Empty Rectangle

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An Empty Rectangle is a two-strong-link chain, a type of AIC (alternating inference chain). One strong link comes from a conjugate pair in a row or column. The other comes from two perpendicular nodes inside a box, at least one of which is a grouped node: one corresponds to a row-box intersection and the other to a column-box intersection.

How to spot it

• For each candidate x from 1 to 9, examine its positions in every box. Find a box where all instances of x are confined to one row-box intersection and one column-box intersection. Candidate x does not have to occupy all five cells in the two intersections, but it must occur in both directions rather than being confined entirely to just the row or just the column.
• Outside that box, find a conjugate pair for x in a row (or column) with one end that sees every instance of x in one of the two intersections.
• Candidate x can then be eliminated from any cell that sees both the other end of the conjugate pair and the other intersection in the box.

How it works

• Let the two intersections occupied by x in the box be A and B, each treated as a node for x, with at least one being a grouped node. Because no instance of x lies outside A and B, at least one of these nodes must be true, giving the strong link [A]━[B].
• Let the two ends of the conjugate pair be C and D, giving the strong link [C]━[D].
• If C sees grouped node B, they form the weak link [B]┉[C]. Together, the links form the two-strong-link chain [A]━[B]┉[C]━[D]. By the rules of an AIC, the end nodes A and D also have a strong relationship: at least one of them must be true.
• If no cell in intersection A contains x, then x must occur in B. This prevents C from containing x, which in turn forces D to contain x. The same reasoning works in reverse: if D does not contain x, then some cell in A must contain x.
• Therefore, candidate x can be eliminated from any cell that sees both A and D.

Examples

On the board above, candidate 4 in box 4 is confined to the intersections with row 5 and column 2. These intersections form two grouped nodes: A is R4C2+R6C2, and B is R5C1+R5C3. At least one of A and B must contain 4, so they form a strong link. Column 9 also contains a conjugate pair at R5C9 and R9C9. Since R5C9 forms a weak link with grouped node B, these links produce a grouped two-strong-link chain with end nodes A and [R9C9:4]. Candidate 4 can therefore be eliminated from R9C2, which sees both end nodes.

This is an idealized Empty Rectangle. A box has nine cells, and a row-box intersection combined with a column-box intersection occupies five of them. If the candidate is confined to those intersections, the remaining four cells contain none of that candidate, forming an “empty rectangle.”

An Empty Rectangle only requires the candidate to be confined to one row-box intersection and one column-box intersection; not every unsolved cell in those intersections needs to contain it. The candidate must, however, have positions in both directions—it cannot be confined entirely to only the row or only the column.

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Empty Rectangles can take several forms. On the board above, the pattern lies in box 6, where candidate 9 is confined to the intersections with row 5 and column 8. Under one decomposition, the vertical pair R5C8+R6C8 is a grouped node, while R5C7 is a simple node. The structure can instead use the horizontal grouped node R5C7+R5C8 with simple node R6C8, or even use the overlapping grouped nodes R5C7+R5C8 and R5C8+R6C8. These decompositions do not change the chain analysis.

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Here are some more Empty Rectangle examples.