Takuzu, Binairo and binary puzzles: a short history
Fill a grid with two symbols. Keep every row and column balanced. Never let three of the same symbol sit side by side. Those three rules make a puzzle that now appears in newspapers, puzzle books and phone apps under so many names that it is easy to miss that they are the same game. Sun & Moon belongs to this family. Here is what is known about where it came from, how the versions differ, and why one rule comes and goes.
One puzzle, many names
Its names include Takuzu, Binairo, Binero, Tohu-Wa-Vohu, Binoxxo, Zernero, Tic-Tac-Logic and Sudoku Binary. In English it is often just a binary puzzle or binary sudoku. The symbols change as well: some versions use 0 and 1, some X and O, some black and white squares. Simon Tatham's free puzzle collection calls its version Unruly. LinkedIn added a daily version called Tango in October 2024, with suns and moons.
Who made it first
The early history is thin, and we have not found a dated first publication for every version, so here is only what the sources support.
- Tohu-Wa-Vohu was invented by the Italian puzzle maker Adolfo Zanellati. The name is a Hebrew phrase from the opening of Genesis, usually translated as formless and empty. Simon Tatham's documentation credits Zanellati with the puzzle type itself.
- Binairo was created by the Belgians Peter De Schepper and Frank Coussement, who founded the puzzle company PeterFrank in 2000. They published their first Binairo in the summer of 2009 in Belgium's largest newspaper. By their own account it was dropped that autumn, readers complained, and it came back. Their version also exists in odd sizes, where each row and column holds one extra 1.
- Tic-Tac-Logic, from the publisher Conceptis, uses X and O and traces its origin to the idea of a one-player version of tic-tac-toe.
Where the name Takuzu came from is less clear. We have not found a reliable account, so we leave it there.
The classic rules
- Every row and column holds the same number of each symbol.
- No more than two of the same symbol may sit next to each other in a row or column.
- No two rows may be identical, and no two columns.
The third rule is the one that varies. The standard Takuzu and Binairo rules include it, and Unruly offers it as an option. Tango drops it, and so does Sun & Moon.
Why the no-identical-rows rule exists
The first two rules leave surprisingly little room. A row of six cells with three of each symbol and no three alike in a row can be filled in only 14 ways. A row of eight has 34. We counted every complete 6 by 6 grid that keeps the balance and no-three rules: there are 11,222. Only 4,140 of them also have six different rows and six different columns. So the third rule removes more than three grids in five before a single clue is placed.
That extra pressure is useful to a puzzle setter. With fewer possible answers, fewer givens are usually needed to pin one down. It also gives the solver an extra move: when a row is two cells from finished and one way of filling it would copy a completed row, the other way must be right.
The version with signs
Signs carry information between cells. Two cells joined by = hold the same symbol, and two cells joined by × hold opposite symbols. Fill one end of a sign and the other follows. A sign can help even before that: an = between two neighbours in a row makes them a matching pair, so the cells on both sides of the pair must hold the other symbol. Signs let a setter give information without filling any cell, which keeps a board open while still steering it to one answer.
A sign inside a line carries a little arithmetic too. A × pair always holds one sun and one moon, so a 6 by 6 row with a × pair needs exactly two more of each symbol in its other four cells. For more solving steps of this kind, see our binary puzzle techniques guide.
Why Sun & Moon does not need the rule
A rule is only needed if the puzzle relies on it. Sun & Moon boards start from a complete, valid grid. The generator reads a spread of = and × signs off that answer, then takes away givens and signs one at a time. It keeps a removal only if exactly one solution remains under the game's own rules (balance, no three in a row, and the signs) and a second solver, limited to human techniques, can still finish the board. Because the uniqueness check never assumes rows must differ, a board never needs that rule to be solved, and an answer is free to contain two identical rows, like the grid above. The same human-style solver sets each board's difficulty, so every board can be finished by reasoning alone.
A puzzle mathematicians study
Binary puzzles look gentle, but in general they are hard. In a short paper posted online in 2013, Marzio De Biasi showed that deciding whether a binary puzzle has any solution at all is NP-complete, which means no known method stays fast as boards grow large. Researchers in the Netherlands, Putranto Utomo and Rusydi Makarim, have compared three ways of solving them by computer: a backtracking search, a satisfiability solver, and algebra over the numbers 0 and 1. On a 6 by 6 or 8 by 8 board none of that worries a human, because the rules are local: many steps come from looking at two or three cells at a time.
Try the sign version
Sun & Moon has 300 boards at 6 by 6 and 100 at 8 by 8, plus a daily 6 by 6 board that is easiest on Monday and hardest on Saturday. If you want to know more about why every board has one answer, read why good logic puzzles have exactly one solution.