In this post, KrazyDad wins a skirmish in our friendly difference of opinion on Insane guessing. Jim’s Insane 465 convinces me to put a cluster color on trial, as I have done only for confirmation of multiple solutions. But the cluster is not from an ordinary Medusa coloring. It comes from an ALS aided nice loop, with no external victims! The theory of nice loop coloring is introduced in this post. The Insane 465 solution path also includes an Andrew Stuart unit forcing chain that would normally be missed in a human search.
In basic solving on Insane 465, a 7-wing on the last line generates a second, but dead, one.
The bv scan fails despite a reasonable bv field. Fishing is poor and X-chains don’t connect. I start two Medusa clusters , but get no no traps and bridges. For most numbers, patterns are too numerous for LPO analysis, even with the aid of the coloring.
However, there is a tantalizing situation in the South box – an internal Death Blossom lite with no victims. It led me to search for a breakdown of this cliquey relationship. I found a unit forcing chain. One candidate in the S box, if true, removes all 4-candidates from c6!
To find this as an Andrew Stuart unit forcing chain, one would somehow have to notice that all four of the 4-candidates in c6 see the r8c5 candidate. It is false regardless of which of them is true, and one of them is.
But let’s get real. Only a computer code would find this unit forcing chain that way.
There is something else new in this example. Notice the weak link in the C box between the 2-group and 4-candidate. It only works because the 2-group, if true, forms a naked pair with the 5-candidates. We can make further use of that phenomenon on Insane 465.
There are extensions from the nice loop, and group envelopes left on the grid.
There are more examples of links created by the presence of a third number. One is r23c4, which can be considered an ALS node as well.
This nice loop has no victims, but it supports the fight in another way. Moving clockwise around the loop, if a tan node is true, all of the tan nodes are true and all of the aqua nodes are false, by AIC logic. Moving counterclockwise, the colors reverse. If any aqua node is true, all are true, and all tan nodes are false. The presence of single bv on a nice loop guarantees that one of these possiblilities, clockwise or counterclockwise, is true. Alternate nodes form a cluster, the logical equivalent of a Medusa coloring cluster.
The nice loop cluster can be combined with Medusa clusters in all of the usual ways. For example, a bridge condition from r4c4 is
not(blue and tan) => green or aqua.
Having no traps or bridge, I have nothing better than to put the nice loop cluster on trial. By that I mean assuming one color is true and calculating the consequences. This should be a last resort, after reasonable advanced and extreme methods are exhausted. It is a concession to the puzzle that I have not uncovered its secret, and to KrazyDad Jim that some Insane puzzles may, indeed, require at least this form of guessing.
In the blog, I have used trials of Medusa clusters to reveal multiple solutions and to test for missing alternatives in Sue de Coq. I define these as trials, as opposed to guesses, or trial and error, or T&E, in that a set of candidates, defined by prior analysis, and likely to be decisive, is tested. Trials reveal logical constraints while avoiding the backtracking trial and error better done by a computer.
But we really are not done just yet. With many candidates agreeing on the other color, there is a possibility of another solution. In fact cluster color trials is the way we have discovered multiple solutions. So we must give the tan party a chance as well. How about if you do that? I will checkpoint your effort in the next post. Go as far as you can prior to trals. We will also embark on Insane 475, so if you want to get started, it’s not too soon.