#design-theory

A Hadamard matrix of order $n$ is an $n\times n$ matrix $H$ with entries $\pm 1$ whose rows are mutually orthogonal, so that $H{H}^T=nI$. Such a matrix can exist only if $n$ is 1, 2 or a multiple of 4. Prove that a Hadamard matrix exists for every multiple of 4.

Fifteen schoolgirls go for a walk every day for a week, in five rows of three. Arrange the rows for the seven days so that no two girls walk in the same row more than once.