#212 - 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, \(C_1, \dots, C_{50000}\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=212. Published Saturday, 11th October 2008, 06:00 am. Solved by 1,654 members at time of mirroring.
Why this is useful
Numerical Computing. Matrix methods and linear-recurrence acceleration are the machinery behind covariance work, PCA, and lattice/transition models (Phases 4, 5, 8).
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.13 Matrix Exponentiation and Linear Recurrence Acceleration · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 3.1 Vectors, Multivariable Functions, and Level Sets
Recommended stepping-stone problems: #147 · #184 · #335
Concepts: geometry number-theory brute-force-reduction
Likely techniques: matrix-exponentiation
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- What exactly is the input to problem 212? Is it a bound (50000), a supplied dataset, or a definition you must generate from?
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- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
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- Which geometry fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
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Which step of your solution were you least confident about, and what evidence would settle it?
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- Related problem: #147 · #184 · #335
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