Sum chain

A sum chain is a killer-Sudoku argument that links multiple cages through their interactions with units and shared cells. Each link in the chain is an arithmetic constraint — usually a 45-rule application or a cage-splitting decomposition — and the chain propagates information from one end to the other through the intersection structure of the cages. The result is an elimination or placement that no single cage's reasoning would have produced alone.

A worked example

Suppose three cages overlap a single box. Cage A has 3 cells in the box and 2 outside, with sum 24. Cage B has 2 cells in the box and 3 outside, with sum 22. Cage C has 4 cells in the box and 0 outside, with sum 20.

The box's digits must sum to 45. Cages A's 3 in-box cells, B's 2 in-box cells, and C's 4 in-box cells together cover all 9 cells of the box (assuming no other cages reach into it), so their combined in-box sum is 45.

Cage C contributes 20. So A's in-box plus B's in-box must equal 25. That's a 5-cell sub-region whose total is fixed; combined with the cage totals (24 and 22), we can derive constraints on each cage's split between its in-box and out-of-box cells. The deduction propagates: A's out-of-box cells must total 24 - (A's in-box), B's out-of-box cells total 22 - (B's in-box), and the values are linked through the in-box constraint.

The chain continues. The in-box totals constrain the out-of-box cells, which constrain the units those out-of-box cells belong to, which apply 45-rule arguments to those units, which constrain other cages, and so on. A long sum chain can run through five or six cages before producing its first elimination.

When you'll see it

Sum chains appear on expert killer puzzles where the cage layout has substantial overlap — multiple cages sharing units, boxes with several partial cages, irregular cage shapes. The signature is a stuck state where simpler killer techniques (45 rule, innies-and-outies, cage splitting) have all fired and the puzzle hasn't fully unblocked. Working out the sum chain explicitly often takes pencil and paper — manual solvers track partial sums in a side calculation — but the eliminations the chain produces are usually substantial.

A note on bookkeeping

Sum chains share the bookkeeping problem of all chain techniques: keeping the partial deductions straight across many steps. Most expert killer solvers develop a notation habit — writing each cage's in-region and out-of-region totals at the cage's edge, updating them as new constraints arrive, and tracing through the implications cell by cell.

The Sudoku Mountain killer solver tracks sum-chain logic implicitly through its cage-aware deduction trace; manual solvers tend to write it out.