Box Line Exclusion

A  general concept known as box/line exclusion or box/line interaction covers slink and aligned triple sweep action, and a bit more. In Sysudoku Speak, it’s a box/line, or bxl in a trace.

Consider the intersection of a box and a line – what I call a chute.  The section of line outside of the box is the line remainder; the section of box outside of the line, the box remainder.

Now that we are all introduced, here is the box/line rule:  If candidates of a number are known to be absent from one remainder, they can be removed from the other remainder. The reason is, the absence guarantees a placement of the number, a true candidate of the number, in the chute. Simply elegant.

But wait a minute, guaranteed placement in the chute is the reason a slink or aligned triple can sweep other boxes along the line. This is why some writers label the sweeping action of aligned slinks and triples as box/line interactions.

More in spirit of the box/line exclusion, as you eliminate the last candidate of a number from one remainder, you can immediately remove all candidates of that number from the other remainder. Aligned slinks and triples mark a known absence from the box remainder. So a sweep by either is a box/line exclusion.

In a less encountered side of box/line exclusion, as we remove the last candidate of a number from a line remainder, we can remove all candidates of the number from the corresponding box remainder. Said another way, when a number’s line candidates  are confined to a box, all other candidates of that number in the box can be removed.

A teasing example, from Andrew Stuart’s The Logic of Sudoku, is repeated in my post   Box/Line Reduction of January 2012. Andrew presents box/line as something to discover amid the number scanned candidates. The keypad placements tell you nothing.

That’s the hard way. In the natural course of line marking, you would find the same boxline much easier to spot.

As you line mark the 4f: middle line, you  discover the lack of 1-candidates in the r2 remainder, and the confinement of the box candidates to a box slink.

Then any N box remainder 1’s, say the one already marked in column 6, would be removed, and three more of the number scanned 1’s would never be placed.

Here is a striking boxline in early line marking of puzzle 50, from the review of Tom Sheldon’s Sudoku Master Class.

The 5 slink in the East box is actually a boxline. The box remainder is unmarked, but we know it contains no 5 because the line remainder has no 5 candidate. The slink eliminates two NE candidates as a hidden dublex, or as a 5-wing, producing NE5 and NW5.  Do you know where?

Box/lines and subsets are fundamental to Sysudoku Basic, and in follow up of later eliminations and confirmations. Subsets, and an aide for spotting them, are the subject of a final Basic page.