X-Cycles are an advanced Sudoku technique that involves building alternating chains of strong and weak links for a single candidate digit, forming a closed loop or cycle across the grid. A strong link exists between two cells in a unit (row, column, or box) when the candidate appears in exactly those two cells — meaning one of them must contain that digit. A weak link connects two cells that can see each other but where other candidates also exist in the unit. By alternating between strong and weak links, you trace a path that eventually loops back to its starting cell.
When an X-Cycle forms a continuous loop with an even number of links, it behaves like a generalized X-Wing. Every cell connected by a strong link into the cycle is guaranteed to contain the digit or not, in an alternating pattern. Any cell outside the cycle that can see both an "on" and "off" cell in the cycle can have the candidate eliminated. This is called a Nice Loop, and it provides powerful elimination capabilities that simpler fish patterns cannot achieve.
X-Cycles also come in discontinuous forms. When a cycle has an odd number of links with a strong-link discontinuity, the candidate at the discontinuity can be confirmed as true. When there is a weak-link discontinuity, the candidate at that point can be eliminated. Understanding X-Cycles requires solid comfort with chain notation, coloring techniques, and logical link analysis. This technique bridges the gap between simple coloring and full-blown Alternating Inference Chains, making it a cornerstone of expert-level Sudoku strategy and pencilmark analysis.
Try It Yourself
Walk through each step of the x-cycles technique on a real puzzle. Follow the instructions and try entering the correct value when prompted.
We are building an X-Cycle on candidate 9. First, identify all cells in the grid where 9 is a candidate. Focus on column 2, where 9 appears in only a few cells, creating strong links.
Step-by-Step Guide
Select a candidate digit and identify all cells where it remains as a candidate.
Map out strong links — pairs of cells in a row, column, or box where the candidate appears exactly twice.
Map out weak links — connections between cells that can see each other but share the unit with additional candidate cells.
Begin building a chain by alternating strong and weak links from a starting cell.
Continue extending the chain until it loops back to the starting cell, forming a cycle.
If the loop is continuous (even number of links), eliminate the candidate from any cell that sees both an ON and OFF node in the cycle.
If there is a strong-link discontinuity (odd loop), the candidate at the break point is confirmed as the solution.
If there is a weak-link discontinuity (odd loop), eliminate the candidate from the break-point cell.
Picture a circle of dominoes standing on end -- push one and they alternate between falling left and right all the way around. If the circle comes back perfectly, you know exactly which dominoes fall which way, and anything caught in between gets knocked out.
X-Cycles are sound because strong links enforce that exactly one of two cells holds the digit, while weak links enforce that at most one does. In a continuous even-length cycle, these constraints assign every node a definite ON or OFF state that propagates consistently around the loop. Any cell outside the cycle that sees both an ON node and an OFF node is constrained to not hold the digit, because it would conflict with whichever node is ON. In discontinuous odd-length cycles, the contradiction at the break point forces either a placement or an elimination at that node.
When to use: Use X-Cycles when simple coloring has not resolved a digit and you notice strong and weak links for that digit forming a loop-like path across multiple rows, columns, and boxes.
Common Mistakes to Avoid
Confusing strong and weak links, leading to an invalid chain that produces incorrect eliminations.
A strong link means the candidate appears in exactly two cells in that unit. A weak link means there are additional candidates. Double-check each link type before completing the cycle.
Attempting to close the cycle with two consecutive weak links, which breaks the alternating pattern.
In a continuous nice loop, strong and weak links must strictly alternate. Two weak links in a row invalidate the chain. Look for a different path.
Applying continuous loop elimination rules to a discontinuous cycle (or vice versa).
Count the links. An even-length loop is continuous -- eliminate from cells seeing both ON and OFF nodes. An odd-length loop is discontinuous -- the rule depends on whether the discontinuity is at a strong or weak link.
More Examples
See x-cycles applied in different puzzle configurations to strengthen your pattern recognition.
X-Cycle alternating chain on a single digit
Highlighted cells show the x-cycles pattern
Practice Puzzles
Apply the x-cycles technique on these mini challenges. Tap a highlighted cell and enter the correct digit.
Quick Reference
- Pattern:
- A single-candidate chain of alternating strong and weak links forming a loop or path
- Action:
- Eliminate based on loop type: continuous loop eliminates from cells seeing both colors
- Look for:
- Strong and weak links for one candidate forming cycles or discontinuous loops