Combined Volume of Cuboids

An axis-aligned cuboid, specified by parameters $\{(x_0, y_0, z_0), (dx, dy, dz)\}$, consists of all points $(X,Y,Z)$ such that $x_0 \le X \le x_0 + dx$, $y_0 \le Y \le y_0 + dy$ and $z_0 \le Z \le z_0 + dz$. The volume of the cuboid is the product, $dx \times dy \times dz$. The combined volume of a collection of cuboids is the volume of their union and will be less than the sum of the individual volumes if any cuboids overlap.
Let $C_1, \dots, C_{50000}$ be a collection of $50000$ axis-aligned cuboids such that $C_n$ has parameters

\begin{align}
x_0 &= S_{6n – 5} \bmod 10000\\
y_0 &= S_{6n – 4} \bmod 10000\\
z_0 &= S_{6n – 3} \bmod 10000\\
dx &= 1 + (S_{6n – 2} \bmod 399)\\
dy &= 1 + (S_{6n – 1} \bmod 399)\\
dz &= 1 + (S_{6n} \bmod 399)
\end{align}

where $S_1,\dots,S_{300000}$ come from the “Lagged Fibonacci Generator”:
For $1 \le k \le 55$, $S_k = [100003 – 200003k + 300007k^3] \pmod{1000000}$.For $56 \le k$, $S_k = [S_{k -24} + S_{k – 55}] \pmod{1000000}$.
Thus, $C_1$ has parameters $\{(7,53,183),(94,369,56)\}$, $C_2$ has parameters $\{(2383,3563,5079),(42,212,344)\}$, and so on.
The combined volume of the first $100$ cuboids, $C_1, \dots, C_{100}$, is $723581599$.
What is the combined volume of all $50000$ cuboids, $C_1, \dots, C_{50000}$?

This problem involves iterating on a random number generator, creating a set of cuboids, then calculating the combined volume of these cuboids. For practical purposes, this problem can be solved by a computer program as it involves handling a large amount of generated data.

Here we can utilize an interval tree data structure to calculate the combined volume. An interval tree can store intervals (just like our ‘cuboids’), and quickly retrieve all intervals that overlap with any given interval or point.

To solve the problem, you’d also need to generate your sequence of numbers using the provided rules. The numbers $S_k$ are defined for a ‘Lagged Fibonacci Generator’. They are defined differently for $k \le 55$ and for $k > 55$.

You can generate the first 55 numbers using the given formula for $1 \le k \le 55$, $S_k = [100003 – 200003k + 300007k^3] \mod{1000000}$. Then, for each $k > 55$, you can generate each number with the formula $S_k = [S_{k -24} + S_{k – 55}] \mod{1000000}$.

Once those numbers are generated, you then use them to specify the parameters for each of your cuboids. The parameters of the cuboids are determined by the modulo operations on the sequence generated by the Lagged Fibonacci Generator.

Once your interval tree is constructed and all of your cuboids are generated, you’d iterate over your 50000 cuboids, and for each one, check if the cuboid exists in the tree (meaning this ‘space’ is occupied by another cuboid already). If it doesn’t exist, add this cuboid’s volume to the total volume and append this interval to the tree.

At the end of the iteration, you will have added up each cuboid’s volume only once, even if they overlap. The total volume calculated will be the combined volume of all 50000 cuboids.

To note, performing this calculation by hand isn’t really feasible due to the large numbers of calculations and checks involved.

The question does not provide enough context to provide a hard-coded solution, but understanding the broad strokes of the method may help you program a solution yourself. This can be done in any high-level programming language such as Python, C++, Java etc. based on your familiarity with the language, with the interval tree data structure being a key component.

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