Binary Puzzle Symmetry and the Transformations That Work

Begin with a completed legal board

We checked the classic rules against BinaryPuzzle.com and independently tested the examples below. Each row and column needs three of each digit, no horizontal or vertical triple, and a pattern different from every other line in the same direction. Diagonals impose no restriction.

The original answer below satisfies all three requirements. Its first two rows start with 00, which is allowed. Looking down the crossing columns is essential before accepting the arrangement; individually legal rows can still create a vertical triple.

• Review the classic Binary Puzzle rules

Rotate the entire board

To rotate a binary puzzle 90 degrees clockwise, send the cell at row r, column c to row c, column 7 minus r on this six-by-six board. R1C1 moves to R1C6. The original bottom row becomes the new first column, read downward.

A rotation sends each old row or column to a whole new line, possibly read backward. Its digit counts stay unchanged. Any three consecutive cells remain consecutive, so a forbidden triple could appear afterward only if one existed before. Distinct lines stay distinct because reversing both cannot make different strings equal.

The same reasoning covers 180-degree and 270-degree turns. For an even square of side n, replace 7 with n plus 1 in the coordinate rule. These claims concern the classic square board and its three rules.

Reflect the whole grid without reordering its parts

A left-to-right reflection reverses every row and reverses the order of the columns. Here R1C1 moves to R1C6, while R2C1 moves to R2C6. Counts and consecutive triples survive for the same reasons as rotation. Uniqueness is preserved within each direction too.

Top-to-bottom reflection works too. Reflection across the main diagonal transposes the grid, exchanging rows and columns. Together, the rotations and reflections give eight geometric operations, including leaving the board as it is. You can use any one on a legal completed square.

Swap every zero and one together

Now keep the coordinates fixed and replace every 0 with 1 and every 1 with 0. Each balanced line still has three of each. A triple after the swap would correspond to the opposite triple before it. Two different patterns also remain different when every bit is inverted.

This global digit swap can follow any of the eight geometric operations, giving sixteen operations to try. The example produces sixteen distinct completed grids under them. Other boards can have symmetries that make some results coincide, so sixteen operations does not guarantee sixteen different answers.

Test an arbitrary row swap against adjacency

Exchange only rows 1 and 2 of the original answer. Each row remains individually legal. Column totals remain balanced because each column contains the same six digits in a different order. Row uniqueness survives; distinct columns also stay distinct under the same reversible reordering.

Vertical adjacency fails. Column 4 becomes 100011, with 000 at rows 2 through 4. Column 5 becomes 011100, with 111 at those same rows. Those two runs are enough to reject the result even though the count and uniqueness checks pass.

Some row swaps happen to work on particular boards. There is no general permission to exchange any two rows. Arbitrary column swaps have the corresponding risk horizontally. Reversing the entire row order preserves adjacency; exchanging selected rows can create new neighbors.

Move the fixed clues with the answer

A playable puzzle includes fixed clues as well as a completed answer. The twelve-clue example below has exactly one classic solution, the original answer. A centered dot represents a blank. We counted completions independently before using it here.

Apply the same operation to every clue position and to the answer. For a digit swap, invert the printed clues while keeping blanks blank. For a rotation, move both filled and empty cells. Rotating only the solution while leaving the clue sheet fixed does not generally produce a matching pair.

Check one clue through its new coordinates

Rotate that puzzle 90 degrees clockwise. The 1 at R1C5 moves to R5C6, and the 0 at R3C6 moves to R6C4. Compare those positions with the rotated answer above. The rotated clue pattern looks different, but it still has twelve givens and exactly one solution.

Every allowed operation is reversible. It pairs each original completion with one transformed completion, and the inverse recovers the original. The number of solutions therefore stays the same when clues move consistently. It covers ambiguous clue sets too, rather than requiring a unique starting puzzle.

Separate equivalent boards from fresh challenges

Takuzu symmetry describes these equivalences without requiring the printed clues to look symmetric. A clue layout can be lopsided and still define a valid puzzle. Rotating it preserves that asymmetry in a different orientation.

A transformed puzzle preserves corresponding rule deductions, although reading direction and recognition can affect a human solve. Keep transformed copies labelled with their source if you make practice sheets. For a new construction with independently chosen clues, follow the puzzle-making guide and check its solution count separately.

• Make a Binary Puzzle from a checked answer

Rules and example checks

BinaryPuzzle.com supplies the classic rule background. The grids, coordinate examples and preservation arguments here are original worked examples. Independent enumeration checked every operation on the twelve-clue puzzle and confirmed the two triples caused by the row exchange.

• Read the classic rules on BinaryPuzzle.com