THE CORE MECHANIC
Each row and column has a target sum shown on its edge. Delete numbers from the grid so the remaining numbers in each row and column add up exactly to that target. A deleted cell contributes 0 to any sum. You cannot move numbers — only keep or delete.
Each cell is independent: a number is either kept or deleted. Your job is to determine, for every cell, which choice is forced by the sum constraints.
FIND FORCED KEEPS AND FORCED DELETES
Two extreme cases give free information:
- Target equals total: if the row target equals the sum of all numbers in the row, you must keep every number. None can be deleted.
- Target is zero: if the row target is 0 (or equivalently, the target equals the sum minus all numbers), you must delete everything in that row.
These are rare at the start but common once other rows/columns are partially resolved. Always recheck after each deletion.
WORK THE MOST CONSTRAINED ROWS AND COLUMNS
For each row and column, calculate: sum of all numbers (S), target (T), and the amount that must be deleted (S − T = D). That "deletion amount" D tells you how much must disappear.
If D equals exactly one number in the row — say D = 7 and there is exactly one 7 in the row — that 7 must be deleted and everything else must be kept. This is one of the most common forced moves in Sumplete.
Tip: also check if the only way to reach D is by keeping exactly one specific subset. For short rows (2–3 cells) this is often obvious. For longer rows, focus on numbers that appear only once in the row.
CROSS-REFERENCE ROWS AND COLUMNS
Sumplete's real depth comes from the interaction between row and column constraints. A cell belongs to both a row and a column — a decision forced by one constraint immediately affects the other.
When you determine a cell must be kept (say, because its row forced it), subtract its value from the column target and the column total. The column now has less to account for. If that column was already near its constraint boundary, the new information may force another cell in that column.
This cross-propagation is the main solving engine. Work the row with the fewest remaining unknowns, apply its forced moves, propagate to its columns, then apply those columns' forced moves, then back to rows. Repeat.
THE SOLVING LOOP
- Compute the deletion amount (sum − target) for every row and column.
- Find any row or column where the deletion amount equals exactly one number or equals zero or equals the total. Apply the forced keep or delete.
- For each kept/deleted cell, update the remaining sum and deletion amount for the intersecting row and column.
- Go back to step 2 with the updated information.
A well-formed Sumplete resolves entirely by this loop without guessing. If you're stuck, look for a row or column where only one combination of keeps/deletes produces the correct deletion amount — this is rarer but solvable by enumeration when the row is short.
THE BEGINNER MISTAKE
Beginners delete numbers that look "large" or "out of place" without checking the deletion amount. Random deletion never produces a unique-solution puzzle — and it makes backtracking painful.
The discipline: never delete (or keep) a cell unless a row or column constraint forces it. Calculate the deletion amount first. If no constraint forces a decision yet, move to a different row or column — the answer will come from a forced move somewhere else, not from a guess here.