Captured Candidates in Killer Sudoku

Captured candidates let you erase a digit from a whole unit without knowing which cell of a cage will hold it. The trick is to look at every way a cage sum can be made and find a digit that turns up in all of them.

What the pattern is

A cage with several possible combinations still tells you something, as long as those combinations overlap. If every valid split of the sum includes a 9, the cage must contain a 9 somewhere, even though you cannot say where.

Now check the geometry. If all the cage cells that could take that digit lie inside one unit, that unit's copy of the digit is spoken for. Every other cell of the row, column or box loses it as a candidate. The cage has captured the digit.

It is the same shape of deduction as a pointing pair, with a cage playing the part of the box.

When to look for it

• On cages of two to four cells, where the combinations are few enough to list quickly.
• On sums near the top or bottom of their range. Extreme sums have the fewest splits and almost always share a digit.
• On cages that fit inside a single box or a single line, since those are the ones that can capture anything.
• After eliminations elsewhere kill a combination, which can turn an ordinary cage into a capturing one.

How to apply it, step by step

1. Choose an unsolved cage small enough to enumerate by hand.
2. List every combination of different digits that reaches the sum, then drop any that clashes with digits already placed in the units the cage touches.
3. Look for a digit present in every surviving combination. That digit is forced into the cage.
4. Find the cage cells that could still take it.
5. If all of those cells share one row, column or box, remove the digit from every other cell of that unit.
6. Repeat as the grid tightens. Each elimination can shrink another cage's list and expose a new forced digit.

A worked example

Box 6, with a three-cell cage summing to 22 (cage cells marked C):

   .  C  C
   .  .  C
   .  .  .

  Three different digits adding to 22:  5 + 8 + 9  or  6 + 7 + 9.
  Nothing else reaches 22 without repeating a digit.
  Both combinations contain a 9, so the cage holds a 9 somewhere.
  All three cage cells lie inside box 6.
  ->  remove 9 from the other six cells of box 6.

The cage is L-shaped, so it does not fill a row or a column, but it does fit inside one box, and that is enough. Note also that this cage never gets resolved by the sum alone: you never learn which of its cells holds the 9, and you do not need to.

Common mistakes

• Placing the digit. A captured candidate is an elimination, not a placement. You know the cage holds a 9, not which cell does.
• Skipping the clash check. Combinations that need a digit already used in the cage's row, column or box are dead and must be dropped before you compare.
• Assuming any shared unit will do. Every cage cell that could hold the digit must be in the same unit. One cell across the box border and the capture fails.
• Only checking lines. Cages are rarely straight. For an L-shaped or T-shaped cage, the box is usually the capturing unit.

Where you meet it

Captured candidates start to matter on medium Killer Sudoku and stay in constant use above it, especially once innies and outies have trimmed the cage sums for you. They are also the gateway to the harder cage techniques, which all work the same way: enumerate the possibilities, then keep only what they agree on.