Hard Killer Sudoku Combinations
Hard Killer Sudoku combinations are what you fall back on when a cage sum has too many options to look up in a table. Instead of reading off the answer, you generate every digit set the sum allows, test each one against the cells, and keep only the survivors.
What the pattern is
Extreme sums are easy. A three-cell cage of 6 can only be {1, 2, 3}, and a two-cell cage of 17 can only be {8, 9}. Mid-range sums are the problem: three cells adding to 17 have seven possibilities, and four cells adding to 22 have eleven.
The technique closes that gap. A combination is only real if each of its digits can actually be placed in a distinct cell of the cage, given the current candidates. Test the placements, throw away the combinations that will not fit, then collect the digits that survive cell by cell. Anything left over is an elimination.
When to look for it
- A medium-sized cage, three to five cells, whose sum sits in the middle of its range.
- A cage that has lost candidates to other techniques, so the raw combination list is now far too generous.
- A cage sharing a row, column or box with another cage whose digits are already pinned down.
- A cage with one digit already placed, which usually cuts the list in half.
How to apply it, step by step
- Write out every set of distinct digits that reaches the sum in the right number of cells.
- Cross off any set containing a digit that no cell of the cage can hold.
- For each remaining set, try to assign its digits to the cells. If no assignment fits the candidates, the set is dead.
- Gather, for each cell, the digits it took in at least one surviving assignment. Every other candidate goes.
- Intersect with the neighbours. If another cage in the same unit is locked to a fixed pair or triple, those digits are unavailable to your cage's cells in that unit, and more combinations fall.
A worked example
Cage 22 covers E1, E2, E3 and F1, all in box 4. Four cells reaching 22 have eleven possible sets, but E2 is already 1, so E1, E3 and F1 must add up to 21. Follow how each set is tested.
Three digits reach 21 in three ways: 4+8+9, 5+7+9 and 6+7+8. Row E already has a 7 (E6) and a 6 (E8), so E1 and E3 can take neither. That leaves F1 to carry both 6 and 7 in 6+7+8, which is impossible, so that set dies. In 5+7+9, E3 can only be 9 and E1 only 5, so F1 is 7. In 4+8+9, F1 is 4, 8 or 9.
Neither survivor contains a 3, so 3 leaves E1, E3 and F1. F1 can still be 4, 7, 8 or 9, so 5 and 6 leave it too. The placed 1 shrank eleven sets to three, and row E finished the job.
A neighbouring cage can cut the list further. A cage locked to a pair or triple in the same unit removes those digits from your cage's cells in that unit, as a separate move after this one.
Easy versus hard combinations
| Easy combinations | Hard combinations | |
|---|---|---|
| Sum | Extreme for the cage size | Mid-range |
| Options | Exactly one | Three to a dozen |
| Method | Table lookup | Enumerate, then test placements |
| Inputs | Sum and size only | Sum, size, candidates, neighbouring cages |
| Payoff | Instant, once per cage | Repeats after every placement |
Common mistakes
- Allowing repeats. Digits inside a cage are always distinct, so
{5, 5, 7}is never a combination for 17. - Checking the digit set but not the placement. A set can contain only legal digits and still fail because two of them need the same cell.
- Doing it once. Every placement in or near the cage shortens the list. Re-run it whenever the grid moves.
- Starting too big. Cages of six or more cells have huge lists and almost never repay the effort. Work their complement instead.
Where you meet it
This is tough-tier work. Easy and medium Killer puzzles are built around lookup-only cages, but from hard Killer Sudoku onwards the setters choose mid-range sums precisely so that a table cannot answer them. Getting quick at pruning combinations is the single biggest speed gain available in Killer.