Cage Splitting

Cage Splitting is a Killer Sudoku technique that extends the rule of 45 to cages that straddle a unit boundary. Where innies and outies pin down a single stray cell, cage splitting deals with a whole cage that lies partly inside a unit and partly outside — breaking it into known partial sums.

The idea

Every row, column and box sums to 45. Suppose a unit is filled by several complete cages plus one cage that crosses its boundary — some of its cells inside the unit, some outside. Add up the complete cages; subtract from 45; and you have the exact sum of the crossing cage's cells that lie inside the unit. You've split the cage's total into an inside part and an outside part, each now a small, known sum you can attack with cage combinations.

A worked example

A column contains three full cages summing to 31, plus two cells of a fourth cage whose total is 20 (its other two cells sit in the next column). The column must total 45, so the two inside cells sum to 45 − 31 = 14. That immediately splits the cage: 14 inside, and 20 − 14 = 6 across the two outside cells. Now apply combinations: two cells summing to 6 can only be {1, 5} or {2, 4}, and two summing to 14 only {5, 9} or {6, 8} — a huge narrowing from a cage you couldn't otherwise touch.

Step by step

1. Pick a unit with clean cage boundaries and one crossing cage.
2. Add the complete cages inside the unit.
3. Subtract from 45 to get the inside part of the crossing cage.
4. Split the cage total — the rest is the outside part — and apply combinations to each.

Common mistakes

• Miscounting inside cells. Be exact about how many cells of the crossing cage lie in the unit.
• Forgetting no-repeats. A partial sum still can't use a digit twice within its cage — that often does the real work.

Which Sudoku types is this best for?

The same logic applies across variants, but it pays off more in some than others.