Incomplete sums
Use the known total of a complete 45-, 90- or 135-region to constrain several boundary cells.
What it is
An incomplete sum uses a complete region—one unit (45), two units (90), or three units (135)—whose boundary is crossed by several cages. Subtract the sums of cages wholly inside the region. The remainder is the combined total of the boundary cells belonging to the crossing cages.
How to spot it
Choose a complete row, column, box, or a combination of them, with several cages crossing its edge. Add the fully contained cages and subtract their total from 45, 90 or 135. Then inspect the boundary cells and their cages.
Worked example
Step through the example. First the position, then the cage the argument rests on, then the candidates, and finally what follows from it.
Step 1 of 4: The starting position
- Pattern cells
- 1/1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9
- Digits
- 1, 2, 3, 4, 7, 8, 9
- Follows from it
- 1/4, 1/5, 1/6
Press a marked square to point it out on the board above.
Why it matters
A candidate can remain only if the boundary cells can still make the remainder, using valid combinations for their cages and respecting row, column and box restrictions. The total alone is not enough; all three constraints must agree.
Practise now
The practice puzzle is picked so that this technique is what gets the solve moving again.
Frequently asked questions
When is an incomplete sum useful?
It is useful when several cells lie on the region boundary. With only one such cell, the simpler innie or outie calculation is usually enough.
Can the remaining sum determine individual digits?
Sometimes, but not automatically. A digit is removed only when the remaining total, the crossing-cage combinations and the normal Sudoku rules rule it out.
Put it to use
Where you will meet it
Expert grids are the only place this pattern regularly decides the solve. Bring your notes.
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