Sudoku Assistant’s ALS Toxic Sets

This post reviews Bob Hanson’s illustrations on ALS toxic sets, in Sudoku Assistant – Solving Techniques. For these illustrations, Hanson uses the bent naked n-set examples of the previous post. Strangely enough, this extreme economy brings insights.

Bob Hanson designed his Sudoku Assistant solver to bypass by default the enumeration and analysis of Almost Locked Sets. His reasons are similar to my reasons for placing this task well back in the Sysudoku Order of Battle. For one thing, as Bob points out, there are so many ALS in the typical puzzle. They slow SA down so much, that unless the user requests it, SA bypasses them. Also, they produce toxic sets of various sizes, and the larger they are, the fewer removals they produce.

Hanson wing ALSBob re-uses the 145-wing of the previous post to illustrate the standard restricted common form of the ALS toxic set. You’ll find examples that are not regular XYZ-wings in my ALS Toxic Sets post of two years ago. If two ALS share a number and all candidates of that number see each other, the number is called a restricted common. Shared candidates of any other number in the two sets form a toxic set.

Bob points out that an XYZ-wing is also a pair of simple, related ALS with a toxic set of three candidates. The 1-pair is the restricted common, making the other 4-candidates of the two ALS a toxic set. Of course, we don’t look for this in a cloud of other ALS. It’s much easier to find as an XYZ-wing.

The example shows clearly why the other common candidate must be shared by the two ALS. One of the ALS ultimately gets the restricted common number, but we don’t know which one. The ALS that doesn’t get it cannot give up another number. The victim would take a number from both ALS.

Hanson dbl rc ALSThe second BNS example adds another ALS toxic set wrinkle. Bob uses it to illustrate something new: that a pair of ALS can share two restricted common links and thus become a single locked set, in which every number defines a toxic set. Bob doesn’t say why, but it’s because, in the solution, each ALS loses one restricted common number and gains the other. We don’t know which is which, but we know that neither ALS can give up a number.

In this case, the 5 eliminations are relevant, being directly attributed to the double link of the ALS. The 7-candidate that Bob left out also gets clobbered. As a glider pilot, Bob is a lot more careful.

In the next posts, I will be reviewing more ALS applications from Bob’s report. His interest in them may come from the fact that enumeration of ALS comes as a byproduct of the computer algorithm finding locked sets. That scratchpad list leading up to the locked set, or failing to find one, contains all of the possible ALS of the unit. It’s all there in Subsets by Susets. Pretty neat, eh?

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Hanson’s Bent Naked n-Set

This post examines Bob Hanson’s Bent Naked Subset rules in Sudoku Assistant – Solving Techniques, and provides a more relevant rationale – and  a more appropriate name – for his technique. Also, XYZ-wings not covered by the BNS are identified, as well as types of BNS not covered by XYZ-wings.

Last post, I objected to the use of the term “subset” in Bob’s name for this formation. I’ll use the term n-set instead, the n standing for the number of values in the set of cells. In the name Bent Naked N-Set ( or BNS) “Bent” refers to the two intersecting units, and “Naked” further restricts the BNS to n cells containing only candidates of n numbers. The cells of the intersection of the units is the “hinge”and the naked set cells in each of the two subunits outside of the intersection are the “wings”. Bob describes two types of BNS. I’ll label them BNS1, having exactly one of the n numbers common to the two wings, and BNS0 having no numbers common to the wings. One or none numbers in common is the requirement for toxic set eliminations

Hanson 145-wingFor his only example of a BNS1, Bob uses this 154-wing.

Then, where you’d expect a proof of this assertion to be, Bob explains why this simple XYZ-wing removes 4r6c3. His explanation lacks the generality of his BNS1 rule, and is a little off base even for the wing, stating that either value of r2c3 forces 4r6c3 out. Using a forcing chain argument to explain an XYZ-wing, to illustrate a BNS? All backwards.

Bob’s unproved BNS1 assertion reads:

“If a bent naked subset contains one and only one candidate k that is present in both of its nonintersection subdomains, k can be eliminated as a candidate in any cell that sees all the possibilities for k in the subset.”

