Naked Quadruple
Description
Find four cells in the same row, column or 3x3 block whose complete candidate lists contain exactly four different digits between them. These cells form a Naked Quadruple. Keep their notes, and remove those four digits from every other cell in that group.
This extends the idea of a Naked Triple. Each cell may have two, three or four candidates, and the lists need not match. For example, {4,8}, {4,8,9}, {4,7}, {7,9} use only 4, 7, 8 and 9 altogether. None of these cells contains all four notes, yet together they form a quadruple.
Why? Four cells in one group must contain four different digits. Their notes allow only these four, so each digit must be used once, in some order. The other cells in the group cannot contain any of them. You do not need to know which digit goes in which of the four cells.
A Naked Quadruple eliminates candidates. It may reveal a Hidden Single or a Naked Single, but it does not guarantee a placement. Several more eliminations may be needed before a cell can be filled. The illustrations below show immediate singles; the demo also follows a longer continuation.
We use pencil notes throughout. Green candidates stay and red candidates are removed. Yellow highlights show every other occurrence of the digits being discussed, and the relevant group is shaded.
Example 1. Four cells, four digits
Example 2. A quadruple in a row
You may recognize the second position from Hidden Triple. The triple of 4, 5 and 9 in the other three empty cells removes exactly the same four notes. Both views are valid; apply the eliminations once, then inspect the updated notes.
The demo starts with an immediate Hidden Single, then shows a different quadruple followed by Claiming Candidates and a Naked Pair before a Naked Single appears. The next technique depends on the resulting position.
Demo
How to find Naked Quadruples
- Use complete, up-to-date pencil notes. Check singles, intersections, pairs and triples first.
- Choose a row, column or block. Look at cells with two, three or four candidates.
- Combine the entire candidate lists of four cells. They must use exactly four different digits altogether. The lists can differ, and no cell needs to contain all four digits.
- Keep every note in those four cells. Remove their four digits from every other cell in the shared group.
- Recheck for singles. If none appear, continue eliminating candidates using the updated notes.
Highlighting the four digits helps you follow their positions, but always check for other notes in the proposed cells too. If there is nothing to remove elsewhere in the group, recognizing the quadruple makes no immediate progress.
Count all the different digits
{4,8}, {4,8,9}, {4,7}, {7,9} form a Naked Quadruple when the four cells share a group. Their combined lists contain exactly 4, 7, 8 and 9. No two of these cells form a Naked Pair, and no three form a Naked Triple: all four cells are needed to establish this naked subset.
{1,3,7,9}, {7,9}, {5,7}, {1,5} do not form a Naked Quadruple. There are five different digits: 1, 3, 5, 7 and 9. Having at most four notes in each cell is not enough. The demo's final check shows this case on a real position.
All four cells must share one row, column or block, though they need not be adjacent. Four cells scattered across the grid do not form a quadruple just because their lists use four digits.
Where to remove candidates
Remove the quadruple's digits only from other cells in its shared group. Keep the quadruple's own notes and every unrelated candidate. You do not need a copy of each of the four digits to remove; even one elimination can help.
Unlike a pair or triple, four cells cannot all lie in both a block and one row or column: a line crosses only three cells of a block. A quadruple therefore shares just one group. A cell outside that group is not restricted by this rule just because it sees some of the four cells.
Naked and hidden subsets
A Naked Pair reserves two digits for two cells; a Naked Triple reserves three for three. A quadruple uses four for four. All remove their digits from other cells in the shared group. Hidden subsets instead remove other digits from their own cells.
The remaining empty cells can express the same restriction. With six empty cells, a Naked Quadruple has a complementary Hidden Pair; with seven, a Hidden Triple; with eight, a Hidden Quadruple. With only five, the remaining cell is already a Hidden Single. Count empty cells, not filled ones.
In this lesson's prepared positions, the selected group has seven empty cells. Its complementary Hidden Triple makes exactly the same eliminations as the quadruple. If that triple is easier for you to see, use it. These are two explanations of one deduction, so applying one does not leave the other as another move in that group.
Try it yourself
These prepared positions come from our puzzle library. The links preserve earlier candidate eliminations. No single, Pointing, Claiming, pair or Naked Triple is available at the start; an equivalent Hidden Triple remains valid. Find four cells whose complete lists use four digits, then remove those digits from the other cells in their group.
- Sudoku #1 — find a quadruple in a row, then check for a single.
- Sudoku #2 — find a quadruple in a column, then inspect the changed notes.
- Sudoku #3 — find a quadruple in a column, then continue eliminating candidates.
Hint shows the Naked Quadruple in each exercise. Apply it once, then request the next hint to follow the continuation. The first two exercises reveal a Hidden Single immediately. The third needs a Naked Triple before a Hidden Single appears. These continuations are examples, not a fixed sequence for quadruples.
Naked Quadruple hints are available in Expert and Extreme games, with pencil notes. All three practice links open directly in pencil mode.
See also
- Start with Naked Triple for the same reasoning with three cells.
- Compare Hidden Triple for the complementary view of the seven empty cells in these examples.
- Review Claiming Candidates and Naked Pair to follow the demo's further eliminations.
- Review Hidden Single and Naked Single to recognize placements after eliminations.
- Review Sudoku Rules for why the four cells must contain different digits.
- Play Expert Sudoku online to combine quadruples with singles, intersections and smaller subsets.