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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A Logical Fallacy of Multiple Solution 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 may violate the strong link 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 earlier, diligent readers discovered my miscopied puzzle has 63 computer solutions, and my additional solving error that made it appear to have the unique solution 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 extend 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, and the cluster is extended to the two 4’s. The South box AIC slink and coloring extension are similarly verified.

Now 2r9c2 sees both colors.  It is false whichever color is true. These eliminations generate a clue 2r3c2.  Also, both 12 and 67 traps are valid. AIC strong links have to be accepted as part of the slink network represented by coloring.

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.

Next we notice a large number of solutions placing 1r7c2. This marks them as solutions that do not recognize the SW AIC slink elimination. In the solutions that do, blue places 9 and green places 4.

Are you getting the drift? Although these solutions agree with the givens and clues derived to this point,


they do not accept the restrictions imposed by the slink network derived from them. And why should they? Backtracking tree search finds all solutions, without regard to any strong links between candidates.


It turns out that all 60 of the “extra” solver solutions have the same type of logical inconsistency. Here is the set placing 1r3c2 where the 2 clue has to be:

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.

So what do we make of this? 

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.

But is a unique solution automatically consistent with the slink network of Sysudoku coloring and AIC?  The answer, thankfully, is still “yes”, based on the slink definition:

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

Nothing else, please. Not “same value”. Not “in the same unit”. Not ” A => not B” as well.

It’s surprising that I never thought to use an AIC slink as I did here, but doubly surprising to discover that logically compromised Sudoku solutions could exist.  Raising the issue invites a more serious examination. If this embarrassing case is verified, more direct cases will be found. Human solving power is demonstated, but does this mean that the concept of a Sudoku solution has to be qualified in barely detectable manner? 

No.  Going back to the definition above, (not A) means “not in the solution”. It does not mean “not in any solution”.  For multiple solution puzzles, the strong link definition is meaningless.

That delivers us from this dilemma.  We can have the AIC slink and the honor of Sudoku as well. Coloring is as simple as I thought.

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.

On a lighter topic, 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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A Very Hard Finale with Ardv2 398

A hard basic leaves it to an unusual ALS to shake things loose, enabling an XY and a UR to wrap it up.  Maybe the hardest of the pre-selected Very Hard.


Line marking looks easy, until you look at the completed grid.








Fortunately, and ALS-XZ comes to the rescue.  With 9’s in common, one ALS has the true 9, and the other, the true 6, and 6r7c7 sees all 6’s in both ALS. The 5 clue removes 5r1c7, leaving a 146 naked triple in r1.

There the action stops.






Turning to coloring,  a second cluster fails to produce a bridge. Doubling back to update the bv map, I realize that the 5 clue’s removal of 5r7c8 creates a decisive XY along the altered railway. The removal wraps tan and confirms yellow. Yellow wraps orange, and its yellow and red.



Yellow also sets up a productive UR, Type 2, removing a bunch of 6’s.  The resulting E1 clue confirms blue, and there’s nothing left to say, except that Ardv2 398 was exceptional in its display of advanced methods.

Next post concludes the Ardson Very Hard v.2 review with a redo of the corrected Ardv2 38. Just in time for the 4th.

But more fireworks come a week after. The side issue about multiple solution puzzles, brought on by my loading typo on 38, and prolonged by a missed pencil mark, is taken up again in the next post. 



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A Little Harder Ardson v2. 358

Next to last, and getting beyond the XY railway, is A.D. Ardson Very Hard, v.2 358. Here is the basic trace, including an X-wing in the close of line marking.

The column 6-wing  was found in line marking closure, when the column 2 6-slink matched the column 9 6-slink from the bypass:

Persistence results in an XY elimination that seems less decisive than it turns out to be.








Moving on to the X-panels, the first one yields a 1-chain ANL (almost nice loop), and the N box removals create a new slink and a slink elimination in the South box. Elsewhere, this is described as a type of box/line interaction.




More decisive action comes on the 5-panel.  My habit is to look for fish before X-chains on a panel. Here I get a kraken 5-wing, where one of the finned fish targets sees the fin via a five link 5-chain. Simpler and better is the 5-chain removing the kraken victim and one more.





The trace runs until there are only two cells of three values holding off a BUG.




Coloring brings an easy wrap of green in r9, and its done.

The review of A.D. Ardson’s Very Hard v.2 closes next time with Ardv2 398. It is the most challenging of the puzzles pre-selected for the review.

Following that, we’ll double back on my miscopied version of Ardv2 38. Is your comment ready?



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Ardson Very Hard v.2 318

The review continues the pattern with Ardv2 318.

Ignore the pencil marks and pick out the givens if you have to, but try the bypass.  It gets close to the edge, leaving three 3-fill lines for box marking.







The full basic trace leaves a Type 2 unique rectangle.







A Type 2 is a form of box/line. The candidate that would remove the single extra value in two corners must go. The regular box/line removes the other 7’s in the North box.






But again, the bv playground allows the XY rail to romp, and a decisive XY chain just happens.

Hang in there, we’ll move on to 358 next week










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