Hidden triple

What is a hidden triple?

A hidden triple is three digits that, within one row, column or box, can only go in the same three cells. Those cells must hold those three digits, one each, so any other candidate in them can be removed.

As with a naked triple, not every cell needs every digit; what matters is that the three digits have no other place in the unit. It is the hardest of the subset patterns to see, because the three cells usually carry several other candidates that bury it.

How to spot a hidden triple

Count, digit by digit, where each missing digit can go in a unit. Set aside any digit with four or more spots. Among the rest, look for three digits whose spots, combined, cover only three cells. A digit with only two spots can be part of it, as long as both spots are among the three cells.

A useful hint: when a unit has many empty cells and a few digits are squeezed into one corner of it, check those digits together. Counting is quicker on paper than by eye, so it helps to write the spots down.

A worked example

Cells are named by row and column: r4c1 is row 4, column 1. Rows count from the top, columns from the left, and boxes are numbered 1 to 9 reading like a book, so box 1 is top-left and box 9 is bottom-right.

This position is from an Expert puzzle our game dealt, with every candidate pencilled in from the digits on the board. Nothing simpler moves it forward: no single, no pointing pair or box-line reduction, no naked pair or triple and no hidden pair. Look at column 3, which has seven empty cells.

Ask where the 2, the 4 and the 8 can go in it. The 2 can only go in r5c3 or r8c3; the 4 only in r6c3 or r8c3; the 8 only in r5c3 or r6c3. Those are three digits confined to the same three cells, r5c3, r6c3 and r8c3, so those cells hold 2, 4 and 8 and nothing else.

Clear the rest: r5c3 loses 1 and 9, r6c3 loses 3 and 7, and r8c3 loses 3. Now look at box 4. The only other cell in it that could hold a 1 was r5c3, so r5c2 is 1.

Why it works

Column 3 needs a 2, a 4 and an 8, and all three can only land in r5c3, r6c3 and r8c3. Three digits need three cells, and there are exactly three, so each of those cells is taken by one of them. A 9 in r5c3 or a 3 in r8c3 would leave one of the three digits with nowhere to go.

Which difficulty levels need it

Expert only. Our game's own solver stops at naked pairs, and any puzzle it cannot finish with those is rated Expert, so no Easy, Medium or Hard puzzle needs a hidden triple.

We measured 3,000 Expert puzzles our generator dealt. A solver that always tries the simplest technique first made a move with a hidden triple in 93 of them. For 5 (0.2%), a hidden triple was the hardest technique the puzzle needed: everything simpler stalls, and adding it finishes the grid. Of the 719 puzzles that the twelve techniques in this guide finish on their own, 8 cannot be finished without it. The rest of our Expert puzzles need techniques this guide does not cover yet, almost all of them chains.

Related: Hidden pair , Naked triple .