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The 1 is placed. What about the highlighted cell?
Level 5 · Outside clues and specials
How to solve quadruples
Digits in a circle on a corner must appear in the four cells around it.
A quadruple is a small circle on the point where four cells meet. Every digit in the circle must appear in at least one of those four cells.
The circle doesn't say which cell holds which digit. Cells that don't get a listed digit can hold anything the other rules allow, and a circle with four digits fills all four cells.
How setters write it
The same rule, worded the ways you’ll see it on puzzles.
- Digits in a circle must appear at least once in the four surrounding cells.
- Quadruples.
Worth memorizing
- The circle doesn’t say where, only that each listed digit is somewhere in the four cells.
- Four listed digits fix the four cells’ contents completely.
- A digit listed twice appears twice, so the two copies must be diagonal to each other.
Solving tips
Four digits fill all four
If a quad shows 2, 4, 6 and 8, those four cells hold exactly 2, 4, 6 and 8 in some order. No other digit fits there.
Repeats sit on a diagonal
If a quad lists 3 twice, two of its cells are 3. Cells side by side share a row or column, so the two 3s must sit diagonally, in different boxes.
Use the box's placed digits
A quad often sits across two or four boxes. If one of those boxes already has its 5, a 5 from the quad must go in the cells outside that box.
Use rows and columns too
If the top row already has a 1 elsewhere, the quad's 1 must go in one of its two bottom cells.
A worked example: a quad with a repeated digit
We name cells by row and column, so row 3, column 3 is r3c3. A quad circle sits where r3c3, r3c4, r4c3 and r4c4 meet. It shows 5, 5 and 7. These four cells are in four different boxes. The top-middle box already has a 5 outside the quad, and row 3 has a 7 outside the quad.
- The circle lists 5 twice, so two of the four cells are 5. Cells side by side share a row or column, so the two 5s must be on a diagonal. They are either r3c3 and r4c4, or r3c4 and r4c3.
- r3c4 is in the top-middle box, which already has its 5. So r3c4 can't be 5. The 5s go in r3c3 and r4c4.
- The 7 must go in one of the two cells left: r3c4 or r4c3.
- r3c4 is in row 3, which already has a 7. So r4c3 = 7.
- The last cell, r3c4, is not limited by the quad. It still can't be 5 (its box has one) or 7 (its row has one).
Practice
2 of 2
Three cells are filled. What goes in the last one?
Going deeper
What each size of clue tells you
A quad with one or two digits leaves at least two cells free. A quad with three digits fills three of the four cells with those digits, and the fourth cell can be anything the other rules allow. That fourth cell may even repeat one of the listed digits, on a diagonal, since the clue only asks for at least one copy.
A quad with four digits fixes the whole 2x2 square. If the four digits are all different, each cell gets one of them. If one digit is listed twice, those two copies sit on a diagonal, and the other two cells hold the other two digits.
Quads can also point along a row or column. Say a quad lists 4, and its top two cells sit in a row that already has a 4. The 4 must go in one of the bottom two cells. Those two cells share a row, so no other cell in that row can be 4.
A repeated digit needs a box border
Two copies of a digit can only sit on a diagonal of the 2x2 square, and those two cells must be in different boxes. If all four cells are in one box, no digit can repeat. So a quad that lists a digit twice always crosses at least one box border.
A digit can't appear three times in a quad. Any three cells of a 2x2 square include two that share a row or column. So the most a quad can hold of one digit is two.
A quad that covers two cells in each of two boxes can repeat a digit. Its two diagonal cells are in different boxes, rows and columns, so nothing stops them from matching.
Quads that share cells
Two quads next to each other share two cells. If one quad lists four digits, those two shared cells must come from that list. Compare it with the other quad's list to narrow them further.
For example, say quad A lists 1, 2, 3 and 4, and quad B next to it lists 8 and 9. The shared cells must hold digits from A's list, so they can't be 8 or 9. B's 8 and 9 must go in B's other two cells.
Common mistakes
Reading the digits in order
The circle lists digits only. It says nothing about positions. A quad showing 1, 3 does not mean 1 goes top left and 3 goes top right. Each digit can be in any of the four cells the other rules allow.
Banning digits that are not listed
If a quad shows 2 and 6, the other two cells can hold any digit the rest of the puzzle allows. The circle only says what must appear, never what can't.
Reading one listing as exactly one
A digit listed once must appear at least once. It could appear twice, on a diagonal across a box border. Only a digit listed twice is sure to appear twice.
In mixed-rule puzzles
Quads work well with other cell clues. With odd-even marks, a quad listing 2, 4, 6 and 8 can't cover a circle, so a circle there breaks the puzzle, and a quad that lists three even digits and covers one circle puts its free cell on the circle. With Kropki dots between quad cells, the dot tells you which listed digits can sit next to each other. With killer cages, a quad's listed digits help decide which cage combination fits. With anti-king, all four quad cells touch each other, so no digit can repeat in the quad at all. Always check each puzzle's rules for how its quads work.
What the video shows
- Digits in a circle at a corner must appear in the four cells around it.
- 1 and 3 both appear. The other two cells are free.
Questions solvers ask
What is a quadruple in sudoku?
A circle at the corner where four cells meet. Each digit in the circle must appear in at least one of those cells.
Can a quadruple digit appear twice?
Yes. A digit listed once must appear at least once, and it could appear twice on a diagonal. If the circle lists it twice, it must appear twice.
Do the other cells around a quadruple have rules?
Only the normal ones. If the circle shows two digits, the other two cells can be any digits the rest of the puzzle allows.
What does a circle with numbers on a corner mean in sudoku?
It is a quadruple clue. The listed digits must appear in the four cells that touch the circle.
What is the difference between a quadruple and a killer cage?
A killer cage gives a total and bans repeats inside it. A quadruple lists digits that must appear in four cells, gives no total, and allows a repeat on the diagonal.
Can a quadruple have more than four digits?
No. Each listed digit needs its own cell, and there are only four cells, so a quad lists at most four digits.
Can a quadruple circle show the same digit three times?
No. Any three cells of a 2x2 square include two in the same row or column. So a digit can appear at most twice in a quad.
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