Lesson 11 of 17 · Subset technique

Naked quads: four squares, four digits

The guide and all sudoku lessons
The full guide Playing on Puzzled
  1. 1The last empty squareFirst move
  2. 2ScanningCore technique
  3. 3Naked singlesCore technique
  4. 4Pencil marksSolving aid
  5. 5Hidden singlesCore technique
  6. 6Naked pairsPair technique
  7. 7Pointing pairsPair technique
  8. 8Hidden pairsPair technique
  9. 9Naked triplesSubset technique
  10. 10Hidden triplesSubset technique
  11. 11Naked quadsSubset technique
  12. 12Hidden quadsSubset technique
  13. 13X-WingCross-line technique
  14. 14XY-WingWing technique
  15. 15SwordfishCross-line technique
  16. 16JellyfishCross-line technique
  17. 17Simple colouringChain technique

A naked quad extends the naked-triple rule. Find four empty squares in one row, column or box whose complete candidate lists use exactly four different digits altogether. Those squares need four different values, so they must take those four digits between them.

The lists need not match. A square can have two, three or four of the digits; what matters is the combined set across all four squares. A fifth digit anywhere in those lists breaks this four-digit argument.

Remove the quad's digits from other squares in that same unit, not from the four supporting squares. A quad narrows candidates; it does not decide the order of its four digits.

The shaded unit shows all its pencil marks. Darker blue squares support the quad, and one green outline marks the removal target. The second example is a labelled reflection of the first. The exercise rotates and renumbers that same board; the counterexample uses a different puzzle.

Example 1: Four different lists in a column

Step 1 of 5

Read the full length of column 6. Look for four squares whose complete lists contain just four different digits in total.

In column 6, row 4 shows 3 and 5; row 7 shows 2 and 3; row 8 shows 2, 3, 5 and 6; row 9 shows 2, 5 and 6. Together these four squares use only 2, 3, 5 and 6. Only one of the lists contains all four digits.

These four squares need different values, accounting for all of 2, 3, 5 and 6 in column 6. The green target at row 3, column 6 is outside the quad. It shows 1, 4, 6 and 7, so remove its 6 and keep 1, 4 and 7. Other removals may be possible elsewhere in the column; this walkthrough follows one target.

Example 2: The same quad reflected into a row

Step 1 of 5

This is the first board reflected across its diagonal, not a new deal. Read row 6 instead of column 6.

This board reflects the first across its diagonal. Column 6 becomes row 6: columns 4, 7, 8 and 9 now hold the four supporting lists. They still use just 2, 3, 5 and 6 between them.

The counting argument is unchanged. Those four squares reserve the four digits in row 6, so the 6 at row 6, column 3 can go. Keep 1, 4 and 7. A quad can also sit in a box: all four supporting squares and the target must belong to the unit you are using.

Example 3: Four squares with a fifth digit

Step 1 of 5

Read row 2. Consider the four marked squares at columns 1, 2, 3 and 5, without ignoring any of their candidates.

In row 2, the marked squares at columns 1, 2, 3 and 5 show 1/5, 1/6/8, 5/6/8 and 2/8. They may look like a group using 1, 5, 6 and 8, until you count the 2 at column 5. Their full combined set is 1, 2, 5, 6 and 8: five digits, not four.

Four squares cannot reserve all five digits. The green target at row 2, column 6 shows 2 and 3. There is no quad here to remove its 2; keep both candidates. The target is actually 2 in this puzzle's unique solution.

Try it yourself

Try the reasoning yourself. Choose an answer to see the explanation.

This variation rotates and renumbers the worked board. The darker blue squares in column 4 form a naked quad. Read all four lists. Which candidate can it remove from the green target at row 7, column 4?

Pick one candidate to cross out in the ringed square, not a digit to place.

How to spot it faster

  • Try singles, pairs and triples first. If they do not help, compare four squares in one unit and write down the different digits across their complete candidate lists.
  • Count distinct digits, not pencil marks. Repeating a 3 in several lists still counts as one digit. All four squares must share a single row, column or box.
  • After finding a valid quad, look outside its four squares but inside that same unit. A target must actually contain one of the quad's digits for a removal to be useful.

Common mistakes

Expecting four identical lists

A naked quad can contain mixed two-, three- and four-candidate squares. The first example has only one square with all four digits. Compare the combined set rather than looking for four matching lists.

Ignoring one inconvenient candidate

Every candidate in every supporting square counts. A fifth digit breaks the four-square, four-digit argument, even if it appears in just one square. Do not erase it merely to make a quad fit.

Questions

Further reading

Try it

Try an expert 9×9 after checking smaller subsets. A random expert deal is not guaranteed to contain a quad. The exercise here is a rotated and renumbered variation of the worked board, so use it to practise the counting rule rather than as an independent test. Free hints still stop at pairs.

Play solo on an expert 9×9