The three-way intersection problem for Latin squares (Q5957695)

Let \(I^3[n]\) be the set of integers \(k\) for which there exist three Latin squares of order \(n\) which have precisely \(k\) cells where all three squares have identical entries and \(n^2-k\) cells where all three squares contain different entries. Further, for integers \(i<j\), let \([i,j]\) denote the set of integers \(\{i,i+1,i+2, \dots, j-1,j\}\). The authors main result is that for \(n\geq 8, I^3[n]=\{0, 1, \dots, n^2-15\} \cup \{n^2-12, n^2-9,n^2\}\). In addition, \(I^3[3]=\{0,9\}; I^3[4]=\{0,1,4,16\}; I^3[5]=[0,7]\cup\{10,25\}; I^3 [6]=[0,19]\cup \{27,36\}; I^3[7]=[0,30]\cup \{37,40,49\}\).