Minimal Binary Puzzle Clues and the Meaning of Minimum

Give minimal and minimum separate jobs

Here, a minimal or irreducible Binary Puzzle has exactly one solution, and erasing any one of its givens creates multiple solutions. A minimum-clue puzzle has the fewest givens among all uniquely solvable puzzles of the specified size and rules. The first claim concerns deletions from one arrangement. The second compares every possible arrangement.

For this experiment we independently enumerated all 4,140 completed classic 6x6 boards, then checked two clue sets against that entire collection. Each row and column contains three zeros and three ones, has no triple, and differs from every other line in the same direction. A row may equal a column.

The enumeration starts with the fourteen balanced six-cell row patterns that avoid triples. It combines different rows, rejects impossible columns, and checks column uniqueness when a board is complete. Counting matches against every surviving board avoids mistaking the first answer found for the only answer.

Keep those rules fixed throughout the comparison. Changing the board size or allowing repeated rows changes the collection of possible answers. A clue minimum only has meaning once that collection is specified.

Puzzle A has eleven essential clues

Copy the first grid if you want to inspect the larger clue set. Dots represent blanks, and every printed digit is a given. Number rows from top to bottom and columns from left to right. R2C2 therefore means row 2, column 2.

Puzzle A has eleven clues and exactly one completion. Its empty first row can look surprising, but columns constrain those cells too. Clues need not appear in every row to determine the whole board. The deletion audit below shows that all eleven are necessary within this arrangement.

Puzzle B determines the same answer with seven clues

Puzzle B has seven givens and exactly one completion. Its last two rows are entirely blank. It uses the same rules and leads to the same completed board as Puzzle A, so the comparison holds even when the intended answer is fixed.

The seven-clue layout uses different positions. You cannot obtain it simply by erasing four clues from A. Every single deletion from A already loses uniqueness. Reaching this smaller layout requires changing which information is supplied.

Erase one clue at a time from Puzzle A

Each row of this table is a fresh experiment. Start with the original eleven-clue puzzle, erase only the listed clue, count all compatible completed boards, then restore that clue before the next test. The numbers are exact totals, including the original answer.

For example, erasing the 1 at R2C2 leaves twelve solutions. Erasing R2C6 leaves two. Both changes destroy uniqueness, even though their totals differ. A larger total does not by itself tell you which clue would make a more useful opening for a human solver.

Repeat the test for Puzzle B

All seven deletions from B also produce multiple solutions. B is therefore irreducible too. The largest count here is forty, after erasing R2C3. Keeping that clue restores the single intended completion.

Checking individual deletions is sufficient to establish irreducibility. Once an erasure permits two answers, deleting still more clues cannot eliminate either answer. Removing constraints only preserves or enlarges the set of compatible boards. That is why no proper subset of either clue set can retain uniqueness.

State exactly what these counts prove

A is minimal and cannot be minimum, because B supplies a smaller uniquely solvable puzzle of the same size. B establishes an upper bound of seven on the minimum clues in Takuzu under these classic 6x6 rules. This experiment does not establish that seven is the minimum.

To prove that final claim, you would also need to rule out every uniquely solvable layout with six or fewer givens, across every valid answer board. Enumerating all completed boards lets us count solutions for a supplied layout. It does not automatically examine every possible choice of clues.

Both examples finish at the answer below. The downloadable record includes the two starting grids and all deletion counts, so you can check the comparison without copying numbers from the page.

• Download the clue grids, answer and deletion audit

Choose clues for the reader you have

When making a puzzle, decide whether you want an irreducible research example or a comfortable solving exercise. An extra clue may be redundant for uniqueness while still giving a beginner a clear first deduction. Neither minimality nor a small clue count guarantees an enjoyable solving path.

Our construction guide covers building a valid answer and testing erasures. For a minimality claim, save the final layout and test every remaining clue. If your solver stops after finding two solutions, record that result as at least two. A timeout is inconclusive. Exact totals require completing the search.

To explore this yourself on paper, copy A and erase R2C6. Try finding two complete legal answers that match everything left. Restore the clue afterward and compare your answers with the shared solution.

• Make a Binary Puzzle and check its solution count
• Look up complete legal row patterns

Rules and verification record

The publisher rules below define the classic constraints used here. The grids and deletion counts are our original exhaustive experiment. These examples certify uniqueness and irreducibility, with no global minimum or human difficulty rating claimed.

• BinaryPuzzle.com classic rules