4-Fills Lead off on a  Dave Green Surprise


In this Sunday 5-star of June 13, 2021, two 4-fills are resolved before the bypass value march begins. Follow-up continues in box marking, leaving little line marking to be done.  Coloring is demanded and it responds, with an unexpected discovery.

The two 4-fill traces show exactly what happens.

In r1, 4 is missing and can’t fill in NE. In r9, 9 is missing and can’t fill in SW. With 9 placed, 457 are missing, and r1c1 can’t be 4 or 5.

The most important take away is, what signaled these moves?

Then if you follow this bypass trace,

and pass this grid to box marking,

an ordinary box marking turns into something else.

 Coloring the almost filled grid, notice how the removal of 1r8c8 allows the blue/green cluster to be completed.

That means there are at least two solutions. Something that isn’t supposed to happen, but when it does, it can be discovered by coloring. In this case, we don’t really know. I might have gotten a given in the wrong place when I dragged them into my template grid.

I meant it when I said “at least”. For the correct answer, use one of the many backtracking solvers that cover the whole tree of possible solutions. I consulted Andrew Stuart’s on his Sudokuwiki site to learn my possibly mistaken version of the Dave Green puzzle has 5 solutions.

Next time, we look at a Sunday 5-star with four 4-fills on the opening grid. Ordinarily, 4-fills are passed over if they don’t resolve immediately, but in this case, let’s fully process each one and see how we like that experience. It appeared in my paper 1/23/22.

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All the Way With Dave


This post goes through Sysudoku Basic with the Dave Green puzzle published in the Akron Beacon Journal on Sunday October 10, 2021. It’s the first in a series of published Conceptis puzzles selected to illustrate basic moves introduced in the later years of Systematic Sudoku, the blog.  In this series, we pause before many significant moves, to give the trace readers the opportunity to see if the move is in their spotting repertoires, and if they know how and why the move works.

Starting on the bypass, is your first move C1? It takes a dublex in the middle colulmns to limit C1 to c5 and a cross-hatch in box E to limit it to r4. That move shows the satisfying challenge of the bypass. It also shows the value of the uncluttered grid as you pit clues and subsets against clues and subsets.

After (C1, W2, NE3), column c7 is a 4-fill. You respond by checking for missing values seeing more than 1 blank cell. In c7 missing values 2 and 6 both see two. SE6 fails to see a third, but 2 does, placing SE2, and leaving a 3-fill in c7.  NE3 also creates a 4-fill in r3. Will the 3-fill or 4-fill be resolved?

All that action can be recorded in the trace, which handles the bookkeeping, and keeps you apprised of exactly where you are. You read a trace by filling out your own grid as you go.

Tracing the fill of c7, the 3-fill does resolve, and its clue SE5 creates another 4-fill in r7. Two of the four missing values place themselves in separate boxes, leaving two naked pairs in the boxes.

Moving ahead, W3 closing a 2 by 2 square in the West box is the next  significant event.

Line c3 missing the square has value 6, not in the square. That places 6 in r4c12  E4, and a new 4-fill in r4.

Continuing through the bypass, the E box fills in, and two more 4-fills are resolved.

Here is the grid passed on to box marking.

Box marking brings in box slinks, marking them with “pencil marks” along the cell top. For trace readers, only the box need be named for each value. Here the combined marking for 1 and 2 in NE creates a naked pair, reserving two cells for a c8 3-fill, promptly resolved.

Continuing in box marking value by value. The 6 list adds the 6 value to the three unfilled cells of r6. That finishes a wall in the C box, starting a plunging follow up for you to complete and trace.

The grid passed on to line marking incudes another resolved 3-fill in the 7’s.

It’s all on the box marking trace:

Line marking completes the candidate field. Each row or each column is assigned a fill string of digits including all missing candidates on that line. The fill string is copied along the line, and digits seen by clues or subsets are deleted. The remaining candidates are placed in the cell, with slink partners positioned to identify the direction of the slink. Lines are filled in increasing order of free cells, so easier fills come first. The trace records the order of line filling.