We’re left to decide if he means one and only one candidate, or one and only one value(number) , in each wing. Bob often says “candidate” when he means “number”. I’m guessing the less restrictive “number”.

Hanson BNW1Is there a proof of Bob’s BNS1 assertion? After showing that an XYZ-wing works, Bob represents a bent naked wing by one of his generalized method diagrams. It represents the naked set candidates in each unit and both units. Bob explains that if all of the candidates of k, the number common to both wings, are removed from A and B, no remaining candidates can be duplicated to make up for the missing number in the naked set.  Do you understand what “duplicate” means here? Let me explain what Bob didn’t. I went around in a puzzled state for a long time.

The “one and only” condition of the BNS1 rule is necessary because Bob’s “subset”, the naked n-set, is not a subset. It does contain n and only n numbers in n cells, but some of those cells are in different units, allowing solved cells to contain the same clue number. This generally allows the true k-candidate to be absent from the naked n-set.

Hanson BNS2With two (or more) common numbers, k1 and k2, the k1 candidates of the n-set are not a toxic set, because in the final solution, a duplicate of k2 true candidate could fill a k1 cell in the other wing. But with only one common number in the wings, no duplicate is available.

This kind of bent naked n-set is a generalization of the regular XYZ-wing whose wings are attached by unit induced winks, including those whose victims see toxic candidates by means of forcing chains.

IN 415 136 boomerangIt does not cover XYZ-wings whose wing-to-hinge winks are constructed from forcing chains, such as this one, found in KrazyDad Insane v.4, b.1, n.5.  Here, 3r2c9 sees the hinge 3r9c4 by a grouped forcing chain, creating the 367 wing that leaves three removals when the smoke clears.




Hanson BNS0Now consider Bob’s other rule, for BNS0. If there is no common wing numbers, the naked n-set behaves like a naked subset. That is, candidates of every number form their own toxic set. That is clearly the case, because all candidates of any n-set number are in the same unit.

The BNS0 is nothing like the XYZ wing, because its wings have no common number.

Hanson BNS2 exBob provides a real BNS0 example. In his example grid, three of the completion candidates are missing, namely theshaded ones 9r4c3, 7r4c8 and 2r9c8. The omissions don’t affect the example. There is a self verifying Sue de Coq Wr4 = 4(1+7)(8+9) +489 that removes 7r4c8.

The BNS numbers are 15789. The wings and intersection are marked in blue, green and orange.Notice that 89r4c8 is left out of the naked set. This in itself shows the naked n-set is really is quite far from a subset.  By the BNS0 rule, 9 has to go because it sees all 9-candidates in the naked set. It was very convenient to be able to leave out the r4c8 cell.

Bob points out that this 9 would also reduce Wr4 to 471, reaching a contradiction. That’s interesting, but not actually relevant, because we are looking for logical methods of making removals, not for candidates that cause contradictions.

To summarize:

A Bent Naked n-Set, or BNS, is a set of n cells with candidates of n numbers, contained in two intersecting units. Two types of BNS produce toxic sets:

In type BNS1, the two wings contain one number in common. The BNS candidates of the common number are a toxic set.

In type BNS0, the two wings have no number in common. The BNS candidates of each BNS number are a toxic set.

I have no generalized BNS examples yet. Before analyzing Hanson’s report, I was not aware of that possibility, and thought that Hanson’s bent naked whatever was simply an XYZ-wing. That’s what out-of-the-box thinkers do for us.

But now that you’re on to it, I’d be happy to publish your examples, with full acknowledgement, of course. You can attach to .

Next we explore Bob Hanson’s views on almost locked sets. Without looking, you could review the Sysudoku posts of July 2012 and anticipate what he’s going to say about the ALS in the example above.

By the way, there’s a new page, titled Find It. There, you will be able to scan titles and key words on a complete list of posts, to find the one back there somewhere, that you want to review again.


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Bent Naked What?

