Kakuro Solving Strategies: From Clues to Deductions

Track both combinations and cell candidates

A run-level combination answers which digits can make the total. Cell candidates answer where those digits can go. Keeping these two ideas separate prevents a common mistake: assigning a digit to a particular cell merely because the digit appears in every valid combination.

For a three-cell run totaling 7, the only distinct-digit combination is {1,2,4}. Every cell is initially allowed any of those three digits. If crossing clues restrict the cells to A = {1,2}, B = {1,2}, and C = {1,2,4}, then C must be 4 because neither A nor B can take it.

After placing 4, A and B remain unresolved. Their combined total is 3, but both orders, 1 then 2 and 2 then 1, remain possible. Record the deduction you have earned and continue through C's crossing run, which may provide the information needed to distinguish A from B.

Reduce the clue before generating combinations

Subtract confirmed digits first, and remove those used digits from the available pool. Suppose a five-cell run totals 25 and already contains 2 and 8. Three blanks remain with a total of 15. Their allowed digits are 1, 3, 4, 5, 6, 7, and 9.

The legal remaining combinations are {1,5,9}, {1,6,8}, {2,4,9}, {2,5,8}, {2,6,7}, {3,4,8}, {3,5,7}, and {4,5,6} before accounting for used digits. Remove every set containing 2 or 8. You are left with {1,5,9}, {3,5,7}, and {4,5,6}; therefore a 5 is required somewhere among the three blanks.

This example also shows why using a sum table mechanically is risky. A table for three cells totaling 15 describes a fresh run. Your partially solved run has additional exclusions. Apply those exclusions before transferring the table's digits into cell notes.

Intersect crossing candidates precisely

Suppose one cell belongs to an across run whose surviving combinations permit {1,4,6,9} there. Its down run permits {2,4,7}. The intersection is {4}, so the cell is 4. The other digits are excluded because each fails at least one of the two constraints.

If the intersection is {4,6}, leave both candidates. More importantly, revisit the whole across run after this reduction. Any combination that requires this particular cell to take 1 or 9 may now be impossible, even if those digits remain legal in other positions.

Treat an empty intersection as an error signal, not as permission to invent another candidate. Recheck clue direction, run length, arithmetic, and previous entries. In a consistent partial solution, every unsolved cell needs at least one way to satisfy both of its runs.

Use distinct-digit bounds

Minimum and maximum totals eliminate options before you enumerate every combination. For three open cells with no other restrictions, the smallest total is 1 + 2 + 3 = 6 and the largest is 7 + 8 + 9 = 24. Restrictions from used digits make the useful bounds tighter.

Imagine three blanks whose available digit pool is {2,4,6,7,9} and whose remaining sum is 15. Their minimum possible total is 2 + 4 + 6 = 12. Their maximum is 6 + 7 + 9 = 22. Both bounds permit 15, but a tentative 9 would leave two cells totaling 6, forcing 2 and 4.

If one of those two cells cannot accept 2 or 4 because of its crossing clue, the tentative 9 has no support and can be removed. This is a local impossibility check: you are testing a candidate against a complete run constraint, rather than committing it as an entry.

Exploit restricted positions within a run

If two cells in the same run are restricted to {2,7}, they occupy those two digits between them. No other cell in that run can use 2 or 7. The cells need not be adjacent, but both must belong to the same uninterrupted run, where repeated digits are forbidden.

Apply the sum constraint after this exclusion. In a four-cell run totaling 20, a reserved pair {2,7} leaves the other two cells totaling 11. Excluding 2 and 7 leaves possible pairs {3,8} and {5,6}. The apparently suitable pair {4,7} is unavailable because the reserved pair already uses 7.

Crossing information may now select one of the remaining pairs or determine its order. Be careful not to extend the exclusion into a separate run simply because it lies in the same row. Kakuro's dark cells divide those regions into independent no-repeat constraints.

Follow a repeatable deduction loop

Good notes should become smaller as information accumulates. After each confirmed entry, update the remaining sums in both directions, remove the digit from the rest of both runs, and discard unsupported combinations. A change in one run can make a previously unhelpful crossing decisive.

When progress stops, return to the most constrained unresolved run. A run with two remaining combinations is usually easier to analyse than one with many. Check whether each combination can actually be assigned to its cells under the crossing restrictions; a valid total alone is not enough.

1. Keep combination sets separate from ordered cell assignments.
2. Subtract solved values and exclude used digits before consulting sum tables.
3. Use the smallest and largest available distinct digits to test bounds.
4. After a candidate removal, recheck the whole run for lost combinations.
5. Before accepting a deduction, state which clue, crossing, or no-repeat rule forces it.

Worked example: follow each deduction

Read rows from top to bottom and columns from left to right, starting at 1. Read each instruction, then compare the orange changes with the previous picture. Keep undecided cells or segments open until the explanation settles them.

? = undecided

1. Follow these changes in order; the picture below includes all of them.

   • Write 1 at row 2, column 2. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 2; row 3, column 2; row 4, column 2; row 5, column 2, the remaining arrangements are (1, 2, 3, 4) or (1, 2, 4, 3); they all give 1 here.

   • Write 4 at row 5, column 2. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 5, column 2; row 5, column 3; row 5, column 4; row 5, column 5, the remaining arrangements are (4, 1, 2, 3) or (4, 2, 1, 3); they all give 4 here.

   • Write 1 at row 3, column 5. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 5; row 3, column 5; row 4, column 5; row 5, column 5, the remaining arrangements are (4, 1, 2, 3); they all give 1 here.

2. Follow these changes in order; the picture below includes all of them.

   • Write 2 at row 4, column 5. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 5; row 3, column 5; row 4, column 5; row 5, column 5, the remaining arrangements are (4, 1, 2, 3); they all give 2 here.

   • Write 2 at row 2, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 2; row 2, column 3; row 2, column 4; row 2, column 5, the remaining arrangements are (1, 2, 3, 4); they all give 2 here.

   • Write 3 at row 4, column 2. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 2; row 3, column 2; row 4, column 2; row 5, column 2, the remaining arrangements are (1, 2, 3, 4); they all give 3 here.

3. Follow these changes in order; the picture below includes all of them.

   • Write 3 at row 3, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 3, column 2; row 3, column 3; row 3, column 4; row 3, column 5, the remaining arrangements are (2, 3, 4, 1); they all give 3 here.

   • Write 4 at row 3, column 4. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 3, column 2; row 3, column 3; row 3, column 4; row 3, column 5, the remaining arrangements are (2, 3, 4, 1); they all give 4 here.

   • Write 4 at row 4, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 3; row 3, column 3; row 4, column 3; row 5, column 3, the remaining arrangements are (2, 3, 4, 1); they all give 4 here.

4. Follow these changes in order; the picture below includes all of them.

   • Write 1 at row 5, column 3. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 3; row 3, column 3; row 4, column 3; row 5, column 3, the remaining arrangements are (2, 3, 4, 1); they all give 1 here.

   • Write 1 at row 4, column 4. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 4, column 2; row 4, column 3; row 4, column 4; row 4, column 5, the remaining arrangements are (3, 4, 1, 2); they all give 1 here.

   • Write 2 at row 5, column 4. Check the cells of the run or outlined cage marked 10. Their values must meet that arithmetic clue and the crossing restrictions. In cell order row 2, column 4; row 3, column 4; row 4, column 4; row 5, column 4, the remaining arrangements are (3, 4, 1, 2); they all give 2 here.

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