n-Fill Rules

As free cells in a  line or box decrease in number, it becomes more likely that one of the missing values can be placed.  Spotting and carrying out this placement depends on free cells and missing values of the n-fill “seeing” each other. A clue placement on a 3-fill,  with three free cells, produces at least one clue, plus a naked pair, but if one of the remaining two free cells is seen by a  missing value, the placement yields two more.  Since the results vary, we just say the 3-fill is resolved.

In Sudoku writing, two candidates of the same value are said to “see” each other when one being a true candidate means that the other is false. This logical happening is called a weak link. If the candidates are in the same box or line, they are weakly linked, but they can be separate units and be weakly linked by an AIC, an alternating inference chain.

In the n-fill rules, free cells and candidate values seeing each other when cell and candidate are in the same line or box. If they see each other, the value cannot be placed as a clue in the cell.

A 3-fill resolution is achieved by one of two rules. And when this happens, one clue is placed and two missing values are placed as two clues or a naked pair in the 3-fill line or box.

Satisfy yourself that both rules work, because a cell can’t be assigned a value it sees, and because in the solution, every line or box contains every value.

With 4-fills, having four free cells and four missing values, two resolution rules are based on the same “seeing” conditions,  but the requirements are harder to meet, so resolutions are less frequent. 

For spotting, look first for multiple free cells and missing values in the same box. With 3 free cells in a box, any missing value in the box is placed in the single free cell outside the box. With two free cells and one missing value in a box, one of the other free cells must see that missing value to place the value in the fourth box. Two cells with two missing values brings a resolution.