Bv are marked and candidates are positioned as each line is filled. When rows or columns are marked , the unfilled lines in the other direction are marked. This is the close and it doesn’t require fill strings.

Here is the line marked grid. Just as the Close is started on c4, a hidden single 7 is discovered in the column being filled. The line marking trace showing the collapse and the solution follow.

The line marking trace on the rows is short here, because so many were already filled. The close finishes line marking on the columns, with the collapse on the first one.

This Sysudoku Basic series of Dave Green puzzles actually started last May 25, 2021 when two  Sunday 5-stars full of 3, 4, and yes, a 5-fill, were posted. You can look them up by post date on the monthly list on the right. That, and the titles page, is a ticket to hundreds of bypass traces.

Next time,  we watch a bypass starting with two 4-fills, and box marking get a 5-star to coloring. The conclusion is a surprise. See if you can DIY this one.

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Beeby’s Elbow Quad in shye’s Catherine Wheel


This post carries us through the Sysudoku process, with the elbow fireworks of the Catherine Wheel, a second 4-set quad example from shye’s forum posts, cited in Beeby’s notes.  

From the givens grill at the end of last week’s post, there are no dublex or cross-hatch clues or subsets for the basic trace. Just a routine box marking, and an uneventful  line marking.

From the line marked grid,

here are the panels showing the elbows of the CW 4-set quad.

Matched elbows on values 1 and 2 on one corner, and another pair on values 4 and 5 on the opposite corner make up a 4-set hidden quad.

As shown last week and earlier for the shye’s other 4-set quad, the removals of 3, 6, 8 and 9 candidates are because the 4 corner cells must solve to the four elbow values.

The 1, 2, 4 and 5 removals are because, in the 4-set, only one of a pair’s two values can be placed in its neck, and placement of the other pair value in the cell opposite its neck makes that value unavailable for either one of its cuffs. An elbow’s value must be placed in one of its three cells. For each elbow pair, one value goes in the neck, and one in a cuff.  

In Beeby’s fireworks action, the 4-set quad removals leave clues (NW3, SE3, NE6, SW8), a boxline NEc8 in 1, a naked quad in NE, and NE2 alone in r2.

After this near collapse,

there is a rare opportunity for remote pairs. In a remote pair chain of even length, a candidate of each value is true in one of the end cells. So seeing the end candidates is toxic.

Now the bv field tells me to get out the crayons. The blue/green trap on 6r6c4 creates a naked pair 37 that wipes out a blue 7 and the blue army with it

I just let the green army spread into the solution.

Don’t quite get follow up traces? Just do your own spread of the green from the blue wrap and watch the trace below doing it with you.

Beeby has its own way of finishing the Catherine Wheel. Instead of the remote pairs, it does finned X wings on 6, 7, 8 and 9, then a 6-wing and that naked pair.

For a change of scenery and to make basic examples available for reference in The Sysudoku Guide, my next posts will feature several Dave Green Conceptis puzzles, 5-star and below, that illustrate basic solving resources in the three stages of Sysudoku Basic.

The first puzzle of the series has significant events in all three stages of Sysudoku Basic, the bypass, box marking and line marking. It collapses in the closing stage of line marking, just prior to advanced methods. Try it your way, then read along, following the trace move by move on your grid. You may discover something missing from your Sudoku practice.

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The UHC Solvers on the Shye Fireworks


As announced last post, a series on Sysudoku Basic is postponed for another look at fireworks. In recent posts, we have been reviewing examples from forum commentator shye’s introduction of fireworks, and proposing a more limited, but  better defined form of firework, the elbow. Another look is prompted by my discovery that both the available DIY oriented solvers used here for recent reviews, known here as Sudokuwiki and Beeby, have quickly added firework options.  First off, let’s see how well they reveal the fireworks of shye’s examples.

