Binairo

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Binairo puzzle

Binairo puzzle

Binairo is one of the most common logic puzzles, which is known by different names in different countries and is often put on a par with a game like tic-tac-toe.

But if in the latter the symbols (crosses and toes) need to be arranged in a row of three pieces, then in the first, on the contrary, they should not be allowed to be grouped in more than two pieces (horizontally and vertically).

Game history

The exact year and authorship of the invention of the game Binairo still causes controversy among fans of the logic puzzle genre. Some players are sure that it was invented by the Italian Adolfo Zanellati, others believe that it was created jointly by two Belgians - Peter De Schepper and Frank Coussement.

There is another version according to which Binairo arose by accident - during the creation of an electronic (single-user) version of Tic Tac Toe in the early 2000s.

One way or another, this game is of Western origin, unlike many popular puzzles ("Kakuro", "Nurikabe", "Kakurasu"), although it has much in common with them, for example, the shape of the playing field (a rectangle, broken into many cells) and the use of binary (black and white) characters.

Confusion about the origin of this puzzle is created by the huge number of its names, which are unique to a particular country. For example, in France everyone knows it as Binero, and this name is a registered trademark of Editions Megastar. In the European Union, this game is also registered as Takuzu and Binairo. In Hebrew the name is Tohu-Wa-Vohu (“Formless and Empty”), and in German it is Eins und Zwei (“One and Two”). In addition, this game is known by the names Tic-Tac-Logic, Zernero and Binoxxo.

Many of the names are explained by the type of symbols and shapes used in this binary puzzle. So, instead of the usual white and black circles, Binairo can use the numbers 1 and 2 (Eins und Zwei), the letters T and V (Tohu-Wa-Vohu) or the symbols X and O (“Cross-so-logic”).

After transferring the puzzle from printed to digital, multiple variations were created, where graphic objects were added to the listed symbols: fruits, icons, coins, and so on. In fact, instead of crosses and toes, or white and black circles, you can use any paired objects in this game. But the classic version is still considered Binairo with two-color circles.

Start playing Binairo right now (for free and without registration)! We believe you will succeed!

How to solve Binairo puzzle

How to solve Binairo puzzle

Binairo is the perfect example of a binary puzzle. From a mathematical point of view, it can be solved in three different ways: Gröbner-based algorithms, SAT solvers and backtracking algorithm.

For most players new to these specific branches of higher mathematics, it is important to know the basic rules of the game and be able to apply logic and deduction in practice!

Game rules

As in most similar games, Binairo does not have strict restrictions on the size of the playing field. This could be a small 4x4 square or a large 30x40 rectangle. The larger the field, the more difficult it is to solve the puzzle, because the player is required to simultaneously comply with three rules:

  • Each row and each column must have the same number of black and white circles.
  • Mugs of the same color cannot be placed three or more in a row. They can only be placed in pairs (2 pieces each), or one at a time.
  • Each row and column must be unique - subject to the first rule (the number of white and black circles must be the same in each row/column).

If we replace the circles with numbers 1 and 2 (as in the German version of the game Eins und Zwei), the last point can be described as follows: on a 5x5 playing field, rows/columns with different order of numbers are allowed: 12212, 12122 and 21122, but repetitions of any of these combinations are not allowed. Therefore, if you see that a double of an already folded row/column may appear on the field in the future, this option can immediately be classified as false.

How to solve the puzzle

As in Sudoku, in Binairo some of the cells of the playing field are initially filled. It is from these filled areas that the solution must begin. Otherwise, it will be simply impossible to solve the puzzle. Beginner players who have not yet mastered the nuances of playing Binairo should follow these tips:

  • Look for circles of the same color on the field, separated by one cell. Since three figures cannot be lined up in a row, an empty cell can be immediately filled with a circle of a different color.
  • If two circles of the same color are located next to each other, they can be immediately surrounded on both sides by circles of a different color.
  • Use the elimination method. For example, in the combination 112XXX (where X is empty cells), the last cell cannot contain a unit, since this is contrary to the game rules. Next to the last one you will have to place a two, and next to it another one, which will lead to the combination 112121 and the number of twos will not be equal to the number of ones.

Since the goal of the game is to have the same number of black and white circles in each row/column, they must consist of an even number of cells: 4x4, 6x6, 10x10, and so on. But there is another version of the game, where there should be more of some circles than others. For example, for every 5 whites there are 6 blacks, or for every 7 blacks there are 8 whites.

Some versions of Binairo also allow duplicate combinations in rows/columns, and without using this technique you simply will not be able to solve the puzzle correctly. These points must be clarified before the start of the game, so as not to waste effort on obtaining a obviously impossible result!