This post explores a relationship between pattern slicing and Medusa coloring, and develops a trial based on this relationship. After extreme resistance to advanced methods, KD Insane v4 b5 n5 falls prey to a coloring cluster generated by a strong link between two patterns, a pattern slink. A Sue de Coq enables a simple wrap to finish a very tough KrazyDad Insane.
Insane 455 basic concedes only three clues in the bypass. Line marking starts with lines of five free cells. The first break is a naked pair.
Below is the grid at the boxline aligned triple.
As line marking is completed, a large ALS combines with a bv for an ALS_XZ.
Arriving at AIC building with nothing but a tight little cluster in the Southwest, I tried Andrew Stuart’s solver to see what I missed.
Nothing in that solver’s repertoire, but it did manage to fashion an unlikely and indecisive ANL starting with a reversed bv XY-chain in the cluster, continuing with a grouped 9-chain ending at 9r7c5, then on the other side, using an AIC hinge to get to a 2-chain and a ALS node to 9-chain back.
There is the above human rationale for that one, but the following solver steps involved unit forcing chains, which Sysudoku deems to be humanly impractical. No person is going to go from cell to cell and line by line, building forcing chains from each candidate in hopes of their arriving at a common candidate.
One unit forcing chain that is especially interesting, is listed in the Sudokuwiki options list as “Quad forcing chains”. Here Sudowiki shows us that any 3-candidate in r6 that is true will force 8r6c4 to be true. Following the coded objectibe, Sudowiki reports it as a removal of 9, but since the quad includes all 3’s in r6, the forcing chains confirm 8r6c4 as a clue. I didn’t carry the removal or confirmation forward, and it. didn’t matter.
Next is another instance in which an inhuman number of computer operations produces a result that a human solver can achieve with a few insightful inferences. Its AI in Sudoku.
First we look at the 1-panel of Insane 455. If you’ve been working on freeforms, you can imagine how impenetrable the enumerated freeforms from c1 are going to be. You would LOL to see the Sudokwiki report that it has found a single orphan on this grid. You do know, from the Guide’s pattern analysis examples, where to start, and after a few freeforms, you know how to finish it.
It’s r1c5 and here’s how it happens. No freeform from c9 into the N box can cross r1c5. Freeforms into C then S, or into S then C cannot cross r1c5. All other freeforms cross r1c5, but cannot cross c1. Those into C then r1c5(solid lines) have no candidates in S to cross c4; those into S then r1c5 (dashed lines) must cross r4c5 then have no way to cross c2.
Now it’s time for a pattern slicing analysis, starting with the 3-panel.
Here are the South to North pink/olive freeforms, three pink and five olive. We seek to determine if there are complete pink and olive freeforms extending from the first two rows, without conflicts. Colors can be assigned to columns 3 and 7 to favor this possibility.
The first assignment restricts the freeforms to one pink and one olive pattern. If no other pink/olive assignments permit slicing, the above is a strong link between patterns, a pattern slink. In fact, a coloring link. With only two possible patterns, if one pattern is false, the other is true.
Instead, the next assignment we try gives another perfect pink olive slice. Now there is the choice: have a trial or work for more evidence. Deciding on the latter, and moving on to the 4-panel,, we get a similar result: more than a single pair.
Except for one thing, an elimination. The candidate 4r6c1 cannot start a 4 pink freeform because all pink freeforms end on r6. Back on the grid, this removal allows a tight XY ANL,
and a boxline removal.
The pink/olive elimination also adds 4-candidates to the blue/green cluster built around the XY ANL.
The AIC hinges are added, but we make no further progress on AIC building. And at this point, Sudowiki has announced no
Returning to pattern analysis, the blue/green cluster and several 6 slinks suggest there may be reason for a pink/olive analysis on the 6-panel.
The start on r8 is more restricted than it looks for a pattern slink. Cell r7c2 and at one 6 in NWc1 must be olive. In the pink/olive 6 map 1 both pink freeforms fail, but there are two olive ones.
Allowing pink a cell in c1, we get a potential cluster in pink/olive map 2.
But switching colors in c5 gives a different pattern pair in pink/olive map 3 below.
That’s three pairs and six possible patterns. Panels 3, 4 and 6 are not restrictive enough to reach a pattern slicing resolution for Insane 455, without trials.
KrazyDad was somewhat right about Insane puzzles requiring guessing. Somewhat, at least, on Insane 455. It defies logic, at least the logic applied so far. But we can get the solution by another kind of logic, a trial. As this post testifies, a trial is not a guess. The theory of trials and examples of many kinds is a subject for the Sysudoku Guide, but here is the place to show a successful pink/olive pattern trial.
The 4 or 6 panel could be used for a demonstration trial, but let’s return to the 3-panel pink/olive maps, and the 3 pink and 4 olive freeforms. In trials of each of these 7, we would be lucky to average 3 trials for a success. However, with pink olive mapping we get to try two patterns at a time. If the pair of patterns on trial contains the true pattern, we have a healthy coloring cluster to move toward a solution. If not, we move toward a contradiction faster with two false patterns on trial.
On the number of trials, we looked at only two of four pink/olive cell maps earlier. The popo and opop combinations each produce a pair of patterns. Here’s how the oopp map produces one single,
and the ppoo map, a pair. That makes three more decisive pair trials and one single, compared to seven single trials.
The successful popo trial is brief. Pink becomes red and olive, orange in the cluster. The arrows show the normal order of development of the cluster by coloring links, revealing that it is not actually connected, and normal development is impossible.
The orphans are marked in black, the traps in blue. The traps bring in E8, C8, N8m and a merger of the two cluster. Orange becomes green, and red becomes blue.
Of course, if neither red nor orange is the true pattern, this trial is going to end soon with a contradiction.
Next, Sue de Coq Er6 = 9(3+4)(2+6) removes 2r6c7 and turns 4r6c7 green trapping 4r6c9.
In the final scene, green is wrapped as two green candidates, if true, would empty r6c9. Blue brings an immediate collapse.
Yes, trials are not undertaken lightly. Guesses require nothing and gain nothing.
Come back next week, after trying out Insane v.4 b.6 n.5. and keep your crayons sharp, especially your pink and olive ones.