Question 11
Let $S$ be the set of points $(x, y, z)$ in $\mathbb{R}^3$ such that $x, y$, and $z$ are positive integers less than or equal to $100$. Let $f$ be a bijective map between $S$ and the $\{1,2, \ldots, 1000000\}$ that satisfies the following property: if $x_1 \leq x_2, y_1 \leq y_2$, and $z_1 \leq z_2$, then $f\left(x_1, y_1, z_1\right) \leq f\left(x_2, y_2, z_2\right)$. Define
$$
\begin{aligned}
A_i & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(i, j, k), \\
B_i & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(j, i, k), \\
\text{ and } C_i & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(j, k, i)
\end{aligned}
$$
Determine the minimum value of $A_{i+1}-A_i+B_{j+1}-B_j+C_{k+1}-C_k$.
$$
\begin{aligned}
A_i & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(i, j, k), \\
B_i & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(j, i, k), \\
\text{ and } C_i & =\sum_{j=1}^{100} \sum_{k=1}^{100} f(j, k, i)
\end{aligned}
$$
Determine the minimum value of $A_{i+1}-A_i+B_{j+1}-B_j+C_{k+1}-C_k$.