Little Sudoku

Techniques › XY-Chain

XY-Chain

A run of two-candidate cells that returns to the same digit at both ends.

A chain of bivalue cells, each seeing the next, linked so that consecutive cells share a digit. If the chain starts with a cell holding {Z,A} and ends with one holding {Y,Z}, then one of those two end cells must be Z — follow the implications either way and you always arrive there.

So any cell seeing both ends cannot be Z.

An XY-Chain is the general form of the XY-Wing, which is just the three-cell case. This is where our Master band lives, and it is the hardest technique this engine will ever require of you — deliberately. Beyond it lie dynamic forcing nets, which are objects of study rather than puzzles anyone enjoys on a phone.

A worked example

Taken from a real puzzle in our bank, at the point where this was the cheapest move available. Pattern candidates are marked in amber; anything the step eliminates is struck through in red.

7 2 9 5 3 1 6 4 8
3 6 4 2 8 7 9 1 5
5 1 8 9 4 6 3 2 7
89 3 5 1 79 28 27 6 4
189 7 6 4 59 28 12 58 3
18 4 2 6 57 3 17 58 9
4 8 1 7 6 9 5 3 2
6 5 7 3 2 4 8 9 1
2 9 3 8 1 5 4 7 6
  1. NudgeA chain of two-candidate cells links back to digit 8 at both ends.
  2. WhereXY-Chain: r4c1, r4c5, r4c7, r4c6, r5c6, r5c7, r6c7, r6c5, r6c8
  3. Do itRemove r6c1 ≠ 8

How to spot it

List the bivalue cells. Start at one, step to a bivalue neighbour sharing a digit, carry the other digit forward, and keep going. When the carried digit matches the one you began with, check what both ends can see.

Practise the XY-Chain →

Where you will meet it

XY-Chain sits at tier 6 of our ladder, which means it first becomes necessary at Master. Puzzles below that band never require it, though it may still appear as a shortcut.

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