Existence of orthogonal Latin squares with aligned subsquares (Q1074586)

''It is shown that for both v and n even, \(v>n>0\), there exists a pair of orthogonal latin squares of order v with an aligned subsquare of order n if and only if \(v\geq 3n\), \(v\neq 6\), \(n\neq 2\), 6. This is the final case in showing that the above result is true for all \(v\neq 6\) and for all \(n\neq 2\), 6. When \(n=6\), the analogous result is obtained for incomplete arrays; the case \(n=2\) having been complete in earlier work.'' This paper completes a series showing that, once again, the obvious necessary conditions for something have proved to be sufficient, but only after a great deal of clever labor. In this case, the authors give us the final stone in the proof that we can have a pair of orthogonal latin squares of order v with an internal pair of orthogonal latin squares of order n whenever v and n satisfy the fairly obvious necessary conditions that \(v\geq 3n\) and \(v\neq 6\) and \(n\neq 2\), 6.