David Badger’s 1001 Hard Sudoki

David Badger, a prolific composer of giant grid Sudoku puzzles, now has a giant book of 1001 standard size puzzles, self rated as “hard”. The publication date, listed in the back,  is June 17, 2017. Here is the standard Sysudoku review of this collection, pre-selecting puzzles 1 to 1001  in steps of 100.

David does many things a little differently, besides grid sizes of 12 x 12, 16 x 16, and 25 x 25 cells and box sizes of 3 x 4, 4 x 4 and 5 x 5.  Instead of extending fill symbols by alphabet, such as the hexadecimal number system {1, 2, …, E,F}, David uses two digit decimal numbers as filling symbols. That must make pencil marking a challenge. His 1001 Hard book carries no copyright notice. Like many of today’s collection books, there is no text of any kind.  Pages are large enough for four grids, with answers at the bottom of each page. The puzzles themselves are formatted in the customary way.

For an overview of the Sysudoku difficulty level, here is the review table:

Two reached advanced, one bypass victim, one box marking victim, and eight collapsing in line marking.  Anything can happen. 

I did 11 pre-selections because puzzle 1001 had a full page rendering, which I took as possibly significant. Turns out, it wasn’t. On 701, the bypass victory depended upon a two 3-fills, and finishes with another.

Here is the grid where the trace ends. Mop up is freestyle.










The advanced track of David’s 301 wasn’t typical .  You might like to examine these three potential UR’s, Dr. Holmes, and ferret out why they fail to eliminate any suspects

From there, investigate the XY railway of his beast, and find an ANL and extension ANL that eliminate a bunch of 7’s.




The extra bv will expand the railway into a  serpentine maize, to which you are to apply Medusa coloring, for a glorious finish.

I’ll checkpoint you on this next time, along with a play-by-play of 101, shown here. You can check me out on the above 101review table report, and find its advanced goodies.


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A Bifurcated Coloring of the AIC Slink

In this post, the AIC slink produces an alternative solution of a puzzle originally solved by a rather extreme pattern coloring analysis. This solution demonstrates how the AIC Slink enhances the already powerful technique of Medusa coloring. The July 11 post introducing the AIC slink has been revised.  It now identifies the condition allowing coloring to be extended via the AIC slink, and names it the not-both condition.

Going back to the KrazyDad Insane review of July 2013,  I thought to have another look at my earlier reliance on the more advanced to extreme methods in that review. Have any innovations or experience since then made any difference?  I was shaken up a little by this data point. I may need to do more.

You have the givens grid, and possibly tried out the bypass route with it.  Anyway, here’s my basic trace.

The bypass makes it look a little different, but the result is the same.

This time, the 5-wing is detected earlier. The icons prevent the addition of 5 candidates in rows 3 and 7 as columns are marked.

Newbies, the font changed when Microsoft changed the definition of the font encoded into four years of slides. In the blog, you’re looking at pictures of those slides.


That earlier attempt amounted to a color trial which I now consider a “last resort”. There are only two possible 5 patterns, and the symmetric slink network of the 5’s allows a 4 candidate into the cluster. The bv slink in that cell is the terminal link for two XY chain slinks between the green 5 and each of the 3 and 7 candidates

Remarkably, the respective 4 candidates are false regardless of the not-both status of the two XY slinks. Take the 5447 slink. If 5 and 7 are both true, 4r9c2 is false. But if 5 or 7 is false (not-both), coloring extends to the slink and 4r9c2 becomes green.  This traps 4r8c3, asserting blue, and removing 4r9c2 anyway.

The same argument applies to the 5443 slink, removing  4r8c3. Thus blue candidates are confirmed.

Note that not-both is decisive in this case, whether it holds or not.


After a modest XY removal at right,




a coloring of the remaining candidates wraps with two orange 3-s in r9.

The red army mops up.



It was startling  to see this sudden collapse of an otherwise intransigent Sudoku, and to wonder how widely the coloring bifurcation of the AIC slink applies.




Next, a review of David Badger’s Sudoku 1001 Hard Puzzles, with preselected puzzles 1, 101, 201, . . . , 1001. The first checkpoint is 701, the only one that falls to the bypass. It’s not a pushover.


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Two of Rebecca Bean’s Less Extreme

Here the Rebecca Bean Extremely Hard review closes with a line marking finish and an easy coloring. The collection is far from extremely hard, if the sample of 12 pre-selected for this review is typical.

On the way to line marking, 9-6 of the previous post gives up an unremarkable bypass, but does offer an unusual box marking.


