hard v.2 158

After a one week pause, we continue the A.D. Ardson v.2 review with puzzle 158.

If you don’t have your copy, just load one from the Calibri font givens here, and do your own solving. Then compare with this checkpoint. It arrives with line marking almost done along the rows, and a productive 4-wing.

The basic trace, gets us to the next grid.

The ample bv field yields no UR’s or XYZ-wings for me, but there is a completely decisive XY chain.

Next is Ardv2 198, from A.D. Ardson Very Hard Puzzles, v.2

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Ardson Not So Hard Ardv2 118

Ardson Very Hard v.2 118 shows off the basic steps, but can’t quite dodge the consequences. A good Sudoku basic workout, and reminding us of the joys of basic solving.

Here is the basic trace, completing box marking with a naked quad.


The grid completes the checkpoint.









The column 6 marking completes a naked triple in r1, which restricts r1c8 enough for the naked pair 26 in the c8 marking, and clues NE9 and SE3, folding up Ardv2 118.

Next time, we move on to Ardson Very Hard v.2 158. That will be in two weeks, May 16, because of unforseen difficulties.

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An Ardson Very Hard Workout

This post checkpoints Ardv2 78, the second review puzzle in A.D. Ardson’s Sudoku Very Hard collection. The puzzle skates by basic and the bv scan, but is tamed on the X-panel.

Along with regular collection reviewing posts, I’m revising the first year’s posts 2011-2012 to bring graphics, tracing and solving examples up to date. That makes the opportunity to take a serious look at puzzles like Ardv2 78 especially valuable, as holistic tests of the fundamentals and order of battle of Sysudoku.

Here the policy of subset discovery in basic Sysudoku is rewarded with a naked triple in the closure of line marking.

The trace and the line marked grid illustrate the movement of the focus of attention. The last row to be line marked is r1. The marking fills the last cell of c2 and that is the time to look for subsets in c2. The NW box is locked at the same time. The time to look for subsets in the boxes of the West tower is in the follow up to line marking of c3. Subsets in the N, C and S banks are sought in the follow up of the c8 closure.

The next point of interest in Ardv2 78 is its remarkable resistance to the bv scan, with such a rich bv field. Here are the possible XYZ-hinges rejected.



A related symptom of the same disease is the lack of productive XY-wings along an extensive XY Railway. Here almost all of the duplicated candidates are in the same unit.

But before I go looking for a Berthier t-chain, I need to get out the X-panel and the crayons.

Help is on the way. First the 3-panel, grouped 3-chain gives up two candidates and a clue before its group breaks up. Actually no, that doesn’t work, because the c1 link is a wink. Reader Dov Mittleman caught that one. Removing that, a 4-panel slink chain allows itself to be interpreted as an AIC, for another removal.

Then comes a swordfish in parade formation on the 6-panel.

After some follow up, the 9-panel weighs in with a column-wise swordfish. The fish removal triggers two box/line interactions.







The crayons have the last word. I started with blue 1r5c2, and the cluster just spreads from there. Two green 3’s are forced into r9, and the blue army sweeps the field.

The extensive cluster tells you something about why 78 is “very hard”. A large region of the grid is dominated by blue/green bv. This region can’t distinguish between two solutions.

Did an Amazon drone fly in the window with your volume 2 of 400 Ardson very hard Sudoku puzzles yet? If not, here is a copy of Ardv2 118 for next week.



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A Run Through Ardson’s Very Hard Volume 2

This post begins a review of A.D. Ardson’s Sudoku Very Hard Puzzles, volume 2. The review will cover 10 pre-selected puzzles from 400, every 40 starting with number 38. The review table will follow in the concluding post.

Ardson’s puzzles start deceptively, with an generous set of givens, often with numbers completed through banks and towers.  For one thing, I’ve never seen the bypass start this way. Basic solving is usually balanced between bypass, box and line marking.  In the first review puzzle, Ardv2 38,  basic trace is typical, ending with a 9-wing.

After the removals, the  remaining 9’s form a dead swordfish (no victims). Did you try the UR opportunity, the 56r16c89 rectangle?