Here I object to Bob Hanson’s use of the XY-wing and the XYZ-wing, in Sudoku Assistant – Solving Techniques, to introduce his more complicated concept of “bent naked subsets”. This treatment limits the wings to unit based links.  Also I’m calling him out for bending Sudoku terminology in confusing ways.

What attracted me to examine Bob Hanson’s Sudoku Assistant write up in detail, was the apparent abandonment of convention in some of his ideas. Realizing that he inhabits a different Sudoku world, I was after fresh ideas, and a point of reference from which to judge Sysudoku  infrastructure. I’m not disappointed in that, but on the bent naked subset, I have to draw some lines.

Hanson 3 XY-wingIt’s good to start with a picture. Bob begins his report section Bent Naked Subsets with this representation of an XY-wing:

Here Bob makes the startling assertion that an XY-wing is a naked triple, a bent one. Wait a minute, let me deal with that. To me, naked triples are locked sets, a.k.a. subsets. Sets of cells and candidates are locked with respect to single units, i.e. lines or boxes. Now “locked” means something else? To Bob, yes. The bent naked whatever turns out to be something much more complicated than an XY wing. I’ll get to that, but right now I simply object to explaining something simple as a special case of something complicated, to be explained later.

Bob does make a proper argument that one of the wing 2-candidates must be true, else they are 1 and 3, leaving the hinge with no candidate. In my neighborhood, that makes the XY-wing’s 2-candidates a toxic set, because seeing both of them is fatal. But that does not make the hinge and wings a locked set. Another 3-candidate in r7 is not removed because it sees both 3’s of the XY-wing. If it sees all 3’s of a locked set, it must be removed, because you cannot remove a number from a locked set. So Bob, you must stay after school and write 25 times (first offense):

XY-wings are not naked triples, and bent naked sets are not subsets.

Also, Bob is getting more red marks for order of battle anomalies in his examples. The problem crops up again in Bob’s example of the XY-wing as bent naked whatever:

Hanson 3 removalsIn the full grid of the XY-wing example shown above, now transcribed from keypad to slink marking, there are removals for three naked pairs and a slink mark. Does SA find XY-wings first? That’s strange.

My switch to the slinks and winks reflects the Sysudoku view of the XY-wing as a short XY-chain. Any form of alternating chain has a slink on each end, and the end candidates are a toxic pair. If one end candidate is false, the alternating links make the other end candidate true. The outstanding payoff is that any pair of candidates of the same number along the chain, with slinks pointing toward each other, are a toxic pair. A chain can become several almost nice loops. Student Assistant manages without all that, according to Bob.

Hanson 3 XY chainThe payoff for human solvers is well demonstrated by Bob’s example above. The extension of the XY-wing in both directions, into an XY-chain, gains two more ANL toxic sets, and three more removals.

Finding such XY-chains is made easy by a simple technique, using the bv map. In a table of bi-value cells, curves are drawn connecting candidates in the path as they are ordered in a XY-chains. The curves connect like rails, limiting the direction of travel from each connection.

Hanson 3 railroadThese XY railways can be searched for matching candidates, and corresponding victims. Search for a matching number starts with the bv slink, setting the direction. It ends on a bv slink, making every other matching number along the path eligible to be the matching toxic set partner.

The relevance of XY-chains in this example, where the bv railways reach 19 of the 21 bv, suggests that Sudoku Assistant does not use alternating chains as a solving resource. That may be all right for a computer based machine, but definitely not for a carbon based solvers.

In my next post, I’ll take up Bob’s actual definition of the new entity he inaptly names the bent naked subset. I can’t go along with his treatment of the XY-wing and XYZ-wing as special cases of it. I’ll stand by my earlier definitions of these wings, with a reminder that both wings are limited to unit based links in Bob’s definition of them. That hurts. These wings are standbys of Sysudoku advanced solving, and this blog has extended their use by introducing ER and forcing chain versions of them.

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Bob Goes Fishing

This post examines Bob Hanson’s grid analysis, the Sudoku Assistant approach to regular fish, as described in Sudoku Assistant – Solving Techniques.

