All-even board

What this constraint-satisfaction probability question tests

This is a medium-difficulty probability problem that combines counting, linear algebra over finite fields, and constraint propagation. It asks you to find the probability that a randomly filled grid satisfies global parity constraints—a type of reasoning common in combinatorics and coding theory interviews.

The key insight is recognizing that row and column sum constraints are not independent. Once you fix certain entries, the parity requirements on rows and columns force the values of remaining entries. The problem rewards candidates who can count the degrees of freedom rather than enumerate outcomes. You'll need to think carefully about how many entries you can choose freely before all others are determined by the even-sum constraints.

• Constraint propagation and rank over GF(2)
• Counting free variables in a system of linear equations
• Independence and conditional probability in grid-based problems