Advanced techniques
Strategy & Hints
1. Cage combo cheat sheet
Memorise these — they're the most useful combinations because each has only one possible set of digits, so the cage is fully determined the moment you see the sum and cell count.
| Cells | Sum | Only combo |
|---|---|---|
| 2 | 3 | 1+2 |
| 2 | 4 | 1+3 |
| 2 | 16 | 7+9 |
| 2 | 17 | 8+9 |
| 3 | 6 | 1+2+3 |
| 3 | 7 | 1+2+4 |
| 3 | 23 | 6+8+9 |
| 3 | 24 | 7+8+9 |
| 4 | 10 | 1+2+3+4 |
| 4 | 11 | 1+2+3+5 |
| 4 | 29 | 5+7+8+9 |
| 4 | 30 | 6+7+8+9 |
| 5 | 15 | 1+2+3+4+5 |
| 5 | 35 | 5+6+7+8+9 |
2. The 45 rule — extended
Beyond single rows: the digits in any complete 3-row band sum to 3 × 45 = 135. Same for a 3-column stack. Combine multiple bands or stacks and the arithmetic compounds quickly. This is how experts crack expert-level puzzles in their head.
3. Naked pairs & triples
If two cells in a unit (row, column, box, or cage) have the same two candidates and only those, those two digits are locked in. Remove them as candidates from every other cell in the unit.
Triples are the same idea with three cells sharing three candidates. Slightly harder to spot but devastating when you do — eliminates three candidates from up to six other cells at once.
4. Hidden pairs
Reverse of naked pairs. If two digits can only appear in two cells of a unit (even if those cells have other candidates), the two digits are locked into those two cells. Remove all other candidates from those two cells.
5. X-wing
An advanced elimination pattern. Find a digit that can only appear in two cells of two different rows — and those four cells form a rectangle. Then that digit must occupy two opposite corners of the rectangle, so you can eliminate it from the rows and columns those corners share.
6. Cage-row intersection
A killer-specific trick: if a cage's digits are partially constrained by a row's "missing digits" calculation (from the 45 rule), the intersection gives you exact placements. Underused, very powerful for hard+ puzzles.