Advanced Techniques: Triples, X-Wing and the Wings

The six techniques for Expert and Evil, from triples to Swordfish and XYZ-Wing.

Before you read this page

Every technique here builds on the intermediate techniques: find a relationship in your notes that must hold, then remove the numbers that conflict with it. The diagrams work the same way: green cells are the key cells, bold numbers are the key numbers, and notes with a red slash can be removed.

Expert puzzles use triples, X-Wing and XY-Wing; Evil adds Swordfish and XYZ-Wing.

Naked triple

An extension of the pair: three cells in one area whose notes together contain only three numbers. Those three cells use up those three numbers, so they can be removed from every other cell in the area.

Each cell does not need all three numbers. Cells with {1, 2}, {2, 3} and {1, 3} together still hold only 1, 2 and 3, so the rule still applies.

Hidden triple

Three numbers can only go in the same three cells of an area. Every other note in those three cells can be removed.

X-Wing

Look at a single number d. If d has exactly two places left in each of two rows, and those four cells line up in the same two columns, they form a rectangle.

There are only two ways to place d in those rows: top left and bottom right, or top right and bottom left. Either way, each of those two columns uses up one d, so d can be removed from those columns in every other row. Swap rows and columns and the same holds.

XY-Wing

Find three cells with exactly two notes each: a pivot with {A, B}, and two wings with {A, C} and {B, C}. Each wing shares a row, column or box with the pivot.

• If the pivot is A, the wing that sees it cannot be A, so it must be C.
• If the pivot is B, the other wing cannot be B, so it too must be C.

Either way, one of the wings is C. Any cell that sees both wings cannot be C.

Swordfish

The three-row version of X-Wing: a number d has two or three places left in each of three rows, and together they fall in only three columns. Those three rows use up one d in each of the three columns, so d can be removed from those columns in every other row.

Not every row needs three cells: shapes like 2-3-2 or 2-2-2 work too. Swordfish is hard to spot. Usually you look for it only after everything else has failed, by counting where one number can go in each row.

XYZ-Wing

A variation on XY-Wing: the pivot has three notes {A, B, C}, and the wings are {A, C} and {B, C}. This time the pivot itself might be C, so a cell can only be cleared if it sees all three cells: the pivot and both wings.

When you are stuck

Go back to the simple techniques first: after an advanced move removes a note or two, a new single often appears right away. Every puzzle here can be solved with these techniques alone, with no trial and error or guessing. If you really cannot find the next step, press Hint and it will tell you which technique comes first.