Catalog study 02 · deterministic solving paths
Where do singles stop working?
In our Hard puzzles, the first productive elimination occurs with a median of 39 cells still empty. Explore that transition from filling a digit to removing a possibility.
What we recorded
For each original puzzle, we followed the existing order: naked single, hidden single, locked candidates, naked pair, hidden pair. After every productive step, the search restarts at naked single. We recorded the number of empty cells immediately before the first candidate-elimination step, after all available singles had been exhausted.
| Grade | Paths with elimination | Median empty | Range |
|---|---|---|---|
| easy | 0 / 128 | Not applicable | No elimination step |
| medium | 0 / 128 | Not applicable | No elimination step |
| hard | 128 / 128 | 39 | 20–56 |
| expert | 128 / 128 | 43 | 25–57 |
The first Hard puzzle, at its first bottleneck
In v1-hard-0001, the path makes 11 single placements before step 12. The next deduction is pointing: in the top-left box, digit 9 can occupy only r2c2 or r2c3. Both lie on row 2, so 9 can be removed from r2c9. That is useful progress even though it does not itself fill a cell.
v1-hard-0001 · position before step 12
Read the position as a cell table
| Cell | Value / candidates |
|---|---|
| r1c1 | Placed: 2 |
| r1c2 | Placed: 7 |
| r1c3 | Placed: 3 |
| r1c4 | Placed: 1 |
| r1c5 | Placed: 8 |
| r1c6 | Placed: 5 |
| r1c7 | Placed: 4 |
| r1c8 | {6, 9} |
| r1c9 | {6, 9} |
| r2c1 | Placed: 5 |
| r2c2 | {1, 8, 9} |
| r2c3 | {4, 6, 8, 9} |
| r2c4 | {4, 6, 7} |
| r2c5 | {4, 6, 7} |
| r2c6 | {4, 6, 7} |
| r2c7 | Placed: 3 |
| r2c8 | Placed: 2 |
| r2c9 | {1, 6, 8, 9} |
| r3c1 | {1, 4, 8} |
| r3c2 | {1, 8} |
| r3c3 | {4, 6, 8} |
| r3c4 | Placed: 3 |
| r3c5 | Placed: 9 |
| r3c6 | Placed: 2 |
| r3c7 | Placed: 5 |
| r3c8 | {1, 6, 8} |
| r3c9 | Placed: 7 |
| r4c1 | Placed: 3 |
| r4c2 | {2, 5, 8, 9} |
| r4c3 | {2, 4, 5, 8, 9} |
| r4c4 | {4, 5, 8, 9} |
| r4c5 | {2, 4} |
| r4c6 | Placed: 1 |
| r4c7 | Placed: 6 |
| r4c8 | Placed: 7 |
| r4c9 | {2, 4} |
| r5c1 | Placed: 7 |
| r5c2 | {2, 5, 8} |
| r5c3 | {2, 4, 5, 8} |
| r5c4 | {4, 5, 6, 8} |
| r5c5 | {2, 3, 4, 6} |
| r5c6 | {4, 6, 8} |
| r5c7 | Placed: 9 |
| r5c8 | {1, 4} |
| r5c9 | {1, 2, 3, 4} |
| r6c1 | {4, 9} |
| r6c2 | Placed: 6 |
| r6c3 | Placed: 1 |
| r6c4 | {4, 7, 9} |
| r6c5 | {2, 3, 4, 7} |
| r6c6 | {4, 7, 9} |
| r6c7 | Placed: 8 |
| r6c8 | Placed: 5 |
| r6c9 | {2, 3, 4} |
| r7c1 | Placed: 6 |
| r7c2 | {1, 2, 5, 8, 9} |
| r7c3 | {2, 5, 8, 9} |
| r7c4 | {4, 7, 8, 9} |
| r7c5 | {1, 4, 7} |
| r7c6 | {4, 7, 8, 9} |
| r7c7 | {2, 7} |
| r7c8 | Placed: 3 |
| r7c9 | {2, 4, 8, 9} |
| r8c1 | {1, 8, 9} |
| r8c2 | Placed: 4 |
| r8c3 | {2, 8, 9} |
| r8c4 | {6, 7, 8, 9} |
| r8c5 | {1, 6, 7} |
| r8c6 | Placed: 3 |
| r8c7 | {2, 7} |
| r8c8 | {6, 8, 9} |
| r8c9 | Placed: 5 |
| r9c1 | {8, 9} |
| r9c2 | Placed: 3 |
| r9c3 | Placed: 7 |
| r9c4 | Placed: 2 |
| r9c5 | Placed: 5 |
| r9c6 | {4, 6, 8, 9} |
| r9c7 | Placed: 1 |
| r9c8 | {4, 6, 8, 9} |
| r9c9 | {4, 6, 8, 9} |
Look first. Reveal one layer at a time.
Find the supporting cells, explain the restriction, then inspect the candidates it removes. The original puzzle link starts from its givens; this panel preserves the intermediate logical state.
Inspect r2c2 and r2c3. Follow these candidates: 9.
In box 1, digit 9 is confined to r2c2 and r2c3. Those cells also belong to row 2. Wherever this digit is placed in the source unit, it occupies that intersection, so it cannot appear in the rest of the target unit. This is Pointing.
The productive elimination
- Remove 9 from r2c9.
After singles
When singles run out, look for a digit confined to the intersection of a box and a line. Remove the candidates that this pattern rules out, then check for new singles. Several eliminations may be needed before another cell can be filled.