Cage Compare in Killer Sudoku

Cage compare in Killer Sudoku puts two cages side by side and uses the difference between their sums to tie their outside cells together. Neither cage says much alone; the relation between them says a great deal.

What the pattern is

When two cages straddle the same unit, the 45 rule ties them into one equation. Every row, column and box totals 45, so once you have added the cages that lie wholly inside a box, the rest of the box belongs to the two straddlers. Substitute each cage's printed sum into that equation and the inside cells cancel out. What remains is a simple relation between the cells sticking out — usually something as tidy as "these two cells add to 10".

That relation is a two-cell constraint on cells that share no row, column or box, which is not something the grid could have told you any other way. Combine it with each cell's candidates and both lists shrink.

When to look for it

• A box, row or column with exactly two cages crossing its boundary and whole cages covering the rest.
• Two cages that share cells, or whose combined cells almost fill a unit.
• Two cages sitting in the same unit where one is already locked to two or three combinations, so you can test whether each one leaves the other anything at all.
• A stalled grid where innies, outies and cage splitting have all failed because there are two straddlers rather than one.

How to apply it, step by step

1. Add the cages lying entirely inside the unit, plus any placed digits they do not cover.
2. Subtract from 45. The remainder is the total of the leftover cells, all of which belong to the two straddling cages.
3. Replace each straddler's inside cells with "cage sum minus its outside cells".
4. Simplify. The inside terms cancel and you are left with an equation in the outside cells only.
5. Test every pair of candidates against that equation and delete the ones that never fit. Remember that a digit repeated inside one cage is illegal, even when the two cells do not see each other.

A worked example

Box 7 (r7c1..r9c3) totals 45

  cages completely inside box 7 ............ 27   (6 cells)
  leftover inside cells .................... r7c1  r7c2  r9c3  = 18

  cage A  15 = r7c1 + r7c2 + r6c2      (r6c2 sits outside, in box 4)
  cage B  13 = r9c3 + r9c4             (r9c4 sits outside, in box 8)

  (15 - r6c2) + r9c3 = 18       ->   r9c3 = r6c2 + 3
  r9c3 = 13 - r9c4              ->   r6c2 + r9c4 = 10

  r6c2 {2,4,6}      r9c4 {5,6,7,8}
  pairs making 10:  2+8 and 4+6      (6+4 fails: r9c4 has no 4)

  -> 6 removed from r6c2;  5 and 7 removed from r9c4

The two cells are in different boxes, different rows and different columns, so nothing but the arithmetic connects them. Yet once the box's inside cells cancel, r6c2 and r9c4 must add to exactly 10, and only two of the twelve candidate pairings survive. As a bonus, the surviving cases give r9c3 as 5 or 7 and confirm that cage B never repeats a digit.

Cage compare versus cage splitting

Common mistakes

• A third straddler. With three cages crossing the boundary the equation has too many unknowns to simplify usefully.
• Dropping placed digits. Solved cells inside the unit that no complete cage covers must be added to the inner total.
• Ignoring cage distinctness. When you test a pair, check that the implied inside values do not repeat a digit already used in the same cage.
• Assuming the outside cells must differ. They usually do not see each other, so an equal pair is legal unless a shared unit or cage forbids it.

Where you meet it

Cage compare is the heaviest of the tough-tier Killer techniques, because it asks you to test one cage's options against another's. It becomes necessary around hard Killer Sudoku and stays useful well beyond, especially in grids where every box has two cages leaking over its edge. Reach for it after the simpler 45-rule methods have run dry.