It’s a Type 4. One slink partner among the extras is true, so the other UR partner must be absent, else a deadly rectangle is guaranteed.

After a fruitless bv scan and X-panels produced only an anemic 9 – cluster, I put in AIC hinges, and extended the cluster via forcing chains. The chain 9r9c1 to 4r9c2 is an AIC slink! The next slink coloring 4r7c2 green traps 12r7c2.  Likewise the AIC slink 9r7c4 to 2r8c5 traps 67r9c5.

After these removals, 2r9c2 is trapped, because it sees blue and green. Another way to look at it, is that both blue and green erase 2r9c2.

It now becomes evident that green forces two 4’s into r9. Blue collapses Ardson v2 38 quickly.

An eventful basic, a classic UR, and SOB alternative but an unusual move to extend a desperate little cluster. Then a tricky trap for the extension that surrenders the green army. Not a bad start for the review.

OK, I’ll preview two more, giving you time to get your copy of A. D. Ardson’s Sudoku Very Hard v. 2 to stay ahead of me through the review. It’s number 78 for next week and 118 the week after. If you do ‘em differently, but logically, comment on the reporting post. But don’t tell me anything ahead of time. Let me suffer.

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A Bypass 3-Fill on the Bus

This post reports an extension of the Sysudoku bypass to include 3-cell fills, and describes the recently republished collection of 534 puzzles, Michael Rios’ Mensa Sudoku, © 2005.

It’s great to take flight, and take in what you can absorb of another culture.  Margie and I recently had that privilege, with Virginia, an accomplished mother hen and lecturer on Mayan history and culture, and Hector, a helpful and confidence inspiring bus driver, on Caravan’s tour of Guatemala. It was a bus sojourn. The country is not that large, but there is a lot of riding time along mountainous roads between stops. And of course, I came armed with the latest Sudoku book, one that popped up on the Barnes and Noble shelf just before the trip. It was Mensa Sudoku, by Michael Rios, and by the same publishers as Peter Gordon’s Puzzlewright Guide to Solving Sudoku, © 2006 Sterling Publishing, under the imprint of Puzzlewright Press. The “Guide” originally was the “Mensa Guide”, but that association was dropped before my confirming review  of Gordon’s solving advice and Frank Longo’s more deserving puzzle collection within it.

Michael describes the collection as starting out easy and getting harder as you go. Being wary of my pen to paper capability on a bus, I only wanted to find the point the puzzles began to require box marking, and let PowerPoint take it from there, back home on my laptop.  I started doing the most knee acessible series 1,2, 5, 6, . . . then dropped back every fourth 13, 17, 21,  .  .  ., then to  every 20th: 201, 221, 241, .  .  . , still on the bypass. This series continued to 381, then closed to every 10, 391 to 531. To my surprise, the bypass solved them all.

As to the possible significance of this, I realized that in the airports, plane and bus, I had slipped into the habit of going for the 3f: (three cell) fills in the bypass, rather than deferring them until line marking. And on this collection at least, this habit paid off handsomely.

Not only that, but I was spotting these line fills quickly with an obvious trick, well suited to the bypass, which I institutionalized immediately with this maroon and grey plaque:

That is quick and simple enough to add to the bypass. I’ll revise earlier posts to do that. This means the 3-fill line joins the wall, the square, and four cornered box as triggers for immediate box marking action. Try it out.

Let’s now have a look at the bypass 3-fill in Mensa 531. You might have started with:


That’s a good start, recognizing the C wall implies E2, and picking up the 3-fill line r6.

The forth line on the c8 fill depends on which qualifying feature you use: r6c8 sees 3 and 8, but also, r3c8 and r5c8 both see a 3.

 Sweet, but actually, we’ve already missed one. You could go back to r4 as a 3-fill generated by C1 and see how many 3-fills turn up in what could the toughest of the review puzzles.

 So, the bottom line is that Michael Rios’ Mensa collection is a basic collection, good for travel, but lacking in advanced challenges. But that’s OK. There’s a time and place for everything.

