#478 - Mixtures
Let us consider mixtures of three substances: A, B and C. A mixture can be described by a ratio of the amounts of A, B, and C in it, i.e., \((a : b : c)\). For example, a mixture described by the ratio \((2 : 3 : 5)\) contains \(20\%\) A, \(30\%\) B and \(50\%\) C.
For the purposes of this problem, we cannot separate the individual components from a mixture. However, we can combine different amounts of different mixtures to form mixtures with new ratios.
For example, say we have three mixtures with ratios \((3 : 0 : 2)\), \((3: 6 : 11)\) and \((3 : 3 : 4)\). By mixing \(10\) units of the first, \(20\) units of the second and \(30\) units of the third, we get a new mixture with ratio \((6 : 5 : 9)\), since:
\((10 \cdot \tfrac 3 5\) + \(20 \cdot \tfrac 3 {20} + 30 \cdot \tfrac 3 {10} : 10 \cdot \tfrac 0 5 + 20 \cdot \tfrac 6 {20} + 30 \cdot \tfrac 3 {10} : 10 \cdot \tfrac 2 5 + 20 \cdot \tfrac {11} {20} + 30 \cdot \tfrac 4 {10})
= (18 : 15 : 27) = (6 : 5 : 9)\)
However, with the same three mixtures, it is impossible to form the ratio \((3 : 2 : 1)\), since the amount of B is always less than the amount of C.
Let \(n\) be a positive integer. Suppose that for every triple of integers \((a, b, c)\) with \(0 \le a, b, c \le n\) and \(\gcd(a, b, c) = 1\), we have a mixture with ratio \((a : b : c)\). Let \(M(n)\) be the set of all such mixtures.
For example, \(M(2)\) contains the \(19\) mixtures with the following ratios:
\[\begin{align} \{&(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1), (0 : 1 : 2), (0 : 2 : 1),\\ &(1 : 0 : 0), (1 : 0 : 1), (1 : 0 : 2), (1 : 1 : 0), (1 : 1 : 1),\\ &(1 : 1 : 2), (1 : 2 : 0), (1 : 2 : 1), (1 : 2 : 2), (2 : 0 : 1),\\ &(2 : 1 : 0), (2 : 1 : 1), (2 : 1 : 2), (2 : 2 : 1)\}. \end{align}\]Let \(E(n)\) be the number of subsets of \(M(n)\) which can produce the mixture with ratio \((1 : 1 : 1)\), i.e., the mixture with equal parts A, B and C.
We can verify that \(E(1) = 103\), \(E(2) = 520447\), \(E(10) \bmod 11^8 = 82608406\) and \(E(500) \bmod 11^8 = 13801403\).
Find \(E(10\,000\,000) \bmod 11^8\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=478. Published Saturday, 30th August 2014, 07:00 pm. Solved by 254 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.8 Bit Manipulation and State Compression · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.2 Primes, Sieves, and Integer Factorization · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #524 · #837 · #858
Concepts: combinatorics number-theory brute-force-reduction
Likely techniques: bitmask-dp gcd-euclid hashing
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- What exactly is the input to problem 478? Is it a bound (11^8), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single residue (the answer is reduced modulo a given number, so keep everything in modular arithmetic from the start).
- Which objects exactly are in scope, and which are excluded by the wording (strict vs non-strict inequality, 'distinct', 'proper', 'below' vs 'up to')?
- What do the arguments of M(n), M(2) mean, and what is the value's type (count, sum, probability)?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 11^8?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 11^8 and the cost of testing one.
- Which combinatorics 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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- Predict the complexity of your intended method in terms of N = 11^8, and the wall-clock time you expect. Write both down now.
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- Predict the failure mode: what is most likely to break - an off-by-one on the bound, a definition misread, precision, or memory?
- Predict the output of the small case from rung 3 BEFORE running it (the statement says: "For example, a mixture described by the ratio (2: 3: 5) contains 20% A, 30% B and 50% C.") - then run it. A surprise here is worth more than an hour of debugging later.
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Which step of your solution were you least confident about, and what evidence would settle it?
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Where did the combinatorics structure do the real work? Name the single observation that collapsed the search space.
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- Related problem: #524 · #837 · #858
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