X-Cycles

Hard

Strong and weak links for a single digit form cycles; nice loops enable eliminations or placements.

How It Works

For a single digit, build a chain using "strong links" (a unit where the digit is only in two cells) and "weak links" (cells in the same unit). If the chain forms a continuous nice loop, candidates at the weak link positions can be eliminated. Discontinuous loops reveal contradictions, enabling placements or eliminations.

Example

3
38
13
3
6
27
5
48
9
14
25
49
16
8
59
27
36
14
768
12
24
3
14
49
15
2
15
17
38
45
46
9
57
6
6
34
24
3
19
18
3
15
7
9
17
57
26
25
48
16
13
14
49
6
17
7
28
24
28
15
38
37
37
49
1
25
6
24
24
3
28
29
46
45
56
3
17
8
Key cells of the techniqueCells where elimination is appliedCandidate to be placedEliminated candidate (crossed out)

Digit 3 cycle: R1C3-R1C7=R4C7-R4C3=R1C3. Cells seeing the weak-link ends can have 3 eliminated.

Practice with a Real Puzzle

This 9×9 puzzle is solver-verified to require this technique on its solution path.

Hard24 givensStrict