Chain Sudoku – Medium

Solving Chain Sudoku: Rules and Strategies

Chain Sudoku is a Sudoku variant. The 9×9 grid is divided not by 3×3 blocks, but by nine drawn chains.

Each row, each column, and each chain contains the numbers 1 to 9 exactly once.

The goal is a fully filled, contradiction-free puzzle. Every move must follow the visible givings and already confirmed entries.

Rules

• Each active cell contains exactly one number from 1 to 9.
• Each row contains the numbers 1 to 9 without repetition.
• Each column contains the numbers 1 to 9 without repetition.
• Each drawn chain contains the numbers 1 to 9 exactly once; normal 3×3 blocks do not count.
• Given numbers are not changed.
• Each final entry must meet all rules simultaneously.

Orientation in the grid

Rows are counted from top to bottom, columns from left to right. In a pair of numbers, the first designates the row and the second the column; (4,7), for example, refers to the cell in row 4, column 7.

The examples each show a specific intermediate state. Blue markings indicate the assumptions used, red markings an exclusion, and green markings a confirmed result.

Basic idea in solving

Follow the curved chains completely. After each entry, check its row, column, and chain again.

After each certain entry, it's worthwhile to do a quick full review of the affected units and additional hints. This creates a chain of small, understandable steps.

1. Complete an almost full row

In the unit Row 1, everything except cell (1,1) is filled. Visible are 1, 2, 3, 4, 5, 6, 7, 9; only 8 is missing among the permissible numbers.

Therefore, 8 goes in (1,1). The move requires no guess: the remaining values of the considered unit are fully visible in the image.

2. Complete an almost full column

In the unit Column 1, everything except cell (1,1) is filled. Visible are 1, 2, 3, 4, 5, 6, 7, 9; only 8 is missing among the permissible numbers.

Therefore, 8 goes in (1,1). The move requires no guess: the remaining values of the considered unit are fully visible in the image.

3. Read the decisive region completely

In the unit Chain 1, everything except cell (1,9) is filled. Visible are 1, 2, 4, 5, 6, 7, 8, 9; only 3 is missing among the permissible numbers.

Therefore, 3 goes in (1,9). The move requires no guess: the remaining values of the considered unit are fully visible in the image.

4. Fully evaluate the drawn region

In the unit Chain 2, everything except cell (1,1) is filled. Visible are 1, 2, 3, 4, 5, 6, 7, 9; only 8 is missing among the permissible numbers.

Therefore, 8 goes in (1,1). The move requires no guess: the remaining values of the considered unit are fully visible in the image.

5. Use an actual region instead of a imagined block

(1,2) and (1,1) both belong to Chain 2. Since 2 is already there, it cannot occur again in the second cell.

Number 2 is eliminated from (1,1). In this puzzle type, the drawn region matters; a normal 3×3 block would have no significance for this deduction.

6. Combine row and drawn region

The cell (1,2) lies in both row 1 and chain 2. The open numbers of both units are compared.

Only 2 fits in both sets, so it is entered in (1,2). This switching between rows, columns, and drawn regions is the core of solving.

7. Merge multiple units in a cell

For (1,1), several units are still incomplete. After exclusion from all related rows, columns, and regions, only 8 remains possible.

The candidate set for (1,1) is thus {8}. Such singletons often only appear when all types of units are checked simultaneously.

Typical solving process

1. Immediately fill in fully determined rows, columns, and relevant regions.
2. Note candidate numbers for open cells that are allowed by all normal units.
3. Apply the puzzle type's extra rule to these candidates.
4. Enter only values with clear justification.
5. After each entry, check all affected units and hints again.
6. Repeat the review until the grid is complete.

Common mistakes

• Using imagined 3×3 blocks: Only rows, columns, and drawn chains matter.
• Missing a chain at a bend: Follow all connected circles to the end.
• Guess too early: Multiple remaining candidates are not yet grounds for a guess.
• Not rechecking after an entry: A single value can alter multiple units and additional hints simultaneously.

Tips for beginners

• Start with units or hints that show the most visible information.
• Write down small candidate sets instead of mixing possibilities in your head.
• Always verify a candidate against every rule relevant to its cell.
• Treat certain exclusions as progress, even if no value is set yet.
• If no move is visible, switch intentionally between rows, columns, regions, and additional hints.

Chain Sudoku becomes clear once normal units and hints are consistently re-evaluated after each certain step. Small, fully justified conclusions are more reliable than early guesses.