XY-Chain

An XY-Chain is a path of cells that each have exactly two candidates. Neighbouring cells see each other and share a digit: the second digit of one cell is the first digit of the next. If the first cell is not its first digit, it must be its second; then the next cell cannot be that digit and must be its own other digit, and so on to the end. When the first digit of the first cell and the last digit of the last cell are the same digit Z, one of the two end cells is always Z, and every cell that sees both ends can lose Z. The Y-Wing is the shortest XY-Chain, with three cells. The XY-Chain is a Master technique in Bare Sudoku.

When to use it

When the board has many cells with exactly two candidates and the wings find nothing. XY-Chains use only those two-candidate cells, so they are easy to follow once you spot a path whose two ends share a digit.

How to spot it

Pick a two-candidate cell and one of its digits as Z. Move to its other digit, then to a cell that sees it and holds that digit along with one other, and continue with that other digit. Whenever you arrive at a cell whose remaining digit is Z, you have a chain: look for cells that see both the first and the last cell and hold Z.

Example

XY-Chain
	c1	c2	c3	c4	c5	c6	c7	c8	c9
r1	1	68	2	46	36	9	5	378	478
r2	46	689	3469	456	7	3456	348	1	2
r3	5	7	34	2	8	1	34	9	6
r4	3	4	7	9	5	2	18	6	18
r5	9	5	8	67	1	67	2	4	3
r6	26	26	1	3	4	8	7	5	9
r7	467	3	469	8	69	4567	149	2	1457
r8	8	1	469	4567	2	34567	349	37	457
r9	247	29	5	1	39	347	6	378	478

Bold digits are given, blue digits were placed while solving, small digits are candidates. Highlighted cells form the pattern, crossed-out candidates are removed, and a circled candidate is the digit to place.

Each cell of the chain r1c5 (6=3) - r9c5 (3=9) - r9c2 (9=2) - r6c2 (2=6) has exactly two candidates. Inside a cell, = joins its two digits: if one is false, the other is true. Between cells, - joins the same digit in two cells that see each other: at most one of them is true.
Suppose r1c5 is not 6. Then it holds its other digit, so the next cell cannot hold that digit and must be its own other digit, and so on along the chain, until the last cell r6c2 is 6.
So 6 is in r1c5 or in r6c2.
A cell that sees both r1c5 and r6c2 cannot be 6. Remove 6 from r1c2.

Common mistakes

Using a cell with three candidates. Every cell of an XY-Chain must have exactly two, or the step from one digit to the other fails.
Linking cells that share a digit but do not see each other. Consecutive cells must be in the same row, column or box.
Removing Z from a cell that sees only one end. The eliminated cell must see both the first and the last cell of the chain.

In Bare Sudoku

Needed from level: Master
In the game, the hint says: XY-Chain

Play this puzzle · Play Sudoku · Sudoku solver

Practice puzzles

In these puzzles, XY-Chain is the hardest technique you need. Each link opens the puzzle in the game; the solver link shows every step.

Frequently asked questions

What is an XY-Chain in Sudoku?

A chain of two-candidate cells in which each cell sees the next and shares a digit with it. When both ends can be the same digit Z, one of them is Z, so cells that see both ends lose Z.

How is an XY-Chain related to the Y-Wing?

A [Y-Wing](y-wing) is an XY-Chain of three cells: the pivot and two wings. Longer XY-Chains apply the same reasoning through more cells.

Can the two ends of an XY-Chain see each other?

Yes. Then Z is removed from the cells that see both ends, as usual. When the chain closes into a loop, even more can be removed; Bare Sudoku handles loops as a separate technique.

Does an XY-Chain use only two digits?

No. Each cell has two digits, but the digits can change from cell to cell, as long as each link shares one digit.

Made by Alparslan Selçuk Develioğlu · Updated 2026-10-04