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Hidden quads

When four digits can only go in the same four cells of a row, column or box, every other candidate can be cleared from those four cells.

What it is

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

It is the hidden pair and hidden triple idea taken one step further. Each of the four digits can appear in two, three or four of the cells. What matters is that none of them appears anywhere else in the unit.

Hidden quads are the rarest of the subset techniques. Many keen players go months without needing one. But they are worth knowing, because when they do appear, nothing simpler will do.

How to spot it

Hidden quads are very hard to see by looking at cells. Count digits instead:

  1. Pick a row, column or box with at least seven or eight empty cells.
  2. For each missing digit, write down which cells it can go in.
  3. Look for four digits whose possible cells, put together, are just four cells.

There is a useful shortcut. In any unit, a hidden quad and a naked set always come together. If a row has seven empty cells and four digits are hidden in four of them, the other three cells must form a naked triple with the remaining three digits. Often the naked set is easier to see. Spotting either one gives you the same result.

A worked example

Look at row 9. It has givens at column 3 (an 8) and column 7 (a 5). So it is missing 1, 2, 3, 4, 6, 7 and 9, and its seven empty cells are in columns 1, 2, 4, 5, 6, 8 and 9.

When you check where each digit can go in row 9, you find that 1, 4, 6 and 9 can only go in columns 1, 2, 5 and 8:

  • 1 can go in column 1 or column 5.
  • 4 can go in column 1 or column 8.
  • 6 can go in column 2 or column 5.
  • 9 can go in column 2 or column 8.

That is a hidden quad. Your notes for those four cells were:

  • Row 9, column 1: 1, 2, 4 and 7.
  • Row 9, column 2: 3, 6 and 9.
  • Row 9, column 5: 1, 3 and 6.
  • Row 9, column 8: 4, 7 and 9.

Remove everything except 1, 4, 6 and 9. That clears 2 and 7 from column 1, 3 from column 2, 3 from column 5 and 7 from column 8.

Now 2, 3 and 7 can only go in columns 4, 6 and 9 of row 9. If one of those three digits is left with only one possible cell, you have a hidden single.

Why it works

Row 9 needs a 1, a 4, a 6 and a 9, and they can only go in four cells. If any of those cells took a different digit, there would be only three cells for four digits. That cannot happen, so the four cells hold exactly those four digits.

A common mistake

The most common mistake is removing the quad digits from somewhere. With a hidden quad, you only clear the other candidates from the four quad cells. The four digits are already absent from the rest of the unit.

The other trap is a fifth cell. If 6 could also go in row 9, column 6, the four digits spread over five cells and nothing can be removed. Check every empty cell in the unit.

When you will need it

Hidden quads are part of the Hard toolkit on sudoku.ie, with triples, naked quads and the X-wing. They are rare on Thursday and Friday Hard puzzles and a little more common on Saturday’s Expert.