THE ONE CONCEPT: DELETION AMOUNT
For every row and column, compute:
D = (sum of all numbers in the row) − (target)
D is how much must be deleted from that row. If D = 0, keep everything. If D equals the total, delete everything. Most rows have D somewhere in between — and that value is your primary tool.
Do this calculation for every row and every column before placing anything. It takes 30 seconds at the start and pays off immediately.
TECHNIQUE 1 — SINGLE-MATCH DELETION
If D equals exactly one number in the row, that number must be deleted and all other numbers in that row must be kept.
Row: 3 · 7 · 4 · 9 · 2 Total = 25 Target = 18 D = 7
Only one 7 exists in this row → delete the 7, keep 3, 4, 9, 2. ✓ 3 + 4 + 9 + 2 = 18.
This is the most common forced move in Sumplete. Scan every row and column for single-match deletions before attempting anything harder.
TECHNIQUE 2 — ZERO AND FULL DELETION
- D = 0: keep every number in this row or column. None can be deleted.
- D = total: delete every number in this row or column. The target is 0.
These are free placements — act on them immediately and update the deletion amounts for intersecting columns or rows.
TECHNIQUE 3 — PROPAGATE AFTER EACH DECISION
Every cell belongs to both a row and a column. When you decide a cell is kept, subtract its value from the column's remaining deletion amount. When you decide a cell is deleted, subtract its value from the column's total but not from D — D stays the same until the kept cells sum correctly.
More precisely: once a cell is determined (kept or deleted), update the deletion amount for the intersecting row and column by adjusting the remaining unknowns. A row that had D = 10 with 4 unknown cells may reduce to D = 10 − 3 = 7 with 3 unknown cells after a 3-valued cell in that row is confirmed as kept.
This cross-propagation is where most of the puzzle resolves. A forced delete in row 2 updates column 4, which now has a single-match deletion, which updates row 6, and so on.
TECHNIQUE 4 — SUBSET CHECK FOR STUBBORN ROWS
When no single-match or zero/full deletion is available, look for rows where only one combination of cells produces exactly D.
For short rows (2–4 cells), enumerate directly: which subsets sum to D? If only one subset works, every cell in that subset is deleted and every other cell in the row is kept.
For longer rows this is harder, but focus on rows where D is very small (close to 0) or very large (close to total) — these have fewer valid subsets.
THE SOLVING ORDER
- Calculate D for every row and column.
- Apply all D = 0 (keep all) and D = total (delete all) rows and columns.
- Scan for single-match deletions (D equals exactly one number). Apply them.
- For each newly determined cell, update D for the intersecting row or column.
- Go back to step 2 — updated D values often create new single-matches.
- If stuck: enumerate short rows with few valid deletion subsets.
Steps 1–5 alone solve most beginner and intermediate Sumplete grids. Step 6 is rarely needed before propagation has thinned the grid significantly.
THE BEGINNER TRAP
Deleting numbers that look large or suspicious without checking D. This is pure guessing — and every incorrect deletion forces you to undo multiple propagated decisions. The puzzle has a unique solution that always follows from the deletion amounts alone. No cell should ever be deleted without a forcing reason.
If you feel the urge to guess, compute D for every row and column first. You'll almost always find a forced move you missed.