Bob Hanson actually takes his generalized method fishing! Out on Sudoku lake, the GN looks like this:

If n rows (columns) contain candidates of a number in only n columns(rows), then those columns(rows) cannot contain candidates of the number in other rows(columns).

To apply GN, set A is the set of N unsolved rows for number k. If n of those rows have candidates in only n columns (Set B), each column contains one and only one true candidate in rows A, and therefore cannot have a true candidate outside or the rows of A.

As Bob points out, the phenomenon of n rows locking column positions for the exclusive use of their candidates is the same as numbers locking unit cells for the exclusive use of their candidates. Lines compete with other lines for positions for their candidates. By numbering the positions along a line and cells along a unit, we make lines competing for positions just like numbers competing for cells. SA uses Bob’s analyzeX for both.

Hanson 2 two swordI’ll use a simple grid analysis example from Bob’s report to show exactly how this works. We can use the Sysudoku Suset algorithm, the scratchpad equivalent of analyzeX. The example is two swordfish removing the same victim.

The sysudokie version is a 5-panel showing the candidates only, and with vertical bars in a free row showing the rows of one swordfish, and dashes in a free column showing the columns of the other swordfish.

Hanson 2 two niceBob’s version is a dot candidate grid with freeforms delineating two swordfish, and two X-chain nice slink loops. The 5’s here just mark the units without of 5-candidates. How does SA find one of these swordfish?




Here is Hanson’s description of the grid analysis process:


Hanson 2 SA grid

The example above shows what SA’s “looking for regions” is like. Candidates of the number 5 lie in 6 rows and 6 columns, but if there are subsets, we know that the maximum size of subset is 3, because 3 rows and 3 columns are assigned 5 clues and 6/2 = 3. For the row analysis, column positions of all rows on a list. Top to bottom, we have

{59, 67, 236, 29, 36, 25}

The scratchpad suset algorithm takes each suset, and combines it with every other one to its right on the list. The combination is just to include all numbers and eliminate duplicates (a set union), while keeping track of the number of numbers. If it forms a suset with n numbers in a list of size n, it has found a Hanson 3 x 3 grid. In this case, we scratch out

59, 67, 236, 29, 36, 25, 5679, 23569, 259

and we have the blue region and swordfish. Believe it or not, this is how I started fishing, but I got tired of playing computer and started looking, human style. I still get out the scratchpad when an almost fish teases my cork, but won’t byte.

That’s all we need to check as far as SA grid analysis finding all regular fish. But does it find anything else? Well, it does find ungrouped nice loops, but we may have to keep reading to discover what else.

Hanson 2 1 gridUnfortunately, the remainder of Bob’s section on grid analysis is a crash landing.

His next example is a 5 x 5 grid that seems to fullfill the step 1 requirement above in columns, with the removal of outlying candidates in r1.

The problem is, you can’t do grid analysis as Bob has described, without locating all of the 1-candidates, so how can you show that with dots left off of most squares? Yes, you can say that candidates outside of the grid are excluded from grid rows, but that comes after you have laid down the grid. Bob, how did you say SA comes up with that grid? By the way, 5 x 5 is too large. Following the first example, since there are three 1-clues, 3 x 3 is big enough.

OK, put in the dots for the full completion “board”. But does SA do this with nothing more than completion? I thought I understood that cross hatching and range checking came first. Not in this example. Cross-hatch it to prove otherwise.

Hanson 2 4x4I wish I could say this section redeems itself with the last example, but it doesn’t. There are three 4 clues, so a 3 x 3 grid is sufficient. There is also a 4-wing, the equivalent of a naked pair. That leaves

(9 – 3 – 2 )/2 = 2,

another 2 x 2 (4-wing) as the only other possible locked grid. The two way jellyfish is dead, i.e. has no victims.

Oh well, Happy 4th!

Next time, we explore Hanson’s bent naked subsets, a novel idea.

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Sudoku Assistant Finds Locked Sets

This posts discloses several significant omissions in Hanson’s discussion of locked sets, including his examples, in the report, Sudoku Assistant – Solving Techniques.

