Box Marking is the second of three phases of Sysudoku Basic. It follows the slink marking bypass, with the task of posting to the grid two types of candidates used, but not written in the bypass: box slinks and aligned triples. A box slink is a pair of candidates forming a strong link because they are the only candidates of their value in the box. An aligned triple is three candidates of the same value occupying the intersection of a box and a line and the only candidates of that value in the box.
In the bypass, aligned box slinks and aligned triples join clues as “sweepers” in dublexes and crosshatches. In box marking, we mark them for use in advanced solving. In doing so, we often spot combinations of pencil marks that imply more clues, and more slinks, and sometimes a collapse of the puzzle into a solution.
Here is another plaque on a Sysudoku wall, showing the full map of slink marking positions within the cell
Box slink partners are posted as pencil marks in the same position at the top of two cells of their box. Row slinks are posted in a list on the left bottom corner, and column slinks are listed on the right bottom corner. If a candidate is a partner in multiple slinks, it’s box slink position takes precedent. If it’s both a row and a column slink partner, then in the cell aligned with both slinks, it is shifted along the bottom to the middle.
An aligned triple is marked as a candidate in three aligned cells of the box. If the intersecting line is a row, the three candidates are centered on the bottom edge of the cell; if a column, the right edge of the cell.
An aligned triple sweeps its line in adjacent boxes because one of its candidates must be true in the box, and and therefore in its line.
To illustrate box marking, here is an example from review puzzle II-49, from Sudoku Road to Mastery, by Moito Publishing. The bypass gained only the clue E4 and the unresolved 3-fill c5.
Practice your bypass by checking how most of the dublexes and cross hatches fail to produce a clue in this case.
In box marking, we need to record the box slinks that you saw in your bypass check, plus any subsets, for advanced solving. Also to be marked are the aligned triples, bv and X-wings described on the Sysudoku Basic page.
The box marking trace is normally 9 lists, one for each value. A list is skipped if all boxes contain a clue of that value. The value is written as each list begins, and is omitted within the list. The letter “m”, standing for “marks” denotes a slink pair, and a “t”, standing for “triple”, denotes the aligned triple.
Repeating your bypass check above, you would post your visualized slinks this way, down to the first clue. See the aligned triple in the 1: scan?
The box marking trace below shows how this action is recorded.
In the 9: list, the second slink S9m combines with the previously marked S6m on the 6: list to form a naked pair S69. The resulting clues S1 and S3 should be credited to box marking, since the triggering events are separated so far in time.
It so happens that a collapse follows. You might like to take it from the two clues above, and then compare your path with a Sysudoku trace.
Good. I just happen to have one.
On the last leg down, Wht257 refers to the hidden triple in the W box. The box slinks fix candidates of four numbers on 4 cells. But three of those numbers are confined to three cells. It means that those three cells must hold 2, 5 and 7 in the solution, and 3 cannot be in r4c1, and must go into r4c3.
The example shows that slink marking can generate hidden subsets as well as naked ones.
The West box is a lesson in subsets here. The four cells marked in W are a naked quad, because the only missing number, 9, cannot go into r4c2. The quad then locks it out of r5c1. It is also locked out of the three cells of the hidden triple.
Many times, box marking is simply routine preparation for line marking and advanced methods. But like this Moito example shows, box marking can finish puzzles in interesting ways.
Here is another example, from a Dave Green 4-star, involving a box 3-fill. When a line 3-fill in r4 identifies W8 and naked pair r4s46, 4 and 6 are already banned from r5c1, giving W2 and W46, adding SW4.
Row slinks, column slinks, and other non-slink candidates are identified in the line marking process following box marking, and described on the Line Marking Guide page.