In line marking, the  c4 creates a naked pair in r9, suddenly reducing 5 to one place in the SE box, a hidden single.






The trace pattern of the collapse is kind’a remarkable as well.

Puzzle 11-6 is less flamboyant, but illustrates well the idea of jumping into Medusa coloring when the bv field is generous.

A vigorous bypass runs out of 3-fills, but leaves pairs. Box marking is tame. In line marking, a naked triple calls attention to the hidden single NE6.






The rows finish up. Going directly to coloring, traps add bv and the cluster expands.








In the second round, an uncommon trap, as 2r3c7 is dispatched by blue or green. As 2r4c7 turns green, 8r4c7 is trapped, turning 8r1c7 green to banish the entire green army. That’s enough to finish 11-6.

Even though Rebecca’s collection doesn’t live up to the extreme label, you can certainly  enjoy the 588 other puzzles. You won’t know when the bypass will prevail, or exactly what advanced technique might be required, but some interesting turns can be expected.

Next, I have another example of the AIC slink. It’s a redo of the first puzzle pre-selected for the KrazyDad Insane review, the volume 4, book 1 #5. You’ll find it on the post of 7/16/13. I still consider the KrazyDad Insanes my most challenging collection.

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Bean’s Extremely Hard I

A review of Rebecca Bean’s 600 Extremely Hard Sudoku Puzzles with Answers begins with the review table and two puzzles, 7-6 and 12-6.

The collection is mostly basic, with the half of them solved by the slink marking bypass.

Puzzle 6 is selected from each of the 12 sections. Two of 12 sections reach the advanced level, loaded with bv.


In 7-6, a productive bypass leaves a simple box marking, and …




In 7-6, a productive bypass leaves a simple box marking, and …

The (46) bv shout out, and one string of four define a remote pair(red). Then an XY wing(black) and a longer XY chain (green) start a steep collapse:







Or, without the long XY chain, coloring brings a wrap of green, and blue wins.









On the second puzzle, 12-6, no graphics but a trace with multiple instances of the 3-fill recently added to Sysudoku Basic. The wide, flat trace profile is a typical result.

Next week, the Bean Extremely Hard review closes with puzzles 9-6 left and 11-6 right, below. Try them out.

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Dave Green’s Three Column Bypass 3-Fill

This post traces the bypass solution of Dave Green’s Friday **** of July 7, 2017, an amazing follow up to my recent addition of 3-fill’s to the slink marking bypass. This puzzle, starting with  three column 3-fills among the givens, was displayed in the previous post.  

Over the years, I’ve several times had the feeling that Dave Green, the Sudoku composer of record for my newspaper, is teasing me with puzzles illustrating themes of a recent posts. It’s more likely to be the posts alerting me to ideas in the puzzles.

But the latest episode with Green’s 3-fill is striking. Although the line marking I’ve advocated  since 2011 includes  3-fill lines, I’ve just very recently made the 3-fill rules a part of the slink marking bypass. It’s a conscious effort to enhance human solving by reducing clutter of unexploited pencil marks, and by utilizing the human ability to visualize aligned slinks and their effects without writing them down. So here is Green with a sumptuous platter of 3-fill lines, the likes of which I had never noticed in his column before. Why now?

Anyway, let’s be thankful. Here are two solution traces. The first gets to a bypass collapse point while going for the 3-fill from the beginning. Later we look at what happens when we hold out against the 3-fill until it becomes necessary.  I recommended the first solving path last post, starting with the 3-fill of c3.

Four lines down into the trace, all three 3-fill columns, plus two rows are filled, one as a 3-fill.






More 3-fill opportunities come up as clues are added.

With collapse eminent, the grid is loaded with leftover effects, the single pencil marks in cells,  that have not been tried as causes.  The puzzle has no chance.

Maybe the 3-fill process of identifying the missing numbers, and testing how many of them are seen from each 3-fill cell is not your preferred way of doing the bypass. If you prefer, you can call on the 3-fill only when needed.

To try that in this case, we place the 3-fill last in the effects lists, and start with the regular survey of increasing numbers. That works well, but when it stalls, the 3-fill keeps it going.

Next is a two-post review of Rebecca Bean’s 600 Extremely Hard Sudoku Puzzles with Answers. In this book, the puzzles and answers are put together in 12 sections of 50. For the review I pre-selected puzzle 6 in each section. “Extremely Hard ” isn’t the Sysudoku rating, and maybe you should estimate your own rating, so here are two puzzles that will be checkpointed in the next post, 7-6  and 12-6. The second post will checkpoint 9-6 and 11-6. Enjoy.

