Killer Sudoku – Medium

Killer Sudoku

Killer Sudoku combines the normal Sudoku rules with sum ranges. The grid is a classic 9x9 Sudoku, but additionally, multiple cages are drawn in. Each cage has a small sum. The numbers in this cage must add up exactly to this sum.

Important: Even in Killer Sudoku, guessing is not recommended. The cage sums provide additional exclusions. A cage with two cells and sum 5 can, for example, consist of 1 and 4 or 2 and 3. A cage with three cells and sum 16 can have several combinations, but already set numbers or Sudoku rules can make the missing number clear.

Basic Rules

• Each row must contain the numbers 1 to 9 exactly once.
• Each column must contain the numbers 1 to 9 exactly once.
• Each 3x3 block must contain the numbers 1 to 9 exactly once.
• The dashed cages each have a given sum.
• The numbers in a cage must add up exactly to this sum.
• Usually, a number may not be repeated within a cage.
• A cage can lie within a row, column, or block, but can also span multiple areas.
• A Killer Sudoku is solved when all normal Sudoku rules and all cage sums are fulfilled.

Strategies for Solving

The cages are the most important additional tool. Not only which numbers are missing in a row, column, and block are checked, but also which number combinations fit the cage sum.

1. Evaluate a true cage

In the example puzzle, there is a cage with sum 16 in the top left, covering the first three cells of the first row. If two values of this cage are known, the third value is unique.

The cage contains the values 5, 8, and an empty cell. Since 5 + 8 = 13, the total missing to reach the cage sum of 16 is exactly 3. The empty cell in row 1, column 3 must therefore be a 3.

The conclusion is clear, as it directly follows from the cage sum.

Cages do not have to lie horizontally; they can have arbitrary shapes. In the example puzzle, there is a vertical cage with sum 20 in column 7. Since two of the three values are known, the third value here is also determined by the sum.

This cage already contains 7 and 5. Together they sum to 12. Therefore, 8 is missing to reach 20. The third cell of the cage must be an 8, regardless of the cage's orientation.

This step is also not guesswork.

2. Use the 45-rule

Each complete Sudoku row, column, and 3x3 block contains the numbers 1 to 9. Their sum is always 45. In Killer Sudoku, this property can be used.

These four cages cover exactly the top-left 3x3 block. Their sums add up to 16 + 9 + 5 + 15 = 45. This matches the sum of a complete 3x3 block.

Typical solving sequence

1. First look for cages whose sum imposes strong restrictions.
2. List possible combinations for small cages and for cages with few cells.
3. Check if rows, columns, or blocks determine the order of a cage combination.
4. Use known values within a cage to directly calculate missing values.
5. Use the 45-rule for rows, columns, or blocks.
6. Update candidates after each certainty.
7. Always check cage sum, row, column, and 3x3 block simultaneously.

Common mistakes

• Calculating a cage sum correctly but forgetting the normal Sudoku rule.
• Using a number twice within a cage.
• Deciding on a combination's order too early.
• Considering cages in isolation without combining with rows, columns, and blocks.

Tips for beginners

• Start with cages where one or two values are already known.
• Write down combinations first, not immediate final numbers.
• Check for each cage whether a number is already excluded by row, column, or block.
• A secure cage entry can immediately enable new normal Sudoku deductions.

At first, Killer Sudoku appears as a math puzzle due to the sums. However, it remains a logic puzzle: the sums restrict combinations, and the Sudoku rules determine where these numbers can be placed.