My first post on fireworks was December 28, 2021. It’s been revised to show that in an elbow, a candidate of the elbow value in either the neck cell or at least one of the cuff cells is true.

Shye’s first example illustrates how three supermposed elbows of three values creates a 3-set, a form of hidden triple not confined to one house. Shye’s example actually is two superimposed elbows and a slink in the third value including the intersection, which has the same result. You know that the solution places the 3 values in these three cells, but that’s all. But what generally happens is that the follow up on the removals of extra candidates pins down these placements and more.

After Basics,  when the Fireworks button is pressed Beeby announces :

“Fireworks triple of 123 at r1c9, r1c1 and r9c9, eliminating: -8 r1c9; -9 r1c9; -6 r1c1; -7 r1c1; -9 r1c1; -4 r9c9; -7 r9c9;   Solved!”

That’s appropriate timing for fireworks.

Another success occurs in shye’s second example, in which two matched pairs of elbows frame a rectangle, with neck cells on opposite corners. This makes a 4-set quad of four corner cells. Candidates of 5, 6, 8 and 9 are removed because the 4 corner cells must contain 1 and 2 of one elbow and 3 and 4 of the other elbow. Values of one elbow are barred from the neck of the other elbow, because the two elbows share the same cuff cells. 1r4c6 forces 3 and 4 into the two cuffs, leaving one elbow of the 1,2 pair without its value.

Both solvers spot the 4-set elbow firework. Sudokuwiki’s note can be misinterpreted. It says r9c1 must be 1 or 2, and r4c6 must be 3 or 4, but that is not true generally of matched pairs of elbows. It’s true only in the context of the quad 4-set formation.

Shye’s examples 3 and 4 combine two matched elbows with matching bv that could be included in solvers’  repertoire, but are not as yet.

Shye’s example 5 is closest to the notion of combined fireworks exploding into far reaching removals. Five fireworks, four of them elbows, share a single cell as an row/column intersection. The problem for solver coding is that only one of the five has a true neck candidate, and there is no programmable fireworks rule establishing which one it is, other than trying each firework.

 Each solver had full opportunity to find this firework. For Beeby, it meant calling for it after each move. For Sudokuwiki, it meant following its path until  several techniques past its difficulty list had been tried. Shye’s example 6 is based on a simple trial of an elbow’s intersection candidate, and should not be included.

In notes on Beeby’s fireworks option, Philip Beeby cites the Catherine Wheel, another shye fireworks quad example from the forum thread. Like the second example above, Beeby sets up the four fireworks elbows and makes corresponding removals. This time it’s not an immediate collapse, but the analysis is worth the effort. Full story next time.

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The Elbow n-set of shye’s Roman Candle


This post is about the sixth example of forum commentator shye’s introductory post on fireworks, his name for elimination methods based on the restriction of candidate values in row/box/column combinations. Our series of posts on fireworks defines a particular form of firework, the elbow, with single candidates in the intersecting row and column outside the box. This post accounts for a 7-set, an extended naked set selected by shye, to solve his Roman Candle puzzle.  This post shows how the 7-set can be derived from an elbow, and resolved for an immediate collapse of the Roman Candle.

Here is the basic trace of the Roman Candle.

The Candle gives up nothing in the first stage bypass, which derives clues and subsets from clues and subsets.  The second stage produces slinks (strong links) in each box., and the last stage fills in candidates in every row and column, with bv and line slinks marked.

The candidates grid, with line marking fill strings attached

In looking for fireworks,  we start with the elbows, marked on an X-panel .

About this panel, shye’s report  says, “seven cells for the following placements: a firework on 1s in r5c5b5 along with positions of 3r1, 4c5, 6c1, 7c9, 8r9, 9r5. all other candidates removed.”

