## Box/Line Reduction

In a box/line reduction, or “box/line” for short, a line’s candidates for a number are restricted to one box. That makes these candidates a toxic set, since one of them must be supplied to the line.  We can remove other candidates of the number from the box remainder.

In the last post, we looked at a case in which we are to recognize the box enclosure as part of the completed pattern of candidates.  Sysudoku solvers don’t do box/lines in this inefficient manner.

To illustrate, we’ll use another example, from  Andrew Stuart’s very worthy book on advanced methods, The Logic of Sudoku.  In the grid section below, Andrew’s traditional keypad candidate placement is  translated into slink marking without column marks.  His example shows the removal of four extra 1-candidates by box/line reduction.  But notice that you cannot even slink mark the 1-candidates correctly with the removed candidates on the grid. With the removals, the middle box candidates are going to be promoted to box marks, also serving the line.

In Systematic Sudoku, box/lines are found in line marking, or sometimes earlier, in box marking. That is, before all candidates are found, not after.  The sysudoku version of this example is accurately depicted below.  The middle line has been marked earlier, being among the 4-free lines. The box/line was recognized and the promotion of the 1-candidates occurred then.

In the diagram, we are in the process of marking the bottom line, a 5-free.  In this marking, the box marks of the 1-candidates prohibit the placement of extra candidates in the middle box.  Line marks are entered 1-candidates instead.

We can alter the solving history slightly to illustrate how this box/line could have been recognized in sysudoku box marking.  Here, the box/line is recognized as the box marks are being added in the right box. It  implies new box marks in the middle box.

The principle of the box marking above is that two lines are locked out for the middle box, guaranteeing that marks can be added in the third line chute of the third box.  It holds when there are multiple candidates in the sweeping lines. I call this form of box/line a “hidden dublex”  in the workbook, which is appropriate.  But I’m not going with “naked dublex” for the most common move in box marking.  In tracing, you can just record the new marks of the third line, leaving it to the trace reader to recognize the cause.

The box/line is a basic solving procedure in Systematic Sudoku, a required lookout in box marking and line marking. Sorry  that I didn’t post it back then in this blog.  Andrew has it right up front in The Logic of Sudoku.  The box/line can also occur later in advanced solving.  On any removal, there is the possiblility of leaving  the remaining candidates of the line in a box.

Last post, we argued that, rather than search for special patterns such as the 2-string kite and skyscraper,  we should just try to connect slinks, and build chains, checking for victims of the toxic sets these connections produce.  I have some homework for you .  Try out your analyst skill on the “ W Wing” depicted below. The boxes are bv.  Is it a new pattern we need to know about?

As we move next to X-loops, are you ready for your X-chain final?

Here is the “diabolical” 236 from  Heron and James, SuDoku for Dummies, after reaching a complete line marking.  I’m finding no wings, regular SdC’s or APE’s.  There is an xyt-chain, but we aren’t going there today.  I’ve marked victims of an X-chain in blue.  With these removals, an XY-chain is connected to remove the 9 victim in SW.  These injuries do bring 236 down.

Are you still using  graph paper for your x-panels and bv map?  You can request sysudoku template files from my email sysudoku@gmail.com .  They are not large files.  If you’re not into MS Office files, it may be time for an update.  It’s as if they were made for Systematic Sudoku.

As usual, solutions in the next Tuesday  post.