Bent Quad

Related techniques: WXYZ-Wing · ALS-XZ · Pairs and Subsets

A Bent Quad is also known as a generalized WXYZ-Wing. Both techniques use four cells and four digits, but their structures are different:

a WXYZ-Wing must have a real pivot group and two wing groups;
a Bent Quad needs no pivot and instead counts how often each digit can occur in the four cells.

1. The pattern

A Bent Quad must satisfy all of these conditions:

exactly four structure cells lie across two intersecting rows, columns, or boxes;
their combined candidates contain exactly four digits;
exactly one digit, called Z, occurs in at least two cells that cannot see each other;
every occurrence of each of the other three digits sees every other occurrence of that digit, so each can be placed at most once in the structure.

Four cells and four digits alone are not enough. The restricted/unrestricted condition is essential.

In the hint display:

blue cells are the four structure cells; none is labelled as a pivot or wing;
green candidates are the three restricted digits;
yellow Z candidates are the only unrestricted digit;
red Z candidates see every yellow Z and can be eliminated.

2. Why must Z occur?

Assume none of the four blue cells contains Z. Only the other three digits remain.

Each remaining digit can occur at most once because all of its positions in the structure see one another. Three digits can therefore fill at most three cells, but the structure has four cells. This is impossible.

Therefore:

At least one yellow Z in the four structure cells must be true.

An external red Z that sees every possible yellow Z conflicts with whichever one becomes true, so that red candidate can be eliminated.

3. Why is there no pivot?

The two houses may intersect at a cell that is not part of the four-cell structure. The cells may also form a bend, loop, or other layout without a single center cell that sees the rest.

The proof does not need a center. It only needs four cells, three digits that can occur at most once, and the remaining digit Z. The game therefore does not invent a “best-looking pivot” for this technique.

4. Overlap with other techniques

Bent Quads often overlap with subsets or ALS-XZ. The game keeps the clearest classification:

eliminations directly covered by a naked subset or locked candidate remain basic techniques;
a structure with a valid pivot and two wings is shown as WXYZ-Wing;
Bent Quad is used only for a pivotless structure that satisfies the complete four-cell proof above.

5. A quick search checklist

Find four candidate cells across two intersecting houses.
Confirm their union has exactly four digits.
Find the only digit Z that occurs in cells that do not all see one another.
Confirm every occurrence of each other digit sees all its matching occurrences.
Eliminate any external Z that sees every Z in the structure.

The shortest way to remember Bent Quad is:

Four cells need four digits. If three digits can occur only once each, the fourth digit must occur.