Naked and Hidden Quads

Naked and Hidden Quads are the four-cell version of the subset idea you already use for pairs and triples. Four cells in one unit claim four digits between them, and everything else in that unit has to give those digits up.

What the pattern is

A naked quad is four cells in a row, column or box whose candidate lists, taken together, contain exactly four different digits. Those four cells must use up all four digits, so no other cell in the unit can have any of them.

A hidden quad is the same fact seen from the other side: four digits that can appear in only four cells of the unit. Those four cells must swallow all four digits, so every other candidate inside them is dead.

	Naked quad	Hidden quad
You count	Digits across four chosen cells	Cells available to four chosen digits
Trigger	Four cells, four digits in total	Four digits, four possible cells in total
Eliminations	Those digits leave the rest of the unit	Every other digit leaves those four cells
Cell sizes	Two, three or four candidates each	Any size, often five or six

Neither cell needs all four digits. A quad built from {1,4}, {3,8}, {1,3,4} and {4,8} is perfectly valid as long as the union is exactly four digits.

When to look for it

The unit has five or six unsolved cells left and nothing simpler works.
You can see two or three cells sharing a small digit set, and one more cell would close the group.
A unit is dense with candidates but four digits keep appearing in the same handful of cells. That is the hidden form.

How to apply it, step by step

Choose one unit and list the candidates of every unsolved cell.
For the naked form, take four cells whose combined candidates number exactly four digits.
Strike those four digits from every other cell in the unit.
For the hidden form, count the cells available to each digit, then find four digits whose cells add up to exactly four.
Delete every other candidate inside those four cells and leave the rest of the unit alone.

A worked example

Naked quad

Example

Row 5, candidates cell by cell

c1 {1,4}      c2 {1,2,6}    c3 {3,8}
c4 {3,5,9}    c5 {1,3,4}    c6 {2,4,6,7}
c7 {4,8}      c8 {5,8,9}    c9 {2,7,9}

Cells c1, c3, c5 and c7 use only 1, 3, 4 and 8. Four cells, four digits, so those cells hold one of each in some order. Remove 1 from c2, 3 from c4, 4 from c6 and 8 from c8. Row 5 now reads c2 {2,6}, c4 {5,9}, c6 {2,6,7}, c8 {5,9} and c9 {2,7,9} — and a naked pair on 5 and 9 has appeared for free.

Hidden quad

Example

Column 4, candidates row by row

r1 {2,5,7}    r2 {1,3,7}    r3 {2,6,8,9}
r4 {1,7,8}    r5 {3,4,8}    r6 {1,5,6,9}
r7 {4,7,8}    r8 {2,3,9}    r9 {1,3,4}

Track where 2, 5, 6 and 9 can go. The 2 fits in r1, r3 and r8; the 5 in r1 and r6; the 6 in r3 and r6; the 9 in r3, r6 and r8. Four digits, and between them only four cells: r1, r3, r6 and r8. Those cells take nothing else, so r1 becomes {2,5}, r3 becomes {2,6,9}, r6 becomes {5,6,9} and r8 becomes {2,9}.

Common mistakes

Counting a digit twice. The union of the four cells must be four digits, not four appearances.
Including a cell with a fifth digit "that will probably go away". It either belongs now or it does not.
Applying a naked quad to cells that are not all in the same unit. Four cells spread over two boxes prove nothing.
Stripping the wrong side. Naked quads clear the rest of the unit; hidden quads clear the quad cells themselves.

Where you meet it

Quads become necessary on hard classic Sudoku and above. They are rarer than pairs and triples, and most quads you find will already have been solved by something cheaper — so reach for one only after singles, pairs, triples and the pointing rules have stopped producing.