Binairo

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!