Simple games and magic squares

For each integer k ⩾ 3, we introduce a simple game ⊟k built from a k × k “strongly rigid” magic square. These games are not weighted, yet come very close to being weighted, and thus they 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. In particular, we show that ⊟k is k − 1 asummable but not k asummable (thus strengthening and simplifying an often-referenced, but unpublished, result of R. O. Winder) and that a certain variant of ⊟k is monotonic, strong, proper, has an acyclic “group” desirability relation, and yet is not weighted (thus strengthening a result of E. Einy and answering a question of B. Peleg).