In the spirit of Valentine’s Day, I come across a coloring resource in Berthier’s xyt chain.
The elaboration of Royle 17-20565 in Chapter XVII of The Hidden Logic of Sudoku is a challenge in itself. By the theory of looking for stuff where the light is better, I look for line subsets as the line is marked, and the lines in closure don’t get examined until – you guessed it – closure.
So with this one, I was in closure on column 5 when I spotted this naked quad. Of course the hidden triple is just as available at this point in Sysudoku basic.
Even if you were too busy sending Valentine cards, and declined the homework, you can appreciate how tough the line marking was, and can imagine how much worse number scanning and hidden logic transformations would be.
So with some relief, I followed up, and got the next cleanup bonus.
If you haven’t gone further already, avoid looking at the grid below. Instead, mark the follow up on the naked triple on your own grid, and contemplate where you would start xyt-chains.
The ticket to ride is a pair of linked bv, with a potential victim within sight of the starting candidate.
The SudoRules first xyt (black) makes a single cell repair in an XY eliminating 4r3c8.
This enables an xyt making two XY repairs and rounding two corners, for another elimination, and a hidden single in r6.
The follow up celebration ends with a skyscraper in the 7’s that you can easily spot (slink-wink-slink), then Berthier reports a hidden xy chain for the collapse.
Instead, we choose to invest the bv dividends in Medusa coloring, and manage to color most of the bv with two clusters.
The bridging logic is
Not(orange and green) => Red or blue.
Not( red and green) =>
Orange or blue.
Oops, that means blue is true, because
(red or blue)& (orange or blue) => (red or orange) and blue => blue.
In trying the obvious, we stumble upon an insight: The xyt version of forcing chain logic is a recourse for coloring.
Starting on one of the few remaining uncolored bv, the xyt assumption “if 9 is false” then includes that red is false and orange is true, and in two ways, 9r5c3 is false. No need to go further. 9r5c3 is false, regardless of 9r5c7. That wraps red, with two red 4’s in c3.
Remember the shortcut wink? Coloring applies in all the bv clover examples the Royle 17 series exhibits in THLS.
We dig a bit deeper into the xyt chain next post. This time, our Royle foil is Royle 17-33442. Its another “solved in half the numbers” wonder.