Catalog study 02 · 512 puzzles
Where do singles stop working?
In this catalog's fixed solving path, Hard puzzles reach the first candidate elimination with a median of 39 empty cells.
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
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
View the 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} |
Work through the example
Inspect the supporting cells and candidate removals. This panel keeps the intermediate state; the original-puzzle link starts from the givens.
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.
Candidates to remove
- 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.