What? The seven cells are necks and cuff cells selected from one elbow and four arms from other elbows, and cells containing 4 from c5 and 9 from r5. The 7 cells form a 7-set, 7 cells solving to 7 values, but shye doesn’t say how they are selected.

A clue to that mystery is that the selected elbow arms are slinks. Place them on the grid and what shye is doing becomes clear. He is assuming that 1r5c5, in the neck cell of the single 1-elbow, is true, removing 1’s in its cuff cells. That adds two more bv to the 7-set.

With 4 and 9 removed from r5c5, two connecting slinks are added and the selected bv form an XY nice loop. Sysudoku readers know that nice loop candidates form a coloring cluster. Along with 1r5c5, either the the blue or the green candidates are true.

Taking the blue candidates to be true, here is the collapse trace, and the solution.

Did this extraordinary example tell us anything about DIY fireworks analysis? No. Looking for an elbow with XY chains forming a closed loop is not a practical DIY trial strategy.

This is a cascaded trial. It starts on the guess without evidence that 1r5c5 is true, then constructs the nice loop for a trial of blue vs. green candidates. Was this guess in fact the basis for composing the puzzle? That would explain a lot.

Concluding this post review, how typical are the elbow panels of shey’s six fireworks examples for hard puzzles? My survey of the 12 ultrahardcore right page puzzles of the Stefan Heine review was not encouraging. No elbow matches, no productive cooperation with coloring, and no enhancement of AIC building. It bears out the impression from shye’s forum posts that it takes extensive computer search to find impressive firework shortcuts comparable to his examples.

I had planned to move on to an update of Sysudoku Basic, by means of recent Dave Green Conceptis 5-stars. But that’s on hold. I just discovered that both of my DIY oriented solvers, Andrew Stuart’s Sudokuwiki, and Philip Beeby’s PhilsFolly, have added a Fireworks component, and have their own interpretation of what Fireworks actually are. So naturally, I have to find out, what’s the take of these two expert programmers on this blog’s particular firework, the elbow. Next Tuesday’s post will be on that.

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Hanabi Without Fireworks


This post reports the ultrahardcore solver solutions for shye’s fifth fireworks example in the forum post  http://forum.enjoysudoku.com/fireworks-t39513.html . It shows the difficulty of the 5th example puzzle and the corresponding value of the fireworks solution.

Starting with the common neck and four cuffs highlighted, Sudokuwiki does a grouped 7-chain ANL in black, duplicating Beeby’s finned swordfish in red.

Next, a grouped 5-chain ANL and extension claiming both 5 removals, matched by a swordfish finned for one, and krakened  for the other. Yes, a bit over the top, but in DIY AIC building or 5-panel fishing, you could be dragged there.

Here’s a three on one. A grouped ANL in black, where 9r1c1 sees 9r9c1 and 9r3c23. The removal enables a grouped 1-way in red, where if 9r3c3 is true, 9r9c3 is false, and if 9r3c3 isn’t true, the 1-way AIC  makes 9r9c3 false. To the right, another 1-way. 9r5c7 is true, or it starts an AIC that erases 9r6c9.

Then Beeby finds an unlikely pair of ALS for an ALS_57 on singles.

With AIC and simple ALS exhausted, Beeby finds this ALS aided  9-chain ANL,  grabbing another 9 candidate, and the Ec7 boxline takes out two more.

Next, a grouped 1-way, where 9r4c6 gets creamed, regardless of 9r4c2 being true or false.

A hidden unique rectangle converts 4 candidates into 2 clues.

Then an AIC ANL gets two more 9’s and the NWr3 boxline gets a third.

Now coloring is overdue, with two clusters breaking out. The 4-set placements may be defined, as red or orange is true, but first we have to follow up the trap at r8c1.  The trap merges red and green, because the red and blue 8’s are strongly linked in r8. If blue is false, red is true. 

The merge expansion brings many more traps, but also, an XY ANL  removes two more 9’s.