A subtle and sometimes bungled task in basic solving is the discovery of locked sets or subsets. Bob Hanson defines locked sets by the following two rules. The naked subset rule is “When n candidates are possible in n cells of [a unit], and no other candidates are possible in those cells, then those candidates are not possible elsewhere in the unit.

In these rules, “is possible” can be made more straightforward, now that we have established that, following the completion scan, all unassigned cells contain at least the true candidate. Now we can paraphrase Bob’s naked subset rule as

When n cells of a unit contain the candidates of n numbers, and no other candidates, then the candidates of those n numbers can be removed from the remaining unresolved cells.

Hanson 1 npBob’s example of a naked subset is the naked pair r1np37 in the puzzle we just solved. The picture is a bit distracting, because the naked pair is not determined until the contents of r1c9 are known. In our basic solving, N6 =>r1np37, not the r1np37 => N6 of Bob’s naked subset rule.

Hanson 1 nt237The second naked triple we displayed last post makes a much more arresting example.

The r136c7nt237 removes two 7-candidates, leaving only NW 7-candidates in c7, and a boxline removing two more 7-candidates in NW, and deriving N7 by hidden dublex. It’s right there in the trace.

The corresponding hidden subset rule is

When n cells of a unit contain all candidates of n numbers, then the candidates of the remaining numbers can be removed these n cells.

Bob’s report has no examples of hidden subsets. You’ll find good ones in Subsets and Susets, posted 10/25/11, with a very tough homework example from the Sudocue site. The Suset algorithm explained in that post is a scratchpad version of the “analyzeX” Bob mentions later, which is commonly used in computer solvers to “see” locked and almost locked sets. Suset building is necessary for human solvers only when a multitude of candidates creates uncertainty about the locked sets. Naked subsets are easily recognized. The hidden subset rule describes the best human approach, which is to look for combinations of n numbers confined to n cells.

Bob doesn’t press the point by showing how his naked and hidden subset rules specialize the generalized method. I think it’s important, to emphasize the theme of Bob’s outstanding work on the SA.

For naked subsets, A is n cells of a unit containing candidates of n numbers, and no other candidates. B is all cells of the unit. The B cells outside of A cannot contain candidates of A’s n numbers.

For hidden subsets , it’s a little more subtle. Let N be the number of unsolved cells of the unit. A has all N cells, with n cells containing all candidates of n numbers, and N – n cells containing only other numbers. B contains the candidates of the other N-n numbers. But by the generalized rule, B can contain no candidates of the n numbers, and is therefore identical to A! The remainder of a hidden subset of n cells is a naked subset of N-n cells!

My Subsets by Susets post also covers naked and hidden locked sets occuring together, and that the Suset algorithm need only explore cell combinations numbering half of the unassigned cells to detect any locked set, naked or hidden. In c1 above, we spotted naked triple 237 more easily that the hidden pair 19 that was also there. Sometimes, the smaller number hidden subset is easier to spot than the complementary larger number naked subset.

Bob invites readers to find additional 1, 4, and 7 locked subsets in his completion grid. Your inventory from basic solving expands to include 2 and 3. I challenge you to find them as a Hanson reader by the subset rules, after Bob’s cross-hatch and range checking eliminations. Is it easier than in Sysudoku basic? If you believe so, then you have to stay after school and do the completion exercise I did to derive that grid, to see if the effort is worth it.

It reminds me of copying the geography notebook in Miss Bryan’s fourth grade class. I did it more than once. And Miss Bryan was my next door neighbor!  Now since you were so attentive in his locked subsets class, we get to go fishing with Bob next week.


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Basic Hanson Homework

This post follows up on sysudokie homework comparing Bob Hanson’s basic solving with Sysudoku basic.

In Sysudoku-ville, Bob’s illustrative puzzle of the last post doesn’t escape line marking. The Hanson 1 BM traceleft over hidden single N5 shows up where expected. No doubt you found the eliminations for 1: NWm => NEm at the very beginning of box marking, and 9: Wm => Cm at the very end. Except that 6: Nm => C6 had intervened to make it 9:Wm =>C9.

