Grouped X-Cycles

Very Hard

X-Cycles where nodes can represent groups of cells in a box-row or box-column intersection.

How It Works

In standard X-Cycles, each node is a single cell. In Grouped X-Cycles, a node can represent all cells in a box-row or box-column intersection that contain the digit. This enables finding longer, more complex chains that single-cell nodes would miss.

Example

9
24
3
18
6
17
5
28
47
7
15
8
36
24
19
36
14
2
14
6
25
49
37
28
19
34
57
3
27
14
57
26
9
17
25
8
25
48
9
13
26
14
38
7
35
18
34
6
28
49
15
24
19
37
5
19
17
26
3
24
24
6
19
6
56
25
47
6
38
14
35
19
24
37
47
19
15
68
25
34
Key cells of the techniqueCells where elimination is appliedCandidate to be placedEliminated candidate (crossed out)

Extended chains are found by treating box-row or box-column intersections as single grouped nodes in the cycle.

Practice with a Real Puzzle

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

Hard25 givens