Kazaguruma – Medium

Kazaguruma Solving: Rules, Overlaps, and Strategies

Kazaguruma is a multi-structure of 5 classic 9×9 Sudoku puzzles. Five 9×9 Sudoku puzzles are arranged like the wings of a windmill around a central subgrid.

Kazaguruma is also known under the name Windmill Sudoku.

The visible figure includes a total of 333 active cells within a 21 by 21 grid of rows and columns. The offset outer puzzles connect through shared 3×3 blocks to the center. An entry can thus influence from a windmill wing across the center to the next.

The goal remains familiar: in each individual 9×9 subgrid, the numbers 1 to 9 must appear exactly once in each row, each column, and each 3×3 block. A shared cell has the same entry in all subgrid references.

Rules

• Each active cell contains exactly one number from 1 to 9.
• Each contained 9×9 Sudoku satisfies the normal row, column, and 3×3 block rules for itself.
• Overlapping cells belong simultaneously to all subgrids meeting there.
• Pre-given numbers are not changed.
• Areas without grid cells are not part of the puzzle and are not filled in.
• The puzzle is only solved when each subgrid is complete and free of contradictions.

Orientation in the diagram

For the examples, we count the rows of the entire diagram from top to bottom and columns from left to right. Row 4, Column 7 refers to the cell at the intersection of these two lines.

These global coordinates do not yet specify which Sudoku units apply. It's always necessary to check which 9×9 subgrid the cell lies in. An overlap cell may thus belong to multiple relevant rows, columns, and blocks.

Basic idea in solving

First work with known Sudoku techniques: complete houses, single cell candidates, and hidden single candidates. After each confirmed entry in a shared cell, switch immediately to the adjoining subgrid and update the candidates there.

When progress stalls, overlaps are not an obstacle but additional information. A candidate remains only allowed if it fits every affected row, column, and 3×3 block.

1. Recognize subgrids and their overlaps

Don't start with individual numbers, but with the shape. The blue outline shows the subgrid above. The yellow area is used together with the subgrid in the middle.

The common area lies here in overall rows 7 to 9 and overall columns 7 to 12. Each of its 18 cells has only one value but belongs to the units of both subgrids.

The thick lines help identify the 3×3 blocks. For rows and columns, always consider the complete 9×9 subgrid, not the entire visible width or height.

2. Close a nearly complete unit

In the displayed interim state, we look at row 11 overall of the middle subgrid. Only row 11, column 10 is open there.

The eight visible entries contain all numbers from 1 to 9 except for 9. Therefore, 9 must be in row 11, column 10. The other subgrids are not needed for this first step.

Such complete houses are the fastest entry point. Check them in each subgrid separately, as a long visible line can belong to several different Sudoku rows.

3. Find a single cell candidate

Next, consider row 2, column 8. The cell is not just the last gap of a single unit; its row, column, and block neighbors must be examined together.

The numbers 1, 2, 3, 4, 6, 7, 8, and 9 are already visible in at least one related unit. They are eliminated from candidates. The only remaining candidate is 5; it can be confidently placed.

Blue cells show matching exclusion witnesses. During your own solving, you don't need to mark them: it is crucial to eliminate candidates from all units of the cell.

4. Use a hidden single candidate

Next, examine the 3×3 block of the middle subgrid in overall rows 10 to 12 and columns 7 to 9, specifically looking for the number 4. Several cells in this unit are still empty.

The 4 appears only in row 12, column 8 in the embedded notes. The cell also has 9 as a candidate, but within the considered unit, the 4 has no other place.

Therefore, the 4 belongs to row 12, column 8. The other open cells here have notes: row 10, column 7: 8; row 12, column 7: 9; row 12, column 9: 2.

5. Cut candidates in the overlapping cell

Row 7, column 8 belongs to multiple subgrids. So, it is not enough to take its candidates from just one direction.

The individual tests give candidates 9 for the top subgrid; candidates 4 and 9 for the middle subgrid. Only the common intersection 9 satisfies all involved row, column, and block conditions simultaneously.

The intersection contains only 9. Thus, row 7, column 8 is already solved. Always update shared cells as soon as something changes in any of the involved subgrids.

6. Solve a shared cell confidently

Next is the typical multi-structure step. Row 9, column 12 is part of the top and middle subgrids at the same time.

After all exclusions, only 2 remains in this cell. Place 2 there. For the final solution, all involved subgrids are considered. The completed entry influences each of them equally.

Check not only the unit where you found the entry but also its broader impact on the other side of the overlap.

7. Track influence into the neighboring grid

The 2 is now visible in row 9, column 12. Next, consider row 1, column 12. This cell is in the same unit of the adjacent subgrid with the new entry.

Before the entry, 2 and 5 were possible. Now, 2 is eliminated. Only 5 remains; thus, row 1, column 12 is also solved.

This sequence exemplifies the ease of solving multi-structures: a secure step first, then immediate check of the connected grid. It creates a chain without guesswork.

8. From a block to a row or column

Consider the 3×3 block of the top middle subgrid in overall rows 1 to 3 and columns 10 to 12. There, 5 can only be in row 1, column 12 or row 3, column 12.

All these possibilities are within the same overall column 12. The 5 must appear in this block along that line.

Outside the block, eliminate 5 from the same subgrid line, here at row 4, column 12. The red notes mark these eliminations.

9. Evaluate a naked pair

Consider the row in overall row 1 of the top subgrid. Here, row 1, column 4 and row 1, column 12 share exactly the same candidates 2 and 5.

These two numbers must split the two blue cells. Therefore, 2 and 5 cannot be in any other cell of this unit.

In row 1, column 8, remove the red notes. A pair does not assign either number yet but can prepare new singles in the same unit.

Typical solving process

1. Mentally mark the boundaries of all 9×9 subgrids.
2. Look for nearly complete rows, columns, and blocks in each subgrid.
3. Note candidates when a cell's value is not immediately fixed.
4. Check shared cells from all involved subgrids.
5. Immediately track each new overlap entry into the neighboring grids.
6. Then use hidden singles, block-line interactions, and pairs.
7. Repeat the cycle until all subgrids are complete.

Common mistakes

• Treat the entire visible line as one Sudoku row. Always consider only the nine cells of a specific subgrid.
• Check a shared cell only in the subgrid where it just appeared.
• Erroneously see empty areas outside the drawn figure as cells to fill.
• Isolate a subgrid. Some puzzles only become clear through overlap information.
• Work on a secure overlap entry without updating candidates in the linked subgrid.
• Guess when unsure, although switching to another subgrid might give new clues.

Tips for beginners

• Use different colored border markings for subgrids if the shape initially seems confusing.
• Start where many givens are present, then follow the shared cells.
• Only jot down candidates allowed in all units of the cell.
• After each entry, first check row, column, and block, then the overlap.
• If a region gets stuck, deliberately switch to another subgrid instead of guessing.

Conclusion

Kazaguruma does not require new rules. The real task is to carefully apply familiar Sudoku logic across multiple overlapping 9×9 grids. By consciously tracking boundaries and shared cells, you transform the big figure into a sequence of smaller, verifiable steps.