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 20 have more.

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

1. Write out every set of distinct digits that reaches the sum in the right number of cells.
2. Cross off any set containing a digit that no cell of the cage can hold.
3. For each remaining set, try to assign its digits to the cells. If no assignment fits the candidates, the set is dead.
4. Gather, for each cell, the digits it took in at least one surviving assignment. Every other candidate goes.
5. 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 17 in 3 cells      r7c7    r7c8              r8c8
current candidates      {4,5}   {1,2,4,5,6,7}     {3,6,8,9}

17 from three digits:  179  269  359  278  368  458  467

  179 269 278 368  ->  nothing for r7c7, which needs a 4 or 5   drop
  359              ->  3 and 9 cannot go in r7c8                drop
  458              ->  4-5-8 and 5-4-8 both fit                 keep
  467              ->  4-7-6 fits (r8c8 has no 7)               keep

surviving digits per cell
  r7c7 -> 4,5      r7c8 -> 4,5,7      r8c8 -> 6,8

removed: 1, 2 and 6 from r7c8;  3 and 9 from r8c8

Seven combinations became two, and two cells lost candidates. Note that 6 leaves r7c8 even though 6 appears in a surviving combination — the only way to use 467 puts the 6 in r8c8, because r8c8 has no 7 to take instead.

Now add a neighbour. If another cage in box 9 is a two-cell 17, it is locked to {8, 9}, so no other cell in that box may hold an 8. That kills 458, leaving 467 alone: r7c7 is 4, r7c8 is 7 and r8c8 is 6. One cage plus one neighbour solved three cells.

Easy versus hard combinations

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.