How to Play Kakuro
Master the art of Cross Sums with our comprehensive guide to Kakuro puzzles
What is Kakuro?
Kakuro, also called Cross Sums or Kakro, is a logic puzzle that combines the best elements of crosswords and Sudoku. Instead of words or letters, you fill a grid with numbers, making sure each group of cells adds up to a specific sum.
The puzzle was invented in the 1960s and became popular in Japan before spreading worldwide. The name "Kakuro" comes from the Japanese words "kasan" (addition) and "cross" - literally meaning "addition cross."
Like Sudoku, Kakuro requires no mathematical knowledge beyond basic addition. The challenge lies in finding the right combinations of numbers that satisfy multiple overlapping constraints simultaneously.
Basic Rules
1. Fill With Numbers 1-9
Each empty white cell must contain a single digit from 1 to 9. Unlike Sudoku, you're not filling a complete grid - only the white cells need numbers.
2. Match the Clue Sums
Dark cells with diagonal lines contain clues. The number above the diagonal indicates the sum for the horizontal group of white cells to its right. The number below the diagonal indicates the sum for the vertical group below it.
3. No Repeating Digits in Each Group
Within a single continuous group (horizontal or vertical), you cannot use the same digit twice. However, the same digit can appear in different groups, even if they intersect.
4. Groups End at Dark Cells
A group of cells continues until it reaches a dark cell or the edge of the grid. Each continuous run of white cells is one group with one sum to achieve.
How to Solve Kakuro: Step-by-Step
Step 1: Read the Grid Structure
Before solving, understand the layout. Dark cells with diagonal lines are clues. Empty white cells need to be filled. Each clue tells you what its associated group of cells must sum to.
Example:
A clue showing "17\" means the horizontal group sums to 17
A clue showing "\23" means the vertical group sums to 23
A clue showing "17\23" provides both horizontal (17) and vertical (23) clues
Step 2: Identify Unique Combinations
Some sum-length combinations have only one possible solution. These are your starting points!
Unique 2-cell combinations:
- • Sum 3 = 1+2 (only option)
- • Sum 4 = 1+3 (only option)
- • Sum 16 = 7+9 (only option)
- • Sum 17 = 8+9 (only option)
Unique 3-cell combinations:
- • Sum 6 = 1+2+3
- • Sum 7 = 1+2+4
- • Sum 23 = 6+8+9
- • Sum 24 = 7+8+9
Step 3: Use Intersection Logic
Each white cell belongs to both a horizontal and a vertical group. The number in that cell must satisfy both sum requirements simultaneously.
Example:
If a cell is in a horizontal group summing to 17 (must be 8+9)
AND a vertical group summing to 16 (must be 7+9)
Then that cell MUST be 9 (the only common number)
Step 4: Apply the Process of Elimination
As you fill in numbers, eliminate them as possibilities for other cells in the same group.
Example:
A 3-cell group summing to 15 could be:
- • 1+5+9
- • 1+6+8
- • 2+4+9
- • 2+5+8
- • 2+6+7
- • 3+4+8
- • 3+5+7
- • 4+5+6
If you determine one cell is 9, you eliminate options without 9, leaving only 1+5+9 or 2+4+9
Step 5: Look for Hidden Singles
Sometimes a number can only go in one specific cell of a group, even if that cell has multiple possibilities when viewed alone.
Check if a number appears in all possible combinations for a group. If it does, and you can determine which cell through intersection logic, you've found a hidden single!
Step 6: Work Systematically
Continue applying these techniques across the entire grid. Fill in certain values first, which creates new constraints that reveal more certain values.
- Start with unique combinations
- Move to highly constrained intersections
- Use elimination on partially solved groups
- Look for hidden singles in remaining gaps
- Verify sums as you complete each group
Common Combinations Reference
Memorizing these common combinations will dramatically speed up your solving. Here are the most useful ones:
Two Cells
| Sum | Combinations | Count |
|---|---|---|
| 3 | 1+2 | 1 ✓ Unique |
| 4 | 1+3 | 1 ✓ Unique |
| 16 | 7+9 | 1 ✓ Unique |
| 17 | 8+9 | 1 ✓ Unique |
| 5 | 1+4, 2+3 | 2 |
| 10 | 1+9, 2+8, 3+7, 4+6 | 4 |
| 15 | 6+9, 7+8 | 2 |
Three Cells
| Sum | Combinations | Count |
|---|---|---|
| 6 | 1+2+3 | 1 ✓ Unique |
| 7 | 1+2+4 | 1 ✓ Unique |
| 23 | 6+8+9 | 1 ✓ Unique |
| 24 | 7+8+9 | 1 ✓ Unique |
| 8 | 1+2+5, 1+3+4 | 2 |
| 15 | 1+5+9, 1+6+8, 2+4+9, 2+5+8, 2+6+7, 3+4+8, 3+5+7, 4+5+6 | 8 |
Advanced Strategies
1. Constrained Number Analysis
When a cell belongs to two groups that both have limited combinations, find the intersection of possible numbers. This often reveals the exact value immediately.