Next week, we begin the review of a collection which may be more like KrazyDad’s SuperTough and Insane collections in stretching Sysudoku principles. It’s one of a number of “very hard level” collections by A.D. Ardson published in 2016. The review posts start with Sudoku Very Hard Puzzles, Volume 2, number 38. Perhaps you’d like to have your implementation of Sysudoku for a checkpoint

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More Satisfaction in Saturday 2

This post completes the last hour of the Satisfaction Sudoku tour of Sysudoku, with alternative solution paths beyond the bv scan in Saturday 2. It features a finned X-wing, a coloring wrap, and an alternative freeform pattern analysis with a decisive set of four orphans.

As we leave the bv scan floor we generally have two options. If we have a large number of bv cells, we usually go directly to the coloring room. But if not, we stop off here with the X-panel. The X-panel is a set of grids, one for each number, that shows only the grid position of the remaining candidates of that number. It’s where we look for fish and X-chains on each panel.

Saturday 2 fishing paid off on my first cast, the 1-panel. It has a finned 1-wing in r6 and r9.

Fins are spare candidates that spoil the fish, because if the victim of the X-wing is true, there is an extra 1-position for r6. But if the victim “sees” the fin, it is removed nevertheless, because it is true, the spare r6 position is erased along with the fish position.

Seeing the fin is fatal for a finned fish victim. Here the seeing is by virtue of being in the same box. That’s a finned fish. With forcing chains a victim outside of the fin box can possibly see the fin. If so, it’s called a kraken fish.


But don’t rock the boat. The finned fish is not the last misfortune to befall Satisfaction 2 out on the lake. The finned 1-wing removal hatches two more fish, an unfinned 1-wing(red) on c3 and c4, and a 1-swordfish(blue) in r4, r5 and r7.  They have the same victim, the former fin.

Now let’s move on down the hall to the coloring room. We tend to come  directly here when there are many bi-value cells. We call them ”bv”, singular or plural.

Coloring is a powerful tool, when the grid is ready. It is an easy and effective way to mark networks of strongly linked candidates, the clusters. At Sysudoku, we use Medusa coloring, the version of coloring that incorporates those slinks between bv partners. You know, the ones that make an XY chain an AIC.  Medusa coloring spreads the slink networks over multiple values.

Candidates of a cluster are marked by one of two colors.  The candidates of one color is true and those of the other color are false. We only have to discover which is which. So what is the catch? It must be hard to do, right?

Wrong. Here is a coloring of Saturday 2, after what’s left of it drags in from the lake. I started with blue on 5r1c5 and then colored 3r1c4, noting that 3r1c5 would be green. Then 4r1c6, because 5r1c6 will be green. Then 5r9c6. Now I fill in those greens, and look for slinks into non bv cells, completing the blue green cluster and using only strong links, exactly two per box or line. Stopped with many bv left uncolored, I dream up two more colors, and start with 5r6c1 red.

Build the cluster carefully, extending it as far as possible, then notice the two red 9’s in r6. In coloring terms, red is wrapped. In r9, true orange means green is false, therefore blue is true. More often, coloring picks off a few candidates before a wrap. The blue diamond eliminations are traps. These are due to seeing both colors of a cluster. Another type of trap is when a candidate finds itsef in a cell with both colors.

Another solving event is a coloring bridge. It occurs when a colors of two clusters see each other, as is the case with orange and green in r9. Orange and green cannot both be true, so logically, one of the opposite colors, red or blue must be true. A trap then occurs when a candidate sees both red and blue. Had that link in r9 been a slink, the two clusters would be merged into one. In Sysudoku, coloring is defined for AIC nice loops, and AIC extensions.

Now for our last stop, let’s take the elevator up to the pattern analysis lab, to see an idea conceived by Andrew Stuart, the author of The Logic of Sudoku, now at work at Sysudoku as a ©PowerPoint weapon . Watch your step there.

We return to Saturday 2 right after line marking, and take another look at the 1-panel. For this demonstration I have to tell you what we mean by a pattern. Some call it a template. A pattern is a set of candidates for a single number with one in every unit lacking a clue (given or uno) of that number. Of course there is one true pattern.

We now know that for every pattern, a freeform can be drawn across the grid touching one of the candidates in every unit that contains any.

