Question 87
Consider $S = \{(x, y, z) \mid x, y, z \in \{1, 2, \dots, 2012\}\}$ as a set of $2012^3$ points in three-dimensional space. For any segment joining two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ in the space, we define its distance triplet to be the ordered triple
$$
(|x_1 - x_2|, |y_1 - y_2|, |z_1 - z_2|).
$$
Alice wants to draw segments in such a way that
a. Each segment joins two distinct points in $S$;
b. Each point in $S$ is an endpoint of at most one segment;
c. For any two segments, their distance triplets are different.
Find the greatest number of segments that Alice can draw.
$$
(|x_1 - x_2|, |y_1 - y_2|, |z_1 - z_2|).
$$
Alice wants to draw segments in such a way that
a. Each segment joins two distinct points in $S$;
b. Each point in $S$ is an endpoint of at most one segment;
c. For any two segments, their distance triplets are different.
Find the greatest number of segments that Alice can draw.