Question 73
Assume that $A_1, A_2, \dots, A_8$ are eight points taken arbitrarily on a plane. For a directed line $l$ taken arbitrarily on the plane, assume that projections of $A_1, A_2, \dots, A_8$ on the line are $P_1, P_2, \dots, P_8$ respectively. If the eight projections are pairwise disjoint, they can be arranged as $P_{i_1}, P_{i_2}, \dots, P_{i_8}$ according to the direction of line $l$. Thus we get one permutation for $1, 2, \dots, 8$, namely, $i_1, i_2, \dots, i_8$. In the figure, this permutation is 2, 1, 8, 3, 7, 4, 6, 5. Assume that after these eight points are projected to every directed line on the plane, we get the number of different permutations as $N_8 = N(A_1, A_2, \dots, A_8)$. Find the maximal value of $N_8$. (posed by Su Chun)
