X-Cycles in Sudoku: A Practical Guide

Build the cycle around one candidate

An X-Cycle follows only one candidate value, such as the heart in Shapedoku or the digit 7 in classic Sudoku. Start by mapping every remaining position for that candidate. Connect two positions with a strong link when one of them must be true, usually because the candidate appears in exactly two cells of a row, column, or box. Connect positions with a weak link when they see each other and therefore cannot both be true. A valid X-Cycle alternates strong and weak links as it travels through the grid and eventually returns to its starting point. The cells do not need to form a neat geometric shape. The cycle may move from a row into a box, then through a column, and back through another box. What matters is the logic of each connection. Before looking for an elimination, verify every strong link by checking that no third candidate was missed in the unit.

Use an even cycle to remove outside candidates

The most common useful X-Cycle contains an even number of links and alternates perfectly all the way around. In that structure, the candidates on the cycle must take alternating truth states. One set of positions is true and the other set is false, although you may not know which set is which. This produces an elimination when an outside candidate sees two cycle positions joined by a weak relationship from opposite sides of the loop. Whichever truth pattern the cycle takes, one of those two visible positions must be true, so the outside candidate cannot survive. Keep the target separate from the cycle itself and check that it sees both required positions. Seeing only one is not enough. In an Extreme Shapedoku puzzle, accurate Notes make this much easier because the candidate map must be complete before the loop can be trusted.

Recognize discontinuous cycles and forced results

Some X-Cycles close with two links of the same type instead of alternating cleanly. This is called a discontinuity, and it can force a direct result at the meeting candidate. If two strong links meet at one candidate, treating that candidate as false would force both neighboring candidates true, creating a conflict in the shared structure. The meeting candidate must therefore be true. If two weak links meet, making the candidate true would force both neighboring positions false and break the strong implications around the rest of the loop, so the meeting candidate is false. These conclusions are powerful, but they are also easy to misread. Trace the cycle in both directions and label every link before placing or removing anything. After one result, stop and rescan for singles, locked candidates, and pairs. X-Cycles are most useful as a precise key. Once they produce a verified result, return to the simpler patterns created by that change.

1. Choose one shape or digit and map all of its remaining candidates.
2. Mark strong links where the candidate has exactly two positions in a unit.
3. Mark weak links between candidates that share a row, column, or box.
4. Trace a closed path that alternates strong and weak links.
5. Use a continuous even cycle for outside eliminations.
6. Use a strong-strong or weak-weak discontinuity only after verifying every link.