Lesson 12 of 17 · Subset technique

Hidden quads: four digits with only four homes

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 hidden quad starts with four digits, not four short candidate lists. In one row, column or box, find four unplaced digits whose possible homes together occupy exactly four squares. Those squares must take the four digits between them, so any other candidates in them can be removed.

The four digits do not have to appear in every supporting square. Some squares may also show several extra marks; those marks are what hide the pattern. Check the whole unit for each tracked digit. One extra home outside the four squares can break the argument.

Compare this with a naked quad. A naked quad starts with four squares that can hold only four digits, and clears those digits from other squares. A hidden quad starts with four digits that have only four homes, and clears other digits from those homes.

We reuse the Naked quads teaching board to show the complementary view of the same restriction. The reflected example and rotated, renumbered exercise are labelled variations, not new deals. Neither view is the only way to make the removal. The counterexample uses a separate puzzle.

Example 1: Find the four homes in a column

Step 1 of 5

Read column 6 and track 1, 4, 7 and 9. Count every home for each digit across the whole column.

Track 1, 4, 7 and 9 down the full length of column 6. The 1s are in rows 3 and 5; the 4s in rows 1 and 3; the 7s in rows 1, 3, 5 and 6; the 9s in rows 5 and 6. Their combined set of homes is rows 1, 3, 5 and 6: exactly four squares.

Those four squares must hold 1, 4, 7 and 9 between them. The green target at row 3, column 6 shows 1, 4, 6 and 7. Its 6 is outside the reserved set, so remove it and keep 1, 4 and 7. Notice that 9 need not be a candidate in the target; the rule applies to the group as a whole.

In the previous lesson, the other four empty squares of this column formed a naked quad of 2, 3, 5 and 6. That view removes the same 6 from this target. Here we follow the four remaining digits instead.

Example 2: Reflect the hidden quad into a row

Step 1 of 5

The first board has been reflected, not dealt again. Read row 6 and follow 1, 4, 7 and 9.

This is the first board reflected across its diagonal. Row 6 now has 1 in columns 3 and 5, 4 in columns 1 and 3, 7 in columns 1, 3, 5 and 6, and 9 in columns 5 and 6. Every home lies in columns 1, 3, 5 and 6.

The hidden quad occupies those four squares, so remove extra digits inside them. At row 6, column 3, remove 6 and keep 1, 4 and 7. This is the same argument in a row; a hidden quad in a box follows the same four-digits, four-homes rule.

Example 3: An extra home breaks the hidden quad

Step 1 of 5

Track 1, 5, 6 and 8 in row 2. Consider columns 1, 2, 3 and 8 as a possible group, but check the entire row.

In row 2, look for a hidden quad of 1, 5, 6 and 8 in columns 1, 2, 3 and 8. All four digits appear among those squares, but that alone is not enough: the 8 also has a home at column 5.

The four digits therefore have five possible homes, not four. They are not all forced into the proposed group. At row 2, column 8, keep the extra 7 as well as 6. In this puzzle's unique solution, the 8 goes at column 5 and the target is 7: removing that 7 would destroy the solution.

Try it yourself

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

This variation rotates and renumbers the worked board. In column 4, track 2, 4, 5 and 8 across the whole column. Their four homes are shaded in darker blue or outlined in green. Which extra candidate can be removed 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

  • Choose one unit and follow digits with only a few possible homes. Compare four unplaced digits and count the different squares they reach. The combined set must contain exactly four squares.
  • Check every home in the whole row, column or box, including a faint mark away from the group. A fifth home can invalidate the four-square restriction.
  • Once the four-home restriction is proved, look inside those squares for candidates outside the tracked set. A square with extra marks is a removal target, not an invitation to choose one of the digits left.

Common mistakes

Erasing the four tracked digits

A hidden quad keeps its four digits and removes other marks from its own squares. This is the reverse direction from a naked quad. The green target in this lesson is one of the supporting squares, not a square outside them.

Mixing homes from different units

All four digits must be confined to the same four squares in one shared row, column or box. Finding two digits restricted in a row and two others restricted in a different column does not make a hidden quad.

Questions

Further reading

Try it

Try an expert 9×9 once smaller subsets stop helping. A random expert deal is not guaranteed to need a hidden quad. The exercise here rotates and renumbers the worked board; it is practice in finding homes, not an independent assessment. Free hints still stop at pairs.

Play solo on an expert 9×9