Coloring doesn’t resolve the 4-set, but the swordfish leaves a single 9 candidate in r6, for C9 wrapping blue. Green candidates are true, and we have the solution of the previous post.

Was doing the X-panels right after line marking and doing the elbows thing worth it? With shye’s rare puzzle, yes.

Next we come to the sixth and last fireworks example of shye’s introductory post. Our post will offer a solution to the embedded  n-set problem that avoids computer search.

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Sharing Hanabi’s Secrets


This post details a fireworks analysis on “Hanabi”, shye’s fifth example. The first four examples used matching elbows alone, along with bi-value cells. This one uses forcing chains to identify the value winning placement in the neck cell shared by four elbows and a firework.

Hanabi allows the bypass a few clues, a naked pair, and two 3-fills.

Box marking adds box slinks (strong links), and line marking, line slinks and bv borders.

For elbows analysis, the line marked grid

is dissected by value in X-panels. X-Panels 1 – 4 display only show empty X-wings, but panels 5 – 8 reveal many elbows.

Fireworks proposer shye selects firework r9, SW, c1 on values 56789. There are no 9 – elbows. Here are the selected elbows and the 9-panel firework:

Returning to our “values’ version of the fireworks principle,

if the four 9 candidates outside the box are false, 9 is placed in r9c1. The four elbows compete with the 9 firework for placement in this common neck cell.  By including the 9 firework, we get five fireworks placing five values on r9c1 and four elbow cuffs.

In his forum post, shye designates r1c9 and r6c4 as base cells and r9c1 as a target cell and states that values missing from the base cells can be removed from the target cell, removing 56r9c1. That’s leaving out a lot.

The definition of shye’s firework base cell is unclear, but for an elbow, let’s say the base cell is the cell outside of the box that sees both cuffs. All the elbows above have the same neck cell. In every elbow above, if the base candidate is true, it is false in both cuffs and is therefore true in the neck of every elbow above.

The set of five cells, four cuff cells and the neck cell, must solve to the four elbow and 9 firework values.

Using a term introduced in Sysudoku to explain BARNs, this is a 5-set, an n-set with n = 5. An n-set is a naked set, but not a subset because more than one house is involved. The most common BARN is a 4-set in a bent area, the intersection of a box and a line.

Highlighting the cuff and neck cells, if the base cell candidate 8r1c9 is true, the two forcing chains shown here force the neck candidate 8r9c1 to be true. That means the base candidates of the other elbows must be false, because if true, they would force their value in the common neck cell r9c1.

Goodbye, 5r6c9, 6r1c4, and 7r6c4.

Starting with this, and working in trial tracing order, here is the trace to the point where we are resisted by

a locked rectangle of 59 bv cells.  However, with 7 already placed, and 9 in its cell, you can place 5r6c1 and break the rectangle to finish the solution.

Next week we continue with the full Sysudoku solver path, because it’s worth seeing, and to highlight the DIY benefit of examining your elbows.

Here is the solution, with some coloring from next week’s final wrap. In shye’s well chosen example, the elbows route is much easier.

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Fireworks 4 Stretches the Matched Elbow Rule.


After a challenging basic, we get two matched pairs on two values again, but this time, linked by a single bv. The second challenge is the application of the matched elbow rule of the last post in a more hypothetical situation. The fireworks result is not an immediate collapse, and we get  interestingly  parallel solution paths of ALS_XZ and BARN.

The basic is easier tracked than performed.

 

The 1-wing removals are made in line marking, before X-panels are constructed.  

Doing the X-panels for the fireworks elbows scan, we note the dead swordfish joining the now dead 1-wing, but what is the significance of having four matched elbows nested around the center cells of four boxes? I’ll say at the end of the post, but more on topic is the two matching elbow pairs on 2 and 8.ic is the two matching elbow pairs on 2 and 8.

Plotting the matching pairs on the grid with curves, there’s a 28 bv on the row and column between the cuff cells of the two matched pairs of elbows.

