#588 - Quintinomial Coefficients
The coefficients in the expansion of \((x+1)^k\) are called binomial coefficients.
Analoguously the coefficients in the expansion of \((x^4+x^3+x^2+x+1)^k\) are called quintinomial coefficients.
(quintus= Latin for fifth).
Consider the expansion of \((x^4+x^3+x^2+x+1)^3\):
\(x^{12}+3x^{11}+6x^{10}+10x^9+15x^8+18x^7+19x^6+18x^5+15x^4+10x^3+6x^2+3x+1\)
As we can see \(7\) out of the \(13\) quintinomial coefficients for \(k=3\) are odd.
Let \(Q(k)\) be the number of odd coefficients in the expansion of \((x^4+x^3+x^2+x+1)^k\).
So \(Q(3)=7\).
You are given \(Q(10)=17\) and \(Q(100)=35\).
Find \(\sum_{k=1}^{18}Q(10^k)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=588. Published Sunday, 29th January 2017, 04:00 am. Solved by 531 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact counting underlies discrete pricing lattices, scenario enumeration, and combinatorial probability (Phase 7).
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Prerequisites
Lessons that prepare you:
1.1 Sets, Functions, and Relations · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.11 Integer Partitions and Counting Structures · 2.1 Functions, Limits, and Continuity · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #154 · #602 · #987
Concepts: algebra combinatorics brute-force-reduction
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Understand the problem
- What exactly is the input to problem 588? Is it a bound (10), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer.
- 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 Q(k), Q(3) 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 10?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10 and the cost of testing one.
- Which algebra fact would, if true, collapse the search - and can you state it as a testable claim before you look for a proof?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says algebra - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10, and the wall-clock time you expect. Write both down now.
- Predict the key data structure: what is stored, keyed by what, and how large will it get at full scale?
- 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: "You are given Q(10)=17 and Q(100)=35.") - 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?
What did you try first, and what specifically made you abandon it - a proof, a timing, or a wrong small-case answer?
Where did the algebra structure do the real work? Name the single observation that collapsed the search space.
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How would your solution change if the bound 10 were multiplied by 1000? Does it survive, or does it need a different idea?
What is the honest complexity of what you wrote (not what you intended), and where is the remaining slack?
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- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #154 · #602 · #987
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