2. Sum Completion
If you've filled all but one cell in a group, simply subtract the filled numbers from the target sum to find the remaining number. Always verify it's in the 1-9 range and not already used.
3. Pencil Marking
For difficult puzzles, write small candidate numbers in each cell. As you discover constraints, erase impossible candidates. When a cell has only one candidate left, fill it in.
4. Sector Analysis
Focus on one area of the grid at a time. Completely solve a corner or edge before moving to other areas. This builds certainty and creates cascade effects.
5. Maximum and Minimum Constraints
For large sums in short groups or small sums in long groups, you're forced to use large or small numbers respectively. Use this to eliminate mid-range possibilities.
Difficulty Levels
Easy
Smaller grids (6x6 to 8x8) with many unique combinations. Most cells can be determined through direct logic with minimal lookahead required.
Medium
Standard grids (10x10 to 12x12) with moderate combinations. Requires tracking multiple possibilities and using elimination across several intersecting groups.
Hard
Larger grids (14x14+) with few unique combinations. Demands careful pencil marking, complex elimination chains, and occasionally trying hypothetical values to test for contradictions.
Expert
Very large grids with minimal initial constraints. Requires advanced techniques like naked pairs, hidden pairs, and multi-level elimination. Can take hours to complete.
Frequently Asked Questions
What is Kakuro?▼
Kakuro, also known as Cross Sums, is a number puzzle that combines elements of crosswords and Sudoku. The goal is to fill empty cells with numbers 1-9 so that each horizontal or vertical group of cells adds up to the clue number shown, with no repeated digits in each group.
What are the basic rules of Kakuro?▼
The basic rules of Kakuro are: 1) Fill each empty cell with a number from 1 to 9, 2) Numbers in each continuous horizontal or vertical group must add up to the clue sum shown, 3) No number can be repeated within the same group, 4) The same number can appear in different groups, and 5) Every puzzle has a unique solution that can be found through logic alone.
How do I start solving a Kakuro puzzle?▼
Start by looking for cells that have only one possible combination. For example, a 2-cell group with sum 3 can only be 1+2. Then look for groups where certain combinations are limited by intersecting groups. Fill in the most constrained cells first, and use the process of elimination as you discover what numbers have already been used in each group.
What's the difference between Kakuro and Sudoku?▼
While both use numbers 1-9 and require logical thinking, Kakuro focuses on addition and number combinations within variable-length groups, whereas Sudoku focuses on ensuring no repeats in fixed 9-cell rows, columns, and boxes. Kakuro has clue sums and groups of different lengths, making it more about mathematical combinations, while Sudoku is more about pattern elimination.
Are there strategies for memorizing number combinations?▼
Yes! The most useful strategy is memorizing unique combinations - like knowing that 16 in 2 cells can only be 7+9, or 17 in 2 cells can only be 8+9. With practice, you'll recognize common combinations instantly. Start by learning all 2-cell combinations, then move to 3-cell combinations for common sums like 6, 7, and 24.
Can I solve Kakuro without guessing?▼
Yes! Every well-designed Kakuro puzzle can be solved through pure logic without any guessing. If you find yourself needing to guess, look for more constraints from intersecting groups, check for unique combinations, or use the process of elimination to narrow down possibilities in each cell.
What makes a Kakuro puzzle hard vs easy?▼
Difficulty in Kakuro depends on several factors: 1) Grid size (larger grids are more complex), 2) Number of unique combinations required (more unique combinations make it easier to start), 3) How many cells intersect multiple constrained groups, and 4) Whether solving requires lookahead or can be done with immediate deductions. Expert puzzles often require tracking multiple possibilities simultaneously.
How can I improve my Kakuro solving speed?▼
To improve speed: 1) Memorize common number combinations for sums 3-45, 2) Practice identifying unique combinations immediately, 3) Develop a systematic approach to scanning the grid, 4) Learn to spot hidden singles (cells with only one possibility), 5) Use pencil marks efficiently to track possibilities, and 6) Solve daily to build pattern recognition and intuition.
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