Simple games and magic squares (Q1894010)

The authors introduce a simple game \(\square_k\) built from a \(k \times k\) ``strongly rigid'' magic square, for each integer \(k \geq 3\). These games provide a uniform sequence of counterexamples to several conjectures that have arisen over the past three decades in the fields of threshold logic, hypergraphs, reliability systems, and simple games.