Unique Loop Type 3
A Unique Loop Type 3 combines the uniqueness argument with naked-subset logic in loop form. When two rescue cells in a longer loop carry extra candidates that pair up with another cell's candidates in a shared house, they form a naked set — and the deadly pattern forces the set to be locked, allowing its digits to be removed from the rest of the house.
What is Unique Loop Type 3?
Type 3 merges the uniqueness argument with naked-subset logic. Two rescue cells in the loop carry extra candidates — not just one shared digit as in Type 2, but two or more. Those extra candidates can combine with the candidates of another cell in the same house to form a naked subset. The deadly pattern guarantees that one of the rescue cells must hold a value from the extra-candidate set, which means the naked subset is complete and locked.
Once the naked set is established, the standard subset elimination applies: any candidate for one of the set's digits in a cell outside the set but inside the same house can be removed. The uniqueness argument is what proves the rescue cells contribute to the set — without it, the rescue cells might both hold deadly-pair digits and the subset would not be forced.
How to spot it
Find a loop with two rescue cells on the same side, each carrying extra candidates beyond the deadly pair. Then look at the house they share and check whether another cell in that house has candidates that overlap with the rescue cells' extras to form a naked set. The set size can be anything from a pair up to a quad. When the set is confirmed, remove the set's digits from every other cell in that house. This is the hardest Unique Loop type to spot because it requires seeing both the loop and the subset simultaneously.
The logic
If neither rescue cell held any of the extra candidates, both would contain only the deadly pair and the loop would become the deadly pattern — two solutions, which is impossible. So at least one rescue cell must hold a value from the extra set. That means the rescue cells, together with the companion cell, collectively contain exactly the naked set's digits within that house. By the naked-subset rule, those digits are locked into the set, and any occurrence of them in other cells of the house can be eliminated. The uniqueness guarantee supplies the premise; the subset rule supplies the conclusion.