Min/Max Boundary Forcing

Every unfilled run has a range of achievable sums given its remaining empty cells. Boundary forcing uses this range to eliminate candidates that would make the remaining sum impossible — without knowing the full combination.

Prerequisites

  • You can read Kakuro clue cells and track which digits are already placed in a run.
  • You are comfortable writing candidates and pruning them after each placement. (Combination Pruning)

The core idea

For a run of N empty cells with a remaining sum of S:

  • Minimum achievable sum: use the N smallest available digits — 1, 2, 3, … (skipping any already used in the run).
  • Maximum achievable sum: use the N largest available digits — 9, 8, 7, … (skipping any already used in the run).

Any candidate digit d in a particular cell is impossible if placing it would make the remaining N−1 cells unable to reach the remaining sum S−d — either too low or too high.

Step-by-step walkthrough

Scenario: a 3-cell across run with clue 14, and the first cell is already filled with 2.

  1. Compute the remaining sum: 14 − 2 = 12. Two empty cells remain, and digits {1, 2} are excluded (2 is placed; 1 would leave nothing for the other cell, but we check formally).
  2. Find available digits: digits {3, 4, 5, 6, 7, 8, 9} are available (not yet used in this run).
  3. Compute the minimum sum for 2 empty cells: 3 + 4 = 7.
  4. Compute the maximum sum for 2 empty cells: 9 + 8 = 17.
  5. Test each candidate: for a candidate digit d in one of the two cells, the other cell must hold 12 − d. Check that (12 − d) is available and in the range [3, 9].
    • d = 3 → other cell = 9. Valid (9 is available and ≠ 3).
    • d = 4 → other cell = 8. Valid.
    • d = 5 → other cell = 7. Valid.
    • d = 6 → other cell = 6. Invalid — repeats 6.
    • d = 7 → other cell = 5. Valid.
    • d = 8 → other cell = 4. Valid.
    • d = 9 → other cell = 3. Valid.
  6. Eliminate 6 from both empty cells. The valid combinations are {3,9}, {4,8}, and {5,7}. Now use crossing runs to narrow down further.

Why it works

Kakuro requires every digit in a run to be distinct. The boundary check is a compact way to ask: "Is there any valid partner digit left for this candidate?" If no valid partner exists, the candidate is ruled out with certainty — no guessing required. This technique is especially powerful in the middle and late stages of a puzzle, when several cells in a long run are already filled and the remaining sum is tightly constrained.

Quick boundary reference

Min and max achievable sums for N empty cells using all available digits 1–9:

Empty cells (N) Min sum Max sum
2 3 17
3 6 24
4 10 30
5 15 35
6 21 39
7 28 42
8 36 44

When some digits are already excluded from a run, the effective min and max shift upward and downward respectively. Recalculate from the remaining available digits.

Common pitfalls

  • Forgetting which digits are excluded: Always subtract the digits already placed in the run before computing bounds.
  • Applying bounds to the full run instead of the remaining cells: The remaining sum and remaining empty cells are what matter — not the original clue.
  • Stopping after one elimination: After removing an impossible candidate, re-check all affected runs. A single pruning often cascades into several more.

Try it now

Start a medium or hard puzzle, enable candidates, and look for any run that already has one or two digits placed. Compute the remaining sum and apply boundary forcing to prune the candidates in the unfilled cells.

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