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Slink vs. Coloring Networks and Illogical Sudoku

This post reports a surprising fact about multiple solution puzzles, that solutions found by backtracking tree search can be logically inconsistent. Specifically, such solutions violate the coloring network implied by the givens.

I happened on this fact by means of a multiple solution puzzle, my mistaken copy of A.D. Ardson’s puzzle 38 from his Sudoku Very Hard Puzzles, v.2.  As reported later, diligent readers discovered my miscopied puzzle has 63 computer solutions, and my additional solving error that made it appear to have the unique solution of the correct puzzle published by Ardson.

Here is the critical grid of my miscopied Ardv2 38. My copying error was the omission of a given 1r7c6. The solving error was identified by my diligently critical reader Guenter Todt. It is the omission of  candidate 2r7c8 in my line marking.

The unique rectangle is Type 3. Since one 5 is true in NEr1, either 6 would bring a too obvious multiplicity on the rectangle.

The AIC’s of the south bank strongly link the Medusa coloring cluster of a dead 9-wing and dead 9-swordfish to additional candidates.  The SW chain creates an AIC strong link between candidates of different values, green 9 and 4r9c2.  The SW slinks and the cell wink convey the AIC inferences both ways: If green 9 is false, blue 9 is true, hence 4r7c2 is false and 4r9c2 is true. Of course 4 coloring depends on the reverse being true: if 4r9c2 is false, 4r7c2 is true, blue 9 is false and green 9 is true. A strong link does exist between these two candidates. The South box AIC slink is similarly verified.

Here I was misled by a faulty view that it is the strong link network that is represented by Medusa coloring.  Strong links allow both terminal candidates to be true. Coloring clusters are stronger, meaning more restrictive.  They require that only one of the oppositely colored slink partners can be  true. Two readers, Dov Mittleman and Guenter Todt reminded me of this fact.

Accordingly, the “coloring” extensions beyond 9 in the grid are incorrect. The colors extend the slink network, but not the coloring network.

The requirement that chain terminals are not both true distinguishes the network types. Coloring requires an exclusive-or between terminals. We’ll call it the not-both requirement. The not-both requirement does not rule out AIC slinks.

In the South box above, the elimination is valid, regardless of the not-both. Either the coloring is valid (not-both true) or  9r7c4 and 2r9c5(both true and not-both false).

“Coloring over” the not-both requirement in the Southwest is a trial. If not-both, the SW and S slink network extensions trap five candidates. In the 2r9c2 trap, green or blue remove 2.  These eliminations generate a clue 2r3c2.  Also, both 12 and 67 traps are valid.


With these eliminations, a blue trial leads to Ardson’s solution and a green trial, to two more solutions. At left are these three solutions.

What are the other 60 solutions? When I asked my friend Gordon Fick, who also reported the 63 solutions from Andrew Stuart’s  solver, he sent them right over.

As you can imagine, a display of 63 solutions in grid form, even the superimposed type of grid here, is not practical for human comprehension.

But there is a way.

Here, stretched out row by row, are the eight solutions containing the blatant paired solutions of the Type 3 unique rectangle. We know we have them all because the 56 and 65 patterns appear nowhere else in the listing in the columns corresponding to the rectangle corners.

It’s notable that even though Andrew’s book and site deal with UR methods thoroughly, the solver includes an option to include these “dirty rectangle” solutions in its list. Solvers not based on human methods, the easily coded back tracking solvers, naturally present these. Unfortunately, this has led some to reject UR and other uniqueness methods, which are based on the expectation that blatantly multiple solutions would not be in a properly composed Sudoku.

Next we notice a large number of solutions placing 1r7c2. This marks them as solutions that do not recognize the “coloring” restrictions of the SW AIC slink elimination. In the solutions that do, blue places 9 and green places 4. The 1r7c2 also accounts for the other two SW AIC slink eliminations.

Note that in these solutions, the SW slink terminals are both true. Not-both fails. Had no solutions been found in a “coloring” trial of the AIC slink, the both true side would be tested.


Here, more solutions place 1r3c2 where the “coloring” says a 2 clue has to be:

But wait.  Five of these solutions are subject to the SW “coloring”, because 4r9c2 is false. These five renegades defy the logic of the slink network derived from the givens and clues they accept, shown here:



And the solutions not joining these groups place 6 or 7 in r9c5 where blue 9 or green 2 ( the top three solutions)  are placed.


These fail the not-both requirement for the South coloring, and but make the eliminations it requires.  They do satisfy not-both for SW coloring, and are consistent with the SW coloring eliminations.

