Cage and Unit Overlap

Cage and unit overlap is what happens when a cage and a row, column or box share cells: the cage's arithmetic tells you a digit must live in that shared area, so the digit leaves everywhere else in the unit. It is the Killer version of locked candidates, with a sum doing the work of a box.

What the pattern is

Every cage sum can be reached only by certain digit sets. If a digit appears in every one of the sets still available, the cage must contain it somewhere. That is a must-have digit.

Now look at where it can sit. If all of its possible cells lie inside one unit, then that unit's copy of the digit is spoken for, and the digit is a dead candidate in the rest of the unit — even in cells the cage never touches.

The logic also runs backwards. If a digit's only remaining places in a unit all fall inside one cage, that cage must contain it, and every combination that lacks the digit can be thrown away.

When to look for it

• A cage of two, three or four cells whose combination list has shrunk to two or three options.
• A cage that spills across a box edge but keeps most of its cells in one line, or vice versa.
• A cage sum near the top or bottom of its range, since extreme sums force high or low digits into every combination.
• A unit where one digit is already down to two or three places and they all belong to the same cage.

How to apply it, step by step

1. List the combinations the cage can still make, given its sum, its size and its current candidates.
2. Find a digit present in every surviving combination.
3. Mark the cells of the cage that can still hold that digit.
4. Check whether those cells share a unit. All in one box, all in one row, or all in one column will do.
5. Remove the digit from every other cell of that unit.
6. Then run it the other way: for each digit in the unit, see whether its remaining places are all inside a single cage, and prune that cage's combinations accordingly.

A worked example

cage 22 in 3 cells:   r4c3 | r4c4  r5c4
                      (r4c3 in box 4, the other two in box 5)

  22 from three different digits:  5+8+9   6+7+9
  every combination contains a 9

  r4c3 candidates {5,6,7,8}   -- box 4 already has its 9 placed

  so the cage's 9 must sit at r4c4 or r5c4
  both cells lie in box 5 and both lie in column 4

  -> 9 removed from the rest of box 5
  -> 9 removed from the rest of column 4

Three distinct digits reaching 22 leave only two options, {5, 8, 9} and {6, 7, 9}, so a 9 is guaranteed somewhere in the cage. The cage's box 4 cell cannot hold it, which pins the 9 to the two cells sitting in box 5. Those two cells also share column 4, so one deduction clears the digit from two whole units at once.

Which direction are you using?

Common mistakes

• Using a digit that is only common to some combinations. It must appear in every surviving one, not most of them.
• Forgetting current candidates. A combination that the sum allows may already be impossible because a cell lost the digit it needed.
• Sloppy geometry. Two cells in the same box and a third in another box means the digit is not locked. Every possible cell must share the unit.
• Eliminating inside the cage. The digit is removed from the rest of the unit, never from the cage cells that carry it.

Where you meet it

The simple form, where a small cage sits entirely inside one box, turns up in easy Killer grids. The overlap version, where you must notice that only part of the cage can carry the must-have digit, belongs to the tough tier and starts to matter at hard Killer Sudoku. Once cages routinely cross box edges, this is the technique that keeps the grid from stalling.