By the matched elbow rule, at least one cuff cell on each matching pair solves to  2 or 8.  The bv switches the true 2 or 8 to the opposite true value, so the cell on the other end must have neither 2 nor 8. By the matched elbow rule, that means the opposite cuff cell has 2 or 8, and the bv forces it to be the same as the true 2 or 8 we started with.  Regardless of which cuff cell we start with, corner cells r1c9 and r9c1 are limited to 2 or 8.

Limiting two neck cells to two values is not necessarily decisive. But solver Beeby takes us through some interesting steps to a solution. 8r1c5 is removed by ALS_98 (in black), or a BARN on 2789.

The 8’s attack continues with an ALS _28, or if you prefer, a BARN on 2368.

The cherry on top is another ALS_28, 

and a rare 5 – pole BARN on 23578, for C8 =>(NW8, SW8).

Then Beeby supplies a finned swordfish.

Asked for another fish, Beeby gets a “sashimi X-wing” with fin at r1c5. I shoulda asked for a simple AIC.

 Finally, the collapse. In the solution, you can verify what you concluded from the four matched elbows on the 1-panel. One set of opposing necks are 1.

Next time, we examine Hanabi, proponent shye’s fifth fireworks example in http://forum.enjoysudoku.com/fireworks-t39513.html.

It suggests there are additional ways in which combinations of elbows can be used to find candidate removals. There are no matching elbows, but elbows of four values share the same box and intersection cell. And there is a firework sharing this cell which is not an elbow.

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Fireworks 3 Matches Elbows and Bv


This post develops a property of a 2 value elbow match, and uses it with matching bv. Before we start on that, here are the elbow panels for the last week’s second example, just in case you want to compare it to your homework. 

The number of elbows is easy to count, and matches are easy to make, just looking at the horizontal or the vertical lines.

Here is the basic trace for the third fireworks example, leading to the line marked grid below.

We may want to do immediate X-panel only for very difficult puzzles, but in addition to a quick fireworks analysis, the panels will access X-chains, regular fish, XYZ, XY railway, freeform pattern analysis, and coloring assessment from the start.

On Fireworks 3 we have one elbow match, values 1 and 9. We could anticipate more as removals come in

Looking back at the line marked grid, there is something to note about the matching values 1 and 9, and their matching elbows. The the elbow cells outside the intersection box, the cuff cells,both see bv of values 1 and 9! These are on r1 and c9. Each elbow cell outside the box sees one of these bv. At least one of these, r1c1 or r9c9 or both, must have a 1 or 9 in the solution. The bv on the same line must have the opposite 1 or 9 in the solution.

That means 9r1c9 sees a true 9, either in an  elbow cell or a bv, hence its removal.

For future reference, let’s encapsulate the critical fireworks property at work here in the matched elbow rule:

The rule doesn’t mention bv. That’s because the rule is a general property of matched elbows that we might apply without matching bv.  Why is it true? It’s because two of the three cells of the elbow, the two cuff cells and the intersection cell of the fireworks rule, solve to the two matched elbow values. If one of the cuffs doesn’t, then the other cuff does.

By the way the removal of 9r1c9 collapses fireworks 3.

The fourth example in shye’s introductory post uses bv in a similar way, but instead of a single pair of matching elbows and two bv, there’s two pairs of elbows and one bv. The firework is easily spotted, but the logic applies the matched elbow rule in a hypothetical way. Next time.

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X-panel Elbows for Fireworks


This post illustrates the use of the Sysudoku X-panel for fireworks analysis, as introduced  in forum corresponent shye’s first two examples.

Here is shye’s statement of the fireworks principle:

What does this actually mean? Given a row, column and intersecting cell, if a value in the cell cannot appear in the row or column outside the box containing the intersection, that value is placed in the intersection cell. This holds because no other candidate of that value in the box can provide that value to both row and column. Shye’s formulation of his fireworks principle uses the word “candidate”, when he means “value”.