Here is the Saturday 2 1-panel with three freeforms drawn from left to right. Satisfy yourself that the blue and green freeforms do indeed mark patterns. Now comes the challenge. Can you branch away from these patterns to mark another pattern?

When you are satisfied about that, look at the red freeform. It cannot get across c1, because the touching rows are taken. So there are only three patterns. One of them is the true one, and all of its candidates have the same color.  In Sysudoku, we sometimes group patterns to color some of the candidates.

More important there is a candidate through which we can get no left to right freeforms. We call them orphans. They have no patterns to take up for them, and they are removed from the grid.

Now look what would happen if we had this pattern information in the bv scan of the previous post. The Sue de Coq knocked out 1r9c6,  and another 1-candidate would have been orphaned. That’s right, 1r4c3!

Sorry to finish on this sad note, but all good things, in fact, all things, must come to an end. I hope you enjoyed the tour and that we will soon be seeing your questions and comments on Sysudoku posts.

Our next project here is a brief review of a collection of puzzles you might have, Mensa Sudoku by Michael Rios. The review will introduce an extension of the Sysudoku box marking. With this enhancement, the bypass solves every puzzle selected for review. You’ll be surprised how simple it is.

Here is the one I’ll use for the introduction of the bypass 3-fill, Mensa Sudoku 531. Try it on your own, with a trace, then compare with the checkpoint next week

The Satisfaction bus just pulled up. Have a great trip home, and do come back and see us.

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BV Scan Satisfaction in Saturday 2

This post continues the visit of Sudoku Satisfaction readers to Sysudoku with a tour of  advanced method principles applied in a Sysudoku solution of Satisfaction’s Saturday 2 puzzle. The roles of Rick Seemel’s Solution Rectangles I – IV in the Sysudoku Order of Battle are identified in this tour.

Welcome back, Satisfaction readers, for the final session of your Sysudoku tour. We hope your Sudoku satisfaction is strengthened and expanded by the tour, and that you will return to explore detailed explanations, examples, templates and reviews available at

Before we resume with some advanced (after inscribing) attacks on Saturday 2, are there any questions?   .  .  .  Yes, what is a fish? And is the X-wing really a fish?

Sysudoku describes a fish as a competition among parallel lines for positions along the line for candidates of a single number. In a regular fish n lines make the case that they must have n or the positions, and other lines must give up those positions. Mathematically, it is another form of subset. Sudoku has many fascinating parallels of this sort.

We say “regular” because there are many irregular forms of fish. I suggest an overview trip to the aquarium in 2012 and following posts.  The X-wing is not well named. The X generally means one number, but that applies to all fish.  The X-wing is a regular fish, n =2, and it doesn’t fit in the wing family.

Oh, I might as well say it now, Satisfaction’s Solution Rectangle I is often an X-wing. Could we have that slide of Friday 2? Thanks.

Here is the Friday 2 3-wing, as we picture it. Columns c1 and c6 must have positions 3 and 8. I realized that as I marked r8, but if I had missed it there, I would not have going over columns in closure.  Columns 3 and 4 must give up 3 positions, leaving only one in c3. Game over.

The diamonds mark eliminations and the fishy icons point in the direction of eliminations. Wouldn’t you like to throw away your eraser pencils now?  I should mention that Satisfaction’s  Example 8-2 SR I might not be count as an 8-wing in the fishing tournament, because it is laced in box marking. I called it a hidden dublex. Since the permitted 8’s in SW and S boxes are blocked from r7, the SE box has to provide one.

On to Saturday 2! This puzzle shows off several of the advanced solving stations at our plant. By the way, “advanced” here doesn’t mean “hard”.  It means “with all candidates”, i.e. after line marking (inscribing).

We’ll just walk it through. Follow me. Please don’t touch the viewing window glass.

Puzzles first come through a unique rectangle screen, which removes candidates that would otherwise generate a glaringly obvious multiple solution. Satisfaction addresses some of the six regular UR types(see Tools on bar menu) , as an SR II.  Saturday 2 came through unaffected.

