Naked quads

Four cells in one row, column or box whose candidates use only four digits between them lock those digits away from the rest of that unit.

A real example from a sudoku.ie puzzle. In column 7, the cells at rows 4, 5, 6 and 8 of column 7 can only hold 1, 3, 7 or 8. These four cells must take the 1, 3, 7 and 8 between them, so no other cell in column 7 can be 1, 3, 7 or 8. Remove 3 from row 1, column 7; 7 from row 2, column 7; 3 and 7 from row 3, column 7; 8 from row 9, column 7. Yellow cells make the pattern. Red marks show what you can cross out.

What it is

A naked quad is four cells, all in the same row, column or box, whose candidates between them use only four different digits. Those four cells must hold those four digits, one each. So no other cell in that unit can hold any of them, and you can remove all four digits from the rest of the unit.

It is the same idea as the naked pair and naked triple, just with four cells. As with triples, each cell does not need all four digits. A cell can have two, three or four of them, as long as no cell has a digit outside the four.

How to spot it

Naked quads are rare and not easy to see, because four cells with up to four candidates each make a busy picture. Some tips:

Look in units with plenty of empty cells, ideally six or more. With fewer, a quad is hardly ever useful.
Focus on cells with two or three candidates. List the digits as you add each cell and stop if the total goes above four.
Use the “opposite” trick. In a unit with eight empty cells, a naked quad and a hidden quad always come together. If the other four cells look simpler, look at them instead.

A worked example

Look at box 5, which covers rows 4 to 6 and columns 4 to 6. It has seven empty cells. Four of them have these candidates:

Row 4, column 4: 1 and 3.
Row 4, column 6: 1 and 6.
Row 5, column 5: 3, 6 and 8.
Row 6, column 4: 1 and 8.

Put together, these four cells use only 1, 3, 6 and 8. So they must hold box 5’s 1, 3, 6 and 8, one each.

Now look at the other three empty cells in box 5:

Row 4, column 5 has 1, 2 and 7. Remove the 1, leaving 2 and 7.
Row 5, column 4 has 2, 3 and 9. Remove the 3, leaving 2 and 9.
Row 6, column 6 has 6, 7 and 9. Remove the 6, leaving 7 and 9.

No cell is solved yet, but the three remaining cells now only use 2, 7 and 9 between them. They are a naked triple. And if one of those cells shares a row or column with other empty cells outside the box, the cleaner notes may well lead to a placement there.

Why it works

Four cells, four digits, all in one box. Each cell needs a different digit, so the four digits are all used up by these four cells. Any other cell in the box that took one of them would leave one of the four cells with nothing to hold.

A common mistake

The usual slip is letting a fifth digit sneak in. If row 5, column 5 also had a 2 in its notes, the four cells would use five digits and the quad would not hold. Count carefully.

It is also easy to mix up which cells to clear. You remove the four digits from the other cells in the unit, never from the quad cells themselves.

When you will need it

Naked quads are part of the Hard toolkit on sudoku.ie, along with triples, hidden quads and the X-wing. They are uncommon. Most Hard puzzles on Thursday and Friday can be done without them, but now and then one is the only way forward, and they also appear on Saturday’s Expert puzzle.