Naked Triple

Summary

Naked Triple is the fourth technique to reach for when Naked Pair can no longer make progress. It extends the same union-based logic from two regions to three: if the union of all candidate rows from regions A, B, and C together spans exactly three rows, then those three regions monopolize those three rows. Because Nyangjari places exactly one cat per row, no other region can place a cat in any of those three rows. Every empty cell in those rows belonging to a fourth region can therefore be marked X.

Unlike Naked Pair, each individual region does not need to span all three rows. Region A might cover rows 1 and 2, region B rows 2 and 3, and region C rows 1 and 3. The union {rows 1, 2, 3} still equals exactly three rows, so the triple is valid. This distributed overlap is exactly what makes Naked Triple harder to spot than Naked Pair, and why it is worth exhausting Naked Pair first before scanning for triples.

When to use

• Apply Naked Triple only after Basic Elimination, Row/Column Lock, and Naked Pair have all run to completion with no further progress.
• A triple qualifies when the union of candidate rows (or candidate columns) from exactly three regions equals exactly three rows (or three columns).
• Because there are more combinations of three regions to check than two, Naked Triple is more costly to find. Always exhaust Naked Pair first.
• After marking new X cells, always re-run Basic Elimination before searching for another triple.
• Check both axes: row-based triples and column-based triples are independent and both must be scanned.

Why it works

Nyangjari's rules state that each row contains exactly one cat. Suppose region A has candidates in rows 1 and 2, region B has candidates in rows 2 and 3, and region C has candidates in rows 1 and 3. The union of their row sets is {1, 2, 3} — exactly three rows. Together, A, B, and C must each place one cat, and all three cats must land somewhere in rows 1, 2, or 3. Since each row accepts only one cat, those three rows are fully claimed by the triple. Any other region's candidate cell in row 1, 2, or 3 is impossible and receives an X mark.

The union must equal exactly three rows. If it spans two rows, you have a Naked Pair (or even a Row/Column Lock). If it spans four or more rows, the triple constraint does not hold and no elimination is possible from this combination alone.

The same logic applies column-wise: if the union of candidate columns from three regions equals exactly three columns, those regions monopolize those columns, and all other regions' cells in those columns receive X marks.

Example

▲ Region a candidates: rows 1·2. Region b candidates: rows 2·3. Region d candidates: rows 1·3. Union {rows 1, 2, 3} = exactly 3 rows → Naked Triple. Other regions' (c) empty cells in rows 1, 2, and 3 are marked ×.

1. Starting state: After previous techniques, region a has candidates on rows 1 and 2, region b on rows 2 and 3, and region d on rows 1 and 3.
2. Union check: {rows 1, 2} ∪ {rows 2, 3} ∪ {rows 1, 3} = {rows 1, 2, 3} — exactly 3 rows. Naked Triple confirmed.
3. Eliminate other regions from rows 1, 2, and 3: Every empty cell in those rows belonging to region c is marked ×. The highlighted candidates in regions a, b, and d are left unchanged.
4. Re-apply Basic Elimination: With the new X marks, check whether any region, row, or column has been reduced to a single candidate and confirm those cat placements.
5. Continue scanning: Search for further Naked Triples — and Naked Pairs — in the updated board before moving to X-Wing.

Watch out for

• Union must equal exactly three — no more: If the union spans four or more rows, Naked Triple does not apply. Four rows would require a Naked Quad, which is beyond Nyangjari's standard technique set.
• Individual regions need not cover all three rows: A triple where A covers rows 1·2, B covers rows 2·3, and C covers rows 1·3 is perfectly valid. The union {1, 2, 3} is what matters, not the per-region spread.
• Naked Pair vs. Naked Triple: If the union of all three candidate row sets happens to equal only two rows, the combination actually contains a Naked Pair — a simpler deduction that should have been found earlier. Always exhaust Naked Pair before searching for Naked Triple.
• Check both axes: After scanning row-based triples, always scan column-based triples. A board that yields no row-based triples may still have a column-based triple.
• Elimination targets other regions only: Never mark the three paired regions' own candidates as X — those are the only remaining positions for those regions' cats.
• Re-run Basic Elimination after every triple: New X marks may create single-candidate situations that cascade into multiple cat confirmations. Never skip this step.
• If Naked Triple stalls, move to X-Wing: X-Wing operates on a single color across two rows and two columns — a completely different search pattern from the region-union approach of Naked Triple.