The next scan is for Sue de Coq, APE, BARN or ALS-XZ removals. Saturday 2 had a Sue de Coq infection requiring the removal of three candidates. “Sue de Coq” is actually the pseudonym of it’s discoverer. The solver writes a logical prescription for the contents of a chute (slot) with two alternate pairs of candidates forced by matching bv (bi-value) cells in the box and line of the chute. In this case,

Sc4 = 2(1+9)(3+4),

that is to say,  2 and (1 or 9) and (3 or 4). Now in c4, cell r1c4 must supply the remaining 3 or 4, so 4r6c4 must go. Likewise, 1 and 9 in S must go.

Other forms of Sue de Coq are detailed in Sysudoku, including one form developed right here, for trials.  What are trials?  A trial determines if a set of candidates, assembled for the purpose, are all true or all false. Its what we do when we don’t know what else to do.

Almost Locked Sets (ALS above) have a role in Sue de Coq.  Two of them can form an ALS-XZ with one or more toxic sets. APE is another way for bv to nibble candidates off of an aligned pair of cells. BARN is a form of bent (dove tailed) naked(plain) set of candidates in a union of a box and line. All of these are installed here at Sysudoku.

We watch now as Saturday 2 goes through the XYZ-wing machine.

An XYZ-wing is a hinge cell with candidates X, Y and Z, with two bv wing cells, XZ and YZ.  The X’s and Y’s are wink (weak link) partners. One of the Z’s is true, because if neither of the wing Z’s is, then they strip XY from the hinge, so the hinge Z is true. This makes the Z’s what we call a toxic set . “Seeing” all three gets you deported. The rectangular form of the XYZ-wing does fit the definition of Satisfaction’s SR IV, but seldom has victims.

Saturday 2 escapes unscathed, but not before showing us a valuable advanced tool, the forcing chain wink. We call this an irregular XYZ-wing, or iXYZ, because one of the wings is attached to the hinge by a chain of links. It is an alternating forcing chain (AIC), in this case an X-chain (single numbered links). In our grid diagrams, dashed curves denote winks and solid curves, slinks. Following the chain from wing to hinge, If 1r9c4 is true, 1r9c3 is false, so 1r7c1 is true, so 1r3c1 is false. The wing 1 sees the hinge 1. It’s a wink. Empty Rectangle(ER) is a special case of the forcing chain wink.

How about a look at Sysudoku’s XYZ control panel? Forcing chain winks in advanced methods is a distinguishing feature of the Sysudoku brand, but Saturday 2 is not impressed. I couldn’t find a 4-cano that “sees” all three 4’s of this iXYZ-wing’s toxic set (the squares). Want to try?

The XYZ map starts with the bv cells, then cells of three or four candidates are admitted to the XYZ map if wing components are available. As each hinge is admitted, the full grid is explored for AIC attaching wings to hinges, and when successful, outside Z numbers to toxic sets. Failing hinges are crossed off. At least my attention is focused on one task at a time.

Another tool of the same nature is the Sysudoku XY Rail for XY chains and loops. Starting with the same map of bv cells, draw all the curves you can through and between bv partners and between cells on winks. XY -chains lie along these curves. There is a slink between bv partners (think about it) and treat every link between partners as a wink. Slinks are winks as well (think about that, too). The result is an alternating chain, the simplest form of AIC.

Saturday 2 provides an excellent example. On any AIC, numbers  that match at the outer ends of slinks form a toxic set. One of them must be true. The reason is evident here. If 3r2c1 is false, then 2r2c1 is true, as is 9r2c6, 1r7c6, as is 4r9c4, as 3r1c4.  And reversing direction, if 3r1c4 is false, 3r2c1 is true. The diamonded 3’s see both.

After you draw it, this XY-chain looks complex. But it’s not. You don’t search for these things. You construct them one link at a time, along a guiding curve,then interpret the results. You only search for victims.

Three- cell XY-chains have an additional name, XY-wings. They work the same, of course. In Satisfaction, these are SR IV .

Before we leave the bv scan floor, I think we should take a short break. The cafeteria is right over there.  Meet you here in ten minutes (next week).

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