Intersection numbers of Latin squares with their own orthogonal mates

Let J ∗ (v) be the set of all integers k such that there is a pair of Latin squares L and L′ with their own orthogonal mates on the same v-set, and with L and L′ having k cells in common. In this article we completely determine the set J ∗ (v) for integers v ≥ 24 and v =1, 3, 4, 5, 8, 9. For v =7 and 10 ≤ v ≤ 23, there are only a few cases left undecided for the set J ∗ (v)

The fine triangle intersections for maximum kite packings

In this paper the fine triangle intersection problem for a pair of maximum kite packings is investigated. Let Fin(v)={(s,t): $\exists$ a pair of maximum kite packings of order $v$ intersecting in $s$ blocks and $s+t$ triangles}. Let Adm(v)={(s,t): s+t\leq b_v, s,t are non-negative integers}, where $b_v=\lfloor v(v-1)/8\rfloor$. It is established that $Fin(v)= Adm(v)\setminus {(b_v-1,0),(b_v-1,1)}$ for any integer $v\equiv 0,1 ({\rm mod} 8)$ and $v\geq 8$; $Fin(v)=Adm(v)$ for any integer $v\equiv 2,3,4,5,6,7 ({\rm mod} 8)$ and $v\geq 4$