Hanson 1 boxed gridThe grid after box marking reveals four more hidden singles.

Having seen the completion grid of the last post, students of human engineering can dig the relatively small number of candidates on the grid at any one time.

But computer solving is different. A horde of candidates is no distraction for Student Assistant.

There was one more naked pair in box marking. SWr8np27.

Hanson 1 LM traceLine marking was moder- ately difficult, Hanson 1 hs gridwith 7 lines of five free cells before the collapse.

A hidden single appeared on the grid. You might look at how it arises in Hanson’s marking.

Two naked triples come up in columns 2 and 7,


Hanson 1 nt237the latter triggering an interesting boxline leading to the collapse.

The challenge of the collapse is that it occurs with five lines unfilled. Be very careful to mark bv and convert eliminations to clues only on marked lines. It is low hanging fruit, but there are briars.

In case you want to retrace your steps, I include the 2-D trace of the last long leg of the collapse, picking up on the left where the previous trace ended on the right.

Hanson 2 collapse traceNext time we continue the review by attending professor Hanson’s class on locked sets. Don’t be late!

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Basic Solving With Bob Hanson

I start my review of Sudoku Assistant – Solving Techniques with the basic solving of Hanson’s Sudoku Assistant, introduced by last week’s post. Basic solving is defined as the process by which we complete the candidate grid, required for advanced solving.

The generalized method of the previous post is Sudoku Assistant’s showcase rule, but Bob advises we should start with hidden singles, which are defined by his general rule of singles:

Hanson singlesThe rule would be more aptly named the “rule of hidden singles”. I hope Bob will forgive the parenthesized paraphrases. Where necessary, Hanson’s terminology and notations will be translated into Sysudoku speak, so that my habitual readers, the sysudokies, will have less difficulty understanding what he is saying.

Bob introduces this rule as a basis for the robot’s first step, cross-hatching, a scan of boxes for hidden singles. He notes that “hidden singles are far easier to find than naked singles”. I second that motion, having rejected Tom Sheldon’s beginning search for naked singles, his completion strategy, in favor of having them emerge in line marking. I therefore associated hidden singles only with lines. I guess I got pretty far without noticing that box marking clues are hidden singles.

Bob’s cross-hatching combines my cross-hatching with the double line exclusion in “clue only” box marking. This is also the Paul Stevens approach. Bob comments that this scan sufficient for easy puzzles, and that many do it without marks. In the Stevens review, we found it lacking in puzzles that talented solvers do solve without marks, noting the “marks in the head” these solvers are carrying. But actually, I believe we should ignore any such comment in Bob’s report about what people do, for reasons I’ll get to shortly.

Hanson 1 dotsIn Bob’s illustration of the cross-hatch, there is nothing wrong with your computer screen. The dots belong in r3c5. It is a way to represent candidates by dots in their keypad positions, and is handy when the grid is packed with candidates. Here Bob is just pointing out the four candidates 4, 5, 6, and 8 that would have occupied r3c5 if the Nhs5 had not been discovered.


The Sysudoku position is that finding box strong links is just as easy as hidden box singles, and should be done at the same time, adding remote pairs and naked locked sets to the Paul Stevens’ claimants for available box cells, thereby generating more clues and fewer candidates. In the Paul Stevens review, I acknowledged that this more comprehensive box marking could be done box by box, as well as the Sysudoku choice of number by number, but I stick with 1 through 9 throughout Sysudoku to maintain a consistent trace.

Bob is saying that all hidden singles are found in the Assistant’s first step. This would include line singles that turn up frequently in line marking, but he says nothing about how they are found. If you are a newbie, Sysudoku line marking is a line by line analysis yielding all candidates, line slinks, bv cells, box and line closed sets, and immediately available X-wings. So I’m looking for some corresponding process on the way to the candidate grid. How and when does Hanson get to the low hanging fruit of boxed remote pairs, aligned triples and naked pairs in basic solving?

