Kakuro Puzzle Tutorial

What is Kakuro?

Kakuro is a number puzzle played on a grid — often called "cross-sums" or "numeric crossword." The rules are simple:

• Fill white cells with digits 1–9
• Each horizontal or vertical run (consecutive white cells) must sum to the clue number shown in the black cell to its left (horizontal) or above it (vertical)
• No digit repeats within a single run

Reading the Grid

┌────┬────┬────┬────┐
│    │ \9 │ \6 │    │
├────┼────┼────┼────┤
│ 3\ │    │    │    │
├────┼────┼────┼────┤
│ 6\ │    │    │    │
├────┼────┼────┼────┤
│    │    │    │    │
└────┴────┴────┴────┘

A black cell with A\B means:

• A = sum of the horizontal run starting to the right
• B = sum of the vertical run starting below
• \B = vertical clue only (no horizontal)
• A\ = horizontal clue only (no vertical)

Minimum Puzzle: 2×2

The simplest possible puzzle:

┌─────┬─────┬─────┐
│     │  \4 │  \3 │
├─────┼─────┼─────┤
│ 3\  │  ?  │  ?  │
├─────┼─────┼─────┤
│ 4\  │  ?  │  ?  │
└─────┴─────┴─────┘

Column clues: left col = 3 down, right col = 4 down Row clues: top row = 4 across, bottom row = 3 across

Solve it:

• Row 1 sums to 4, two cells, no repeats → only option: {1,3} or {3,1}
• Row 2 sums to 3 → only option: {1,2} or {2,1}
• Column 1 sums to 3 → top cell + bottom cell = 3
• Column 2 sums to 4 → top cell + bottom cell = 4

If Row 1 = [1, 3]:

• Col 1 top = 1, so Col 1 bottom = 3−1 = 2 ✓ (Row 2 needs 1+2=3 ✓)
• Col 2 top = 3, so Col 2 bottom = 4−3 = 1 ✓ (Row 2 = {2,1} ✓)

Solution:
┌─────┬─────┐
│  1  │  3  │
├─────┼─────┤
│  2  │  1  │
└─────┴─────┘

Core Strategy 1: Unique Combinations

Some sums have only one possible combination for a given run length. Memorize these — they unlock cells immediately.

2-cell runs (no repeats, digits 1–9)

3-cell runs

4-cell runs

Example application:

┌──────┬────┬────┬────┐
│  \6  │    │    │    │
└──────┴────┴────┴────┘

3-cell run summing to 6 → must be {1, 2, 3} (order unknown, but you know all three values).

Core Strategy 2: Candidate Lists

When a combination isn't unique, list all possibilities and cross-eliminate using intersecting clues.

Example: 2-cell run summing to 8:

• Candidates: {1,7} {2,6} {3,5} (not {4,4} — no repeats)

Now suppose an intersecting column clue for one of those cells forces it to be odd. That eliminates {2,6} → remaining: {1,7} or {3,5}.

Strategy 3: Overlap Cells

When a run has only two possible combinations and both share a digit in the same position → that cell is solved.

Horizontal sum = 4, 2 cells:
  → only {1,3} or {3,1}
  Both contain digit 3 somewhere, but position unknown.

Vertical clue forces the LEFT cell to be odd:
  → left = 1 or 3 (both odd — no help yet)

Vertical clue forces the LEFT cell to be > 2:
  → left = 3, right = 1  ✓ solved!

Strategy 4: Min/Max Bounds

Every run has a minimum and maximum possible sum:

• Min for n cells = 1+2+…+n = n(n+1)/2
• Max for n cells = 9+8+…+(10−n) = n(19−n)/2

If a clue is close to min or max, the digit options are tightly constrained. A 4-cell run summing to 12 (just 2 above min 10) must be {1,2,3,4} shifted slightly — candidates are very few.

Worked Example: 3×3 Grid

┌──────┬──────┬──────┬──────┐
│      │  \16 │  \8  │  \13 │
├──────┼──────┼──────┼──────┤
│  9\  │  A   │  B   │  C   │
├──────┼──────┼──────┼──────┤
│ 13\  │  D   │  E   │  F   │
├──────┼──────┼──────┼──────┤
│ 19\  │  G   │  H   │  I   │
└──────┴──────┴──────┴──────┘

Clues:

• Row 1 (A,B,C): sum = 9
• Row 2 (D,E,F): sum = 13
• Row 3 (G,H,I): sum = 19
• Col 1 (A,D,G): sum = 16
• Col 2 (B,E,H): sum = 8
• Col 3 (C,F,I): sum = 13

Step 1 — Unique combos:

• Row 1 = 9, 3 cells → {1,2,6} or {1,3,5} or {2,3,4}
• Col 2 = 8, 3 cells → {1,2,5} or {1,3,4}

Step 2 — Intersections narrow candidates:

Col 2 contains only digits from {1,2,3,4,5}. Cell B is in both Row 1 and Col 2.

Row 1 candidates for B: any digit in the row combo that also appears in col combo.

• If Row 1 = {2,3,4}, then B ∈ {2,3,4} ∩ {1,2,3,4,5} = {2,3,4}
• If Row 1 = {1,3,5}, then B ∈ {1,3,5} ∩ {1,2,3,4,5} = {1,3,5}
• If Row 1 = {1,2,6}, then B ∈ {1,2,6} ∩ {1,2,3,4,5} = {1,2}

Step 3 — Row 3 = 19, 3 cells:

Combos: {2,8,9} {3,7,9} {4,6,9} {4,7,8} {5,6,8}

Step 4 — Apply column constraints:

Col 1 = 16: A+D+G = 16. Col 3 = 13: C+F+I = 13.

Continue cross-eliminating until all cells are pinned. Each new solved cell reduces candidates in the intersecting run, cascading toward a full solution.

Strategy 5: Parity

Digits 1–9 have known parity. If a run's sum is odd, it must contain an odd number of odd digits.

Example: sum = 7, 2 cells → must be {1,6} {2,5} {3,4}. Each pair contains exactly one odd digit ✓.

If a column intersection forces a cell to be even, and the run only works with an odd value there → contradiction → backtrack and try the other candidate.

Strategy 6: Subset Sum Elimination

When some cells in a run are already solved, subtract their total to find the remaining cell(s).

Run sum = 17, 3 cells.
Cells A and C are known: A=5, C=3.
Then B = 17 − 5 − 3 = 9.

Solving Order (General Approach)

1. Mark all unique-combination runs first
2. List candidates for all remaining runs
3. Apply intersection constraints (row ∩ column)
4. Fill any cell that has only 1 candidate
5. Update affected run candidates → repeat from step 3
6. Use parity / min-max / subset-sum for stubborn areas
7. If stuck: trial-and-error on a cell with 2 candidates

Quick Reference Card

UNIQUE COMBOS TO MEMORIZE:
  2-cell:  3={1,2}         17={8,9}
  3-cell:  6={1,2,3}       24={7,8,9}
  4-cell: 10={1,2,3,4}     30={6,7,8,9}
  5-cell: 15={1,2,3,4,5}   35={5,6,7,8,9}

GOLDEN RULES:
  • No digit repeats within one run
  • Clue = exact sum (not max/min)
  • Digits 1–9 only (no zeros)
  • Intersecting cells must satisfy BOTH runs

Start with 4×4 or 5×5 beginner grids, focus on unique-combination clues first, and build up to the 10×10+ puzzles where the chain of eliminations spans the whole board.