Board puzzles with algebra of binary variables

Conversely, a variable with value of 0 corresponds to an empty cell—no hidden object.

It invites the player quickly establish some equations, and inequalities for the solution.

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.

However, an equation set with binary variables cannot be always solved by applying linear algebra.

Using this fact on the first statement, the equations above can be reduced to A game based on the algebra with binary variables can be visualized in many different ways.

The main equation is written by using the total number of the hidden objects given.

One possible way to determine a partitioning is to choose the lead clue cells which have no common neighbors.

At some cases, the player can set a variable cell as 1 and check if any inconsistency occurs.

The clue cell marked red with value 1 does not have any remaining neighbor that can include a hidden object.

In algebraic form we have two equations: Here a, b, c, and d correspond to the top four grayed cells in Figure 6.

Try-and-check may need to be applied consequently in more than one step on some puzzles in order to reach a conclusion.

This is equivalent to binary search algorithm[3] to eliminate possible paths which lead to inconsistency.

Because of binary variables, the equation set for the solution does not possess linearity property.

tentaizu_4x4_example — Figure 1: An example puzzle on 4x4 board

tentaizu_4x4_example_with_variables — Figure 2: Binary variables are marked

tentaizu_4x4_example_solved_partially — Figure 3: The example solved partially

tentaizu_4x4_example_with_variables_solved — Figure 4: The example solved

tentaizu_4x4_example_partitioned — Figure 5: An example for partitioning

tentaizu_4x4_example_for_inconsistency — Figure 6: An example for try-and-check method