In summary, with my completed trial of the coloring network eliminations from the slink network, I did find 5 logically inconsistent solutions. That proves such solutions exist, at least for solvers not undertaking this slink network trial, like Andrew’s. And certainly for backtracking solvers taking no account of the slink network or coloring network.

So what do we make of logically inconsistent, impossible to derive, solutions?  

The objective of Sudoku is to find the unique placement solution.  

This accidental encounter with the truth about multiple solutions may not surprise you, but it is disturbing, isn’t it?. Composers use backtracking search to check that given patterns have solutions.  Certainly they should run the search long enough to verify single solutions.

Can a unique solution be inconsistent with the slink network and the coloring network? No.

But multiple solutions? The coloring network accommodates multiple solution. It uses traps and wraps to extend itself over the multiple solutions, given enough time and patience.

The slink network, which includes AIC slinks, does not accommodate multiples in this way.

Candidates A and B are strongly linked if (not A) => B and (not B) => A.

In this fundamental definition,  “(not A)” means “not in the solution”. It does not mean “not in any solution”.  For multiple solution puzzles, the definition of the strong link is meaningless.

That delivers us from this “logical inconsistency” dilemma.  We can have the AIC slink and the honor of Sudoku as well. It is the multiple solution puzzle itself that is meaningless. The slink network of its givens is not credible, as this blog has discovered before. Know where? That’s your research assignment.

Coloring is stronger, but demands more than the slink network.  Among the solutions of the miscopied ardv2 38, I found a not-both misdemeanor, where originally I thought I had a coloring felony conviction, but I’ll take it. Three readers had my back, and I was forceful fully relieved of a careless belief.

And, let’s not forget, that when coloring would be decisive, but the not-both cannot be confirmed, then a slink network “coloring” trial is still possible. Either the coloring is valid, or both terminals are true. Here, that trial would have led you to the complete human analysis of the miscopied puzzle’s multiplicity.  Let’s not buy in to computer backtracking solutions as anything superior to human reasoning.

On a lighter topic for next time, how about an outlandish exercise on the bypass 3-fill bars? The bypass 3-fill was added to the Sysudoku bypass last April 11, with the 3-fill rule, in which you fill the cell seen by two of the missing numbers with the third number, or the cell not seen by one of the numbers with that number. That post displayed a case with two parallel 3-fills generating two crossing 3-fills, from Michael Rios’ Mensa Sudoku

My favorite breakfast chore, Dave Green’s Sudoku, just published another 3-fill wonder, in Ohio’s award winning Akron Beacon Journal. It’s the Friday 4-star of July 7, 2017. It features three 3-fill columns. Is Dave a reader?  

Anyway, it’s clear that a tracing convention is required for the bypass 3-fill, and one will be added to the trace page. List the 3-fill effects in parentheses, and place the list first ahead of other effects. That’s it.  Start your trace with c3.



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Closing Very Hard v.2 with the Real 38

To be fair, here is my do over of the real Ardv2 38, the one with a given 7 in r7c6.  My omission of it hatched a productive distraction, having no bearing on the collection it came from.

The basic solving is not much different, and closes with the same 9-wing. But this time, there is a naked pair nestled on the 9-wing, and a naked single in the South box.

As before, the first advanced step is the Type 4 Unique Rectangle. This time it closes r1.

And this time, the AIC slinks are unnecessary. The 9’s coloring is supplemented by an XY nice loop.  One nice about it is, candidates seeing both ends of any link of the loop are false.

The coloring already claims one trap, and the nice loop brings two more, 7r2c9 and 4r7c4.

But there’s more to the nice loop. Every other candidate on a nice loop is true. If a coloring cluster overlaps a nice loop, it tells which loop candidates are true, extending the cluster.







As the cluster grows, more bv are added to the XY arsenal.  This XY-chain resolves a naked pair for another extension.

At this point, green is caught forcing two 1’s into c9. Green is removed and blue readily confirms Ardson’s solution.

In conclusion, the 38 reliance on the Type 3 UR, the 9-wing and nice loop coloring helps to assure that the Very Hard v.2 collection is broadly advanced in difficulty level.  As to being “Very Hard”, that depends more on your own level of understanding, which is bound to grow with mine as we continue to explore human Sudoku solving.

Here is the review table for Ardson’s Very Hard Sudoku v.2:

Now with the review completed, next post will redo the troublesome miscopy of 38. It is a reminder of the profit that attends keeping an open mind about our mistakes.

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