X-Cycles is a chaining strategy where you build a chain of alternating strong and weak links between cells that have the same candidate.

A **strong link** is one between two peer cells with the same candidate, where there are the only 2 candidate cells in that group for this digit.

A **weak link** is similar but there are more than 2 candidate cells in the group for the digit.

**Nice Loop** is a loop which starts with a strong link and ends with a weak link OR starts with a weak link and ends with a strong link. Cells that see both ends of a weak link in the loop, cannot be the solution and can be removed as candidates.

**Weak Loop** is a loop which starts and ends with Weak links. The starting cell where the weak links meet cannot be the solution and can be removed as a candidate.

**Strong Loop** is a loop which starts and ends with Strong links. The starting cell where the Strong links meet is the solution.

Nice Loop logic is that one of the two ends of a weak link is must be the solution, as they are both part of a strong link.

Weak Loop logic is that similarly to above, it sees two cells connected to strong links, one of which must be the solution.

Strong Loop logic is that the cell must be the solution as the other two ends of the strong loops cannot be both the solution.

6-Singles Chain: Off-chain cell [5,3] can see both cell [1,3] and [5,9] which are on the chain and of different colors.

That means that candidates 6 can be removed from cell [5,3].

8-Singles Chain: Cells [4,2] and [4,6] are both blue.

That means that the pink cells are the solutions in the chain. Cells [3,2] and [5,5] take on the value of 8.

5-Singles Chain: Cells [2,2] and [2,4] are both pink.

That means that the blue cells are the solutions in the chain. Cells [4,2] and [6,5] take on the value of 5.

This algorithm is currently not easy to do without paper and pencil. You can take a picture of the puzzle and print it, then follow the instructions below.

- Fill in all the pencil marks in the puzzle.
- For every available digit 1 to 9
- Start with a cell which contains the digit candidate.
- In the peers of the current cell, look for another cell with the same digit candidate.
- If the two cell are the only cells which contain the candidate in the group they share, than this is a strong link.
- If there are other cells with the same candidate in the group, then this is a weak link.
- Take the new cell and repeat the process, but you must alternate the links betweeen week and strong.
- The cycle is finished when you end up back at the starting cell.
- Based on the first and last link types, you perform the actions as outlined in the Description section above.

If you tap on the following links on an iOS device which has the Sudoku Tutor app installed, it will launch the app and open the practice puzzle. Tap hint once the puzzle is open to see the strategy in action.

Back to **Sudoku Solvers**