Question 167
Determine the largest integer $N$ for which there exists a $6 \times N$ table $T$ that has the following properties:
(i) Every column contains the numbers $1,2,3,4,5,6$ in some order.
(ii) For any two columns $i \neq j$, there exists a row $r$ such that $T(r,i)=T(r,j)$.
(iii) For any two columns $i \neq j$, there exists a row $s$ such that $T(s,i) \neq T(s,j)$.
(i) Every column contains the numbers $1,2,3,4,5,6$ in some order.
(ii) For any two columns $i \neq j$, there exists a row $r$ such that $T(r,i)=T(r,j)$.
(iii) For any two columns $i \neq j$, there exists a row $s$ such that $T(s,i) \neq T(s,j)$.