Kakuro & Number Grid Mastery: Combinatorics, Unique Partitions & Block Deduction

Kakuro Magic Partition Table

• 2-Cell Runs:
  • 3 = 1 + 2
  • 4 = 1 + 3
  • 16 = 7 + 9
  • 17 = 8 + 9
• 3-Cell Runs:
  • 6 = 1 + 2 + 3
  • 7 = 1 + 2 + 4
  • 23 = 6 + 8 + 9
  • 24 = 7 + 8 + 9
• 4-Cell Runs:
  • 10 = 1 + 2 + 3 + 4
  • 29 = 5 + 7 + 8 + 9
  • 30 = 6 + 7 + 8 + 9

1. The Power of Unique Integer Partitions

In Kakuro, many clues have only one valid combination of digits, regardless of order. Memorizing these key partitions gives tournament solvers a tremendous speed advantage:

When you see a 3 and an 8 perpendicular to each other, you can cross-eliminate candidates instantly!

2. Cross-Sum Intersect Analysis

Consider a simple cross-intersection:

• Horizontal Clue (2 cells): Sum = 4 → {1, 3}
• Vertical Clue (2 cells): Sum = 16 → {7, 9}

Because the intersection cell must satisfy both clues simultaneously, what happens if the vertical clue was instead Sum = 10 with digits {1, 9}? The intersecting cell must be 1, because 1 is the only number shared between {1, 3} and {1, 9}.

3. Extending Logic to Futoshiki & KenKen

The same numeric elimination principles apply across the entire logic grid family:

1. Futoshiki Inequality Chains (/futoshiki):
   • In a 5x5 grid, a chain like A > B > C > D forces A=5, B=4, C=3, D=2 or A=4, B=3, C=2, D=1. The highest value must sit at the root of the chain.
2. KenKen / Calcudoku (/kenken):
   • Multiplicative cages (e.g., 24× in 3 cells) break down into prime factorization (24 = 2 × 2 × 2 × 3 = 1 × 4 × 6 or 2 × 3 × 4).

4. Play the Number Logic Suite on Puzzle Paradise

• Solve daily arithmetic cross sums in Kakuro.
• Master inequality grids in Futoshiki.
• Test arithmetic operations in KenKen / Calcudoku.
• Build your number intuition with Classic Sudoku and Hidato Pathfinding.
• Check your ranking among worldwide solvers on the Leaderboard.