## Unsolvable 40 is Solved?

In the last post, Unsolvable 40 was staggered by a remarkable combination of coloring and LPO.  Only one pattern, 8b, was removed by LPO conflicts.  But other conflicts combined to force another pattern 8c to the color blue in an existing blue /green  cluster.  This triggered a merger of two existing clusters.  The extended cluster then trapped candidates of a third pattern.  These removals fed back through LPO conflict logic to remove two patterns for 6, arriving at blue patterns 8c, 6d or 6e and green patterns  8d and 6a. The two blue patterns are shown below:

Now a look at pattern restrictions due to reduction of the conflict tables. Shaded for pattern color, the reduced tables show that 9c is blocked by the possible 8 patterns.  We are reminded that 8c => 9d, and that 9e =>5a.

It is disappointing that discarding 9c only removes one indecisive candidate.  For the present,  I set up patterns 8d and 6a for a trial of green.  I mark it in a breadth first fashion, hoping to recover a short sequence to a contradiction, which I can then express as a forcing chain, which we will take up in the blog shortly.  This move is the reason for the question mark in the title of the  post.

To set up the trial, delete blue candidates, drag green ones to the center, and start the breadth first markup trace.  It begins:

Green => 8d and 6a >(N7=>r5np345=>Cnt34=>Cc4np79=>E7=>SE7=>SW7=>red, E5, W2) , getting us to

When red is proved, we could consider this a forcing chain, in that 8c, the blue pattern, also establishes red.  So red is true regardless.

Continuing with red from here, however, leads to

red => r1np13=>NW4=>NE4,

which leaves no 4’s in c7, a contradiction which shows that green is false.

Now blue, with 8c and 9d, should generate the solution.  It does, without requiring a resolution of the 6d vs 6e question.  It turns out to be 6e. The solution, with candidates dragged into the center of the cell to represent clues, is below.

The question mark remains in the title of this post because not everyone accepts trying green first to arrive at blue, as a solving technique.  After my anniversary post and some follow up on our Unsolvable 40 experience,  I’ll take up this question, with a related exploration of forcing chains.

Anniversary?  Yes, it’s true.  My first post was September 29, 2011. So next time, I get to write about how I feel about that.