Box Line Exclusion

Boxline Exclusion

A  general concept known as box/line exclusion or box/line interaction can frequently obtain additional removals from a candidate removal. In Sysudoku Speak, it’s a boxline. In a trace, it’s marked by the code bxl.

In the intersection of a box and a line, let’s call the section of line outside of the box the line remainder and the section of box outside of the line, the box remainder.

Now that we are all introduced, here is:

The reason is, the absence from a remainder guarantees a placement of the value in the intersection, which sweeps the other remainder. Simply elegant.

In Sysudoku Basic actions, we don’t need the boxline rule. When a box value is missing from the box remainder, there’s an aligned slink or triple, sweeping the line remainder.  In line marking, when a value is missing from the line remainder, it has to be present in the intersection, and missing from the box remainder.

Here is a line marking example from Moito’s Sudoku Road to Mastery and Very Hard III-34, posted January 30, 2018.

Because the r7 line remainder is free of 8, the 8’s in SEr7 are to be repositioned to mark an aligned triple.


And also, since the SE box remainder contains no 8,  we don’t need to wait for the marking of r9 to  promote the S box 8’s  to box slink positions.

A teasing example, from Andrew Stuart’s The Logic of Sudoku, is repeated in the post   Box/Line Reduction of January 2012. The line remainder is split in this one.

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

Then any N box remainder 1’s, say the one already marked in row 3, is removed, and with the slink marking in place, three more number scanned 1’s will not be placed as r1 is line marked.

Sysudoku Basic does require the boxline rule in following up Basic action with all their effects, often including collapse of the puzzle. In Basic and Advanced follow up, candidates are being removed, and the boxline rule becomes  a spotting rule.

A spotting rule enhances human solving by focusing attention.  It tells what to look for and when to look for it. As you eliminate the last candidate of a value from a line remainder, remove all candidates of that value from the box remainder. And when you remove the last value from a box remainder, you’re reminded to re-position the candidates of any new slink or aligned triple and remove the value from the line remainder.

The boxline shows up both ways in the continued line marking of Moito’s Very Hard III-34. In a boxline chain, a naked quad creates E8m, and the removal of 8r5c3 leaves an aligned triple in Wc2 and a slink in NWc3. That’s two boxlines eliminating 8r3c2.


Box/lines are fundamental to Sysudoku Basic and Advanced. A second page on boxlines is included in the Sysudoku Advanced section to illustrate how advanced methods create them.

Subsets, and an aid for spotting them, are the subject of a fifth Basic child page.