Empty Rectangle

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An Empty Rectangle is a same-digit two-strong-link chain. One of its strong links comes from a conjugate pair in a row or column. The other comes from two perpendicular grouped nodes inside a box. In that grouped-node strong link, one node corresponds to the intersection of a row and the box, and the other to the intersection of a column and the box.

For a candidate digit $d$, suppose that within a box, all instances of $d$ are confined to one row intersection and one column intersection. Then if there is also a conjugate pair in the corresponding row or column, and one end of that conjugate pair forms a weak link with the row or column intersection inside the box, we get a two-strong-link chain. Any candidate $d$ that forms weak links with both the other end of the conjugate pair and the opposite grouped node can be eliminated.

For example, look at digit $4$ in the board below. In box $4$, all candidate $4$s are confined to the intersections of that box with row $5$ and column $2$. We can group them into two nodes: node $H$ is $R5C1+R5C3$, and node $V$ is $R4C2+R6C2$. These represent the propositions “$4$ is in one of the cells in $H$” and “$4$ is in one of the cells in $V$,” and one of them must be true, so they form a strong link. In addition, column $9$ contains a conjugate pair at $R5C9$ and $R9C9$, giving another strong link. Since $R5C9$ forms a weak link with grouped node $H$, we get a grouped two-strong-link chain whose endpoints are grouped node $V$ and $R9C9$. Candidate $4$ in $R9C2$ forms weak links with both endpoints, so it can be eliminated.

This is a very clean Empty Rectangle. A box has nine cells. A row intersection and a column intersection occupy five of them, leaving four cells behind. If the chosen candidate is confined to the intersections, those remaining four cells contain none of that candidate, creating the empty rectangle.

An Empty Rectangle only requires the chosen candidate in a box to be confined to one horizontal intersection and one vertical intersection. It does not require every unfilled cell inside those intersections to contain the candidate.

The pattern can appear in several forms. In the next example, the Empty Rectangle is in box $6$, where candidate $9$ is confined to the intersections with row $5$ and column $8$. When grouping the nodes, only the vertical pair $R5C8+R6C8$ is a true grouped node, while $R5C7$ is a simple node. But you could just as well group the horizontal pair $R5C7+R5C8$ and pair it with simple node $R6C8$, or even allow overlapping grouped nodes. These different decompositions do not change the chain analysis.

Here are some more Empty Rectangle examples.