Pairs and triples

Hard puzzles add subsets: a few cells that, between them, must hold a few particular numbers. Once you know which numbers those cells take, those numbers are used up for the cells around them.

The idea

When two cells of a region can only hold the same two numbers, one takes each. This is a Naked pair: no other cell of the region can hold either number.

Example

In the example, earlier singles have placed the 2 and the 1 in the two-cell region at the top left. The five-cell region beside it already holds the 2 at row 1, column 3, so it still needs 1, 3, 4 and 5. Its shaded cells at row 2, column 2 and row 2, column 3 both touch the 1 at row 1, column 2 and the 3 at row 3, column 2, so each can only be 4 or 5. Between them they take the 4 and the 5. The region’s other empty cells, ringed at row 1, column 4 and row 2, column 4, therefore cannot be 4 or 5, and those notes are crossed out. They will hold the 1 and the 3.

In the five-cell region that holds the 2 at row 1, column 3, the shaded cells at row 2, column 2 and row 2, column 3 can only be 4 or 5, so between them they take both. The ringed cells of the same region, at row 1, column 4 and row 2, column 4, cannot be 4 or 5.Bold numbers are given at the start and plain ones were placed by earlier steps. Small numbers are notes, and a crossed-out note is one the step removes. Shaded cells are the evidence, and a ring marks each cell the step is about.

More subsets

  • Naked triple. Three cells of a region that can only hold three numbers between them take those numbers, so the region’s other cells cannot.

  • Hidden pair and Hidden triple. When two or three of a region’s missing numbers fit only in the same two or three cells, those cells hold them and nothing else.

  • Subset neighbour. A subset’s cells hold its numbers between them, so a cell outside the region that touches every place one of those numbers could go cannot hold it.

  • Neighbour pair and Neighbour triple. Cells that each touch or share a region with the others hold different numbers. When between them they can only hold two or three numbers, they take them all, so a cell that touches or shares a region with every one of them cannot hold any of those numbers.

Play Tectonic, or go back to how to play.

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