Next, Bob presents the first two special cases of the generalized method rule. He classifies them as row/column range checking, and names them the locked candidate rules, form 1 and 2. In both rules, A is a chute, the intersection of a line and a box. Form 1 says that the absence of a candidate k in the line remainder (elsewhere in A) removes all candidates in the box remainder (elsewhere in B). Form 2 says that the absence of a candidate in the box remainder (elsewhere in A) removes all candidates in the line remainder (elsewhere in B).

Assuming you are still awake, do you recognize these rules? Of course. Form 2 is about the remote pair and aligned triple removals in box marking and later. Form 1 is a boxline removal.

Hanson markedBob’s illustration of the Form 2 (N1m marks) in action answers my question above, and a lot more. Bob draws arrows showing the remote pair r3c23 removing two candidates, leaving two in the NE box. But he draws them on a Sheldon “completed “ grid, which I have reproduced here on a Sysudoku template. Bob’s “marking” is Sheldon’s “completion” strategy.  I long ago rejected it as a human solving tactic, calling it the “number scan”.

In  reviews I normally transcribe a keypad candidate grid into a Sysudoku one, for my readers. But this one has too many unslinked candidates. That’s why keypad candidate grids from elsewhere have such tiny fonts for pencil marks, or have candidates represented by dots. Of course these remedies do nothing to reduce the visual complexity of a sea of candidates.

I did the “completion” of candidates from the original clues myself, then checked my work against Hanson’s. I had made a few mistakes. That is understandable because the process is booooring. Nothing worth thinking about is going on. I made a list of marks 1 through 9 down the right side of the grid. For each row I selected a candidate number (not a clue) in the row and dragged a copy of it into its keypad position in each row cell without that number it its column. Its 10 times more mechanical than line marking.

We don’t really care how Bob does the completion by hand, or how  Sudoku Assistant does it. The fact is, every computer solver has to do the completion of candidates first because the “completed” array, now represented by the keypad numbering in the diagram above, is the way a computer sees or draws the arrows. Think about it.

You can reproduce the computer algorithm yourself, but it is even more boring than my method above. To play computer, make up a 9 by 9 array of lists 1 through 9. That’s 81 lists, you know. Then for every given clue, delete the clue number from the lists of all cells in its row, column and box. You can see how Sudoku Assistant actually does range checking by following the ‘x’s above. The red arrows were “drawn” in the completion scan. In the range checking scan, SA observes that the only 1’s are in NWr3, and consequently, it removes 1’s from cell lists along r3. Were you confusing the 1’s and x’s on the top row with box marks? No, there are no box marks. Box marks are signposts for humans, denoting strong links and why they are strong.

You can see why Sudoku writers, most of whom are programmers, offer nothing better than the completion algorithm in their solving advice. Because of this, the elegance and fun of basic solving that many of my readers have discovered in the elaborate basic solving traces of some puzzles, is compromised or completely obscured by the completion algorithm. Even experts are deprived. And those who are uninformed and naturally gravitate to the completion algorithm get a discouraging false impression of Sudoku solving by brain.

So now we understand how line hidden singles are found. As SA deletes clue numbers from cell lists, it can easily track how many cells are left for each number in each line. When that number gets down to one, SA has a hidden line single. Voila! Just one more thing. SA can do the same for boxes, so why do we need cross-hatching? The answer is, SA doesn’t, but we do. I think Bob’s cross-hatching is a human interpretation of what an inhuman algorithm actually does.

In Bob’s report, he leaves one hidden box single “missed” in the cross-hatching for readers to find. You can find it as SA does, by tracking the candidate lists above. But you’ve already found it, so circle it on your trace.

Next, mark Bob’s first range checking example on your trace, the NWm =>NEm. It’s an ordinary elimination with an aligned slink.

The second “marks elimination” example on the candidate grid above is W9m => C9m. Play computer and carry it out on the candidate lists. What happens to that one in your box marking trace?

Next week I’m taking time out to checkpoint your basic solving of Bob’s illustration puzzle. This post got a bit long for that. Then we will closely scrutinize the next basic subject of his Sudoku Assistant report, locked sets. If you didn’t do that homework already, it’s not too late. There are some nice examples of locked sets waiting to be discovered.

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