## Ultrahardcore 3 Schlagt Forcing Chains

In this post, we continue with Sudokuwiki into forcing chain methods, and their relevance to the ultrahardcore theme. Sudokuwiki stalls, as Stefan predicts.  Then we fashion a trial for a logical, but practical outcome.

On the grid after the red/orange traps of last week,  you’re looking at the three 6’s in c3 for a unit forcing chain. Where do you start?

How about with every forcing chain from 6r1c3? Compile a list of every force out, including the cell and the value forced out. The list starts with r1c3 and every other value in it.

Then from there include all the extended AIC. Start with every wink from r1c3 that is followed by a slink of any kind. You’re lucky there are none. But how about the wink to the other candidates in r1c3, like 8. That gets to three more cells, and to 5, and a bunch more. Does Sudokuwiki include 8 and those chains? I don’t know. Anyway, once you have your list for a second starting value, then you can limit the chain terminations of the third search to the terminations common to the first two lists. That describes the coding of a solver’s exhaustive search for unit forcing chains. With minor modification, it applies to cell forcing chains.

Humans can take advantage of all kinds of special breaks, like watching for 6r3c3 and 6r5c3 to force any of the other values out of r1c3.  Or from  6r5c3, landing on blue, and forcing blue out everywhere, including 3r1c3.   That gives us incentive to find r3c3 guilty of the same.  It is, by virtue of the same blue force out. We have our unit forcing, but there’s more. Two of the 6’s confirm green 3. Does 6r1c3 do that as well? Well, not quite. By the red forcing chain, 6r1c3 forces out 3r7c3, but there’s a third 3 in c3.  But the other two, by confirming 3r8c3, also force out 3r7c3, and we have a second unit forcing chain.

But a finding process is not logical if it is not exhaustive. Humans are not exhaustive in forcing chains, and we can’t be sure  about solvers until we analyze their codes. Just strike the “in forcing chains” for one answer to the Stefan’s “intuition vs. logic” dilemma, and to possible objections to the use of trials here.

We are here to demonstrate what humans can reasonably find and enjoy in Sudoku puzzles. Sudokuwiki was written by Andrew Stuart with much the same intent. It is used here with exceptions and modifications where easier and more likely means are found.

The construction of trials is as logical as other solving techniques in common use. You can logically object to a trial by demonstrating the logic that the trial conceals. In that sense, a use of a trial is inferior, though the construction of the trial is not.

Next on view is a unit forcing with four candidates, a quad forcing chain. The four cooperate, with three forcing chains merging into one. But the point really is, how many possible quad forcing chains are you going to look at before you come upon the next one?

Sudokuwiki’s next is an r5 forcing chain on the three 6 candidates. The inside killer is 6r5c1. The removal above allows 6r5c3 to reach 6r5c1. The 6r5c7 forcing chain is a bit devious, but you know what you need to do.

In a parting cell forcing,  Sudokuwiki uses its new 7 slink in forcing chains from 26r2c1 to 7r2c6. Candidate 7 has the straight shot that might induce you to search out the rest.

Now as Sudokuwiki throws up its electronic hands in disgust, you can lay your plans for a coloring trial. Next week, we’re going to pick a color and do a coloring trial. But remember we have a bridge between the two clusters. Red and blue together is impossible, so orange or green or both is true.