Butterfly Sudoku – Medium

Solving Butterfly Sudoku: Rules, Overlaps, and Strategies

Butterfly Sudoku is a multi-doku consisting of 4 classic 9×9 Sudoku grids. Four offset 9×9 Sudoku grids are stacked so closely that their common areas form a closed 12×12 diagram.

The visible figure includes a total of 144 active cells within 12 rows and 12 columns. Unlike loosely connected multi-dokus, here the four sub-puzzles share broad strips; in the center, a cell belongs to multiple sub-grids.

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

Rules

• Each active cell contains exactly one number from 1 to 9.
• Each contained 9×9 Sudoku grid satisfies the normal row, column, and 3×3 block rules.
• Overlapping cells belong to all intersecting sub-grids simultaneously.
• Given numbers are not changed.
• The puzzle is solved only when all sub-grids are complete and contradiction-free.

Understanding the Diagram

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

These overall coordinates do not specify which Sudoku units apply. It is necessary to check in which 9×9 sub-grid the cell lies. An overlapping cell can belong to multiple relevant rows, columns, and blocks.

Basic Idea for Solving

Start with known Sudoku techniques: complete houses, individual cell candidates, and hidden single candidates. After each certain entry in a shared cell, move to the adjacent sub-grid and immediately update the candidates there.

If progress stalls, overlaps are not obstacles but additional information. A candidate is allowed only if it fits into every affected row, column, and 3×3 block.

1. Recognize sub-grids and their overlaps

Begin not with individual numbers but with shapes. The blue border shows the sub-grid top-left. The yellow area is used simultaneously as the sub-grid top-right.

The common area lies here in the overall rows 1 to 9 and the overall columns 4 to 9. Each of its 54 cells has only one value but belongs to the units of both sub-grids.

The thick lines help identify the 3×3 blocks. For rows and columns, it still always counts the full 9×9 sub-grid, not the entire visible width or height.

2. Complete a nearly full unit

In the displayed stage, we look at row 1 in the overall row of the top-left sub-grid. Only row 1, column 3 is open there.

The eight visible entries contain all numbers from 1 to 9 except 1. Therefore, 1 must be in row 1, column 3. The other sub-grids are not needed for this first step.

Such complete houses are the fastest entry points. Check them in each sub-grid separately, as a long visible line can belong to multiple different Sudoku rows.

3. Find a single cell candidate

Now consider row 2, column 1. The cell is not just the last hole of a single unit; its row, column, and block neighbors must be checked together.

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

Blue cells indicate matching exclusion clues. When solving on your own, you don't need to mark them: the key is to eliminate candidates from all units of the cell.

4. Use a hidden single candidate

Next, we check the 3×3 block of the top-right sub-grid within overall rows 1 to 3 and overall columns 10 to 12 for the number 7. Several cells in this unit are still empty.

The 7 appears in the displayed notes only in row 2, column 10. The cell also has 8 as a candidate, but within the examined unit, 7 has no other place.

Therefore, 7 belongs in row 2, column 10. The other open cells have notes: row 2, column 12: 8; row 3, column 10: 3 and 8; row 3, column 11: 3 and 6.

5. Candidates in the overlap field are intersected

Row 8, column 7 belongs to multiple sub-grids. It isn't enough to take its candidates from just one direction.

The individual checks give candidates 2 and 8 for the top-left sub-grid; 2 and 3 for the bottom-left sub-grid. Only the common intersection 2 satisfies all involved row, column, and block conditions simultaneously.

The intersection contains only 2. Thus, row 8, column 7 is already solved. Always update shared fields as soon as something changes in one of the involved sub-grids.

6. Securely solve a shared cell

Next is the typical multi-doku step. Row 1, column 6 is part of the top-left and top-right sub-grids.

We look at row 1 of the overall row in the top-right sub-grid; only 8 is missing there. So, 8 is entered. The conclusion is reached in one of the sub-grids. Since it’s the same shared cell, the entry also applies to all involved grids.

Don’t just check the unit where you found the entry. Its greatest effect often lies on the other side of the overlap.

7. Track the effect into the neighboring grid

The 8 is now visible in row 1, column 6. Now consider row 1, column 1. This cell shares an adjacent unit with the new entry.

Before the entry, 8 and 9 were possible there. Now 8 is excluded. Only 9 remains; thus, row 1, column 1, is solved.

This exact sequence makes multi-doku puzzles highly solvable: a safe step, immediately followed by a control scan in the connected grid. This creates a chain without guessing.

8. Deduce from a block to a row or column

We consider the 3×3 block of the top-right sub-grid within overall rows 1 to 3 and overall columns 7 to 9. Here, 8 can only be in row 3, column 7, or row 3, column 9.

All these options are in the same overall row 3. The 8 must therefore appear in this line within the block.

Outside the block, the 8 can be eliminated from the same sub-grid line here at row 3, column 10. The red notes mark these exclusions.

9. Evaluate a naked pair

In the 3×3 block of the top-right sub-grid within overall rows 4 to 6 and columns 10 to 12, row 4, column 10, and row 4, column 11, have exactly the same two candidates: 3 and 9.

These two numbers must share the two blue cells below. Therefore, 3 and 9 cannot appear in any other cell of this unit.

In row 5, column 10, the red notes are removed. A pair does not yet set either number, but can lead to new single candidates in the same unit.

Typical solving process

1. Mentally mark the borders of all 9×9 sub-grids.
2. Look for nearly complete rows, columns, and blocks in each sub-grid.
3. Note candidates as soon as a cell is not yet fixed.
4. Check shared cells from all involved sub-grids' perspectives.
5. Follow each new overlap entry immediately into the neighboring grids.
6. Afterward, utilize hidden single candidates, block-line interactions, and pairs.
7. Repeat until all sub-grids are complete.

Common mistakes

• Treat the entire visible line as a single Sudoku row. Valid are only the nine cells of a specific sub-grid.
• Check a shared cell only in the sub-grid where it was found.
• Isolate a sub-grid. Some puzzles only resolve through overlapping information.
• Work on a secure overlap entry without refreshing candidates in the connected sub-grid.
• Guess in case of uncertainty, as a switch to another sub-grid may give new clues.

Tips for Beginners

• Use different colored border markings for sub-grids if the shape initially appears confusing.
• Start where many clues are given, then follow shared cells.
• Write only candidates allowed in all units of the cell.
• First check row, column, and block after each entry, then check overlap.
• If a section stalls, consciously switch to another sub-grid instead of guessing.

Conclusion

Butterfly Sudoku does not require new rules. The actual challenge is to carefully apply familiar Sudoku logic to multiple overlapping 9×9 grids. By consciously tracking borders and shared cells, you turn the large figure into a sequence of smaller, verifiable steps.