## Toxic ALS in Unsolvable 40

Our struggle with Unsolvable 40 continues with a detailed walk through of the search for ALS toxic sets, and an analysis of the its stuck number ALS.

Here is our grid with row ALS superimposed on a thin, as yet unproductive AIC web.

Scanning down rows,  we are looking for a restricted common for each ALS.  ALS r3c456 fails that, but it has two stuck numbers in r3, namely 7 and 8.

This ALS acts as a bv containing either 3 or 9. We know how  bv’s define Sue de Coq chutes.  In SdC NEr3, 4 and 6 cannot both be omitted from the chute, but 3 and 9 can.  Thus the ALS forces the SdC  NEr3 to contain either  1(3 + 9)(4 + 6), removing 3 in r3c1, or 164, making green true.

We might be tempted to say, “Either green or remove the candidate? Then the candidate must be green!”  But alas, it isn’t so. A good analysis tool for cases like this is a table enumerating the different combinations that can occur.  The table shows that green does not imply that 3 is false.  In fact, six combinations can occur, half for green and half for blue. The 3-candidate and green can occur together, so no Sue de Coq removal occurs.

Another case occurs in r7. ALS r7c136 contains stuck 2 and 9 in r7, leaving 1 or 3. This forces SdC SEr7 to be 5(1 +3)(6 + 8) or 568, with similar indecisive results as r3 above.  But in the same row, ALS r7489 sticks 6 and 8 and also leaving 1 or 3. In the solution, one ALS contains 1, the other, 3. Candidates are too spread out along rows to provide restricted commons for ALS toxic sets.

There were fewer column and box ALS so I combined them on the same grid.  Again the multiple candidates along the columns don’t support restricted commons.

It’s a challenge to search for toxic set pairing between row, column and box ALS on separate screens.  It might be practical to combine them on one grid, but here it might require a split window to compare grids.

My only success on the ALS toxic search was this a very modest, and rather ghostly one

The final stage of our LPO prep is to look for removals by ALS aided AIC. This means building onto the AIC by finding ALS on the row, column and box slides to extend the chain.

Maybe you can spot one.  I couldn’t.  Next time,  we’ll apply LPO and hope.