A practical way to apply this fact is to find row/box/column combinations with only single candidates of an intersecting cell value in the row and column outside the box. Then by the fireworks principle above, the two single candidates and the intersection cell mark three cells, one of which contains the true candidate. For a reason soon to be apparent, we’ll refer to this combination of row, column and intersection cells, an elbow. We’ll call the single row and column cells containing the value of the elbow,  the cuff cells. To go along with that, the intersection cell will be the neck of the elbow.

In an elbow the value candidate is true, either in the neck, or in at least one cuff. If the neck candidate is false, at least one of the cuff candidates is true, because the true candidate in the box cannot be in both row and column of the elbow, so a cuff candidate is alone, in the row or the column. Another candidate of interest is one that sees both cuffs. If one of these is true, the neck candidate is also true, because both cuff candidates are false.

Commentator shye refers to elbows as fireworks, also uses that term for other, less well defined forms restricting intersection box values.

Shye’s first example in http://forum.enjoysudoku.com/fireworks-t39513.html illustrates the idea that, if you find three elbows matching three values at the neck and cuffs, you have a type of hidden triple.  The three cells must be reserved for these three values, so candidates of other values are removed.

Here is the Sysudoku line marked grid for shye’s first example, with matching  neck and cuffs for values 1, 2. There is no 3-candidate in c1 outside of the NE box, so value 3 is nevertheless  limited to the three cells of the two elbows, and candidates other than 1, 2, and 3  are removed.

We depict elbows on grids by freeforms because their “legs” are not necessarily strong links.

How hard is it to spot matching elbows? It’s much easier once you have all elbows of a value on an X-panel, with all 9 X-panels side by side. For that, you mark outside singles having outside singles from the same box in the crossing direction. After finding two matching elbows, you could look for a third one – armed elbow to match them. Here’s an elbow map for shye’s first example.

Values 1, 2 and 3 matching elbows, with three cells in common, a hidden triple. Other candidates are removed from these three cells. The 9 x-panels make it easy to verify that these and no more elbows of different values match. In this case, the removals bring an immediate collapse.

For the second example in shye’s fireworks introduction post, the homework of last week, here is a basic trace. This example shows how four elbows on a four cell rectangle,  sharing two pairs of values, makes a hidden quad.

Now in place of the usual line marked grid, here is the corresponding set of X-panels. Find and mark the elbows, including four matching on two values, that allow 1 and 2 in r9c6, SW r1c1 and r4c1; and 3 and 4 in r4c1, C r4c6 and r8c6, removing, 5r4c1, 89r4c6, 951c6 and 56r9c1.

On the line marked grid, the “the matching elbows form a 4-set quad.

Values 1 and 2 must appear in the 3 cuff and neck cells r4c1, r9c6, and r9c1. Similarly, values 3 and 4 must appear in cells r4c1, r9c6 and r4c6. That’s enough to remove 5,6, 8 and 9 candidates from the corner cells.

But also in this 4-set, a candidate of the other pair’s values cannot be true in its neck cell. If 1r4c6 were true, one cuff would have to solve to 3 and the other cuff to 4, so that both values could appear in that elbow pair. But that leaves the other pair missing 1 or 2 in all three cells. Basically, the row/column intersect cells can solve only to one of the pair’s two values.

The “quad” brings an immediate, but long, collapse.

Without the fireworks, this example keeps the ultrahardcore solvers at bay for 30 slides. My general conclusion from these first two examples is that, with X-panels, DIY exhaustive fireworks elbow detection is reasonable to do. A plausible  test of the frequency of results might be to see how many, if any, fireworks matches occur in the 22 ultrahardcore linemarked grids of the right and left page reviews. Volunteers, anybody?

Before you do that, let’s look at shye’s further examples of what to do with them. Next week, we start the new year with shye’s third example.

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