Board puzzles with algebra of binary variables ask players to locate the hidden objects based on a set of clue cells and their neighbors marked as variables (unknowns). A variable with value of 1 corresponds to a cell with an object. Contrary, a variable with value of 0 corresponds to an empty cell—no hidden object. Overview These puzzles are based on algebra with binary variables taking a pair of values, for example, (no, yes), (false, true), (not exists, exists), (0, 1). It invites the player quickly establish some equations, and inequalities for the solution. The partitioning can be used to reduce the complexity of the problem. Moreover, if the puzzle is prepared in a way that there exists a unique solution only, this fact can be used to eliminate some variables without calculation. The problem can be modeled as binary integer linear programming which is a special case of integer linear programming. History Minesweeper, along with its variants, is the most notable example of this type of puzzle. Algebra with binary variables Below the letters in the mathematical statements are used as variables where each can take the value either 0 or 1 only. A simple example of an equation with binary variables is given below: a + b = 0 Here there are two variables a and b but one equation. The solution is constrained by the fact that a and b can take only values 0 or 1. There is only one solution here, both a = 0, and b = 0. Another simple example is given below: a + b = 2 The solution is straightforward: a and b must be 1 to make a + b equal to 2. Another interesting case is shown below: a + b + c = 2 a + b ≤ 1 Here, the first statement is an equation and the second statement is an inequality indicating the three possible cases: a = 1 and b = 0, a = 0 and b = 1, and a = 0 and b = 0, The last case causes a contradiction on c by forcing c = 2, which is not possible. Therefore, either first or second case is correct. This leads to the fact that c must be 1. The modification of a large equation into smaller form is not difficult. However, an equation set with binary variables cannot be always solved