#669 - The King's Banquet
The Knights of the Order of Fibonacci are preparing a grand feast for their king. There are \(n\) knights, and each knight is assigned a distinct number from \(1\) to \(n\).
When the knights sit down at the roundtable for their feast, they follow a peculiar seating rule: two knights can only sit next to each other if their respective numbers sum to a Fibonacci number.
When the \(n\) knights all try to sit down around a circular table with \(n\) chairs, they are unable to find a suitable seating arrangement for any \(n>2\) despite their best efforts. Just when they are about to give up, they remember that the king will sit on his throne at the table as well.
Suppose there are \(n=7\) knights and \(7\) chairs at the roundtable, in addition to the king’s throne. After some trial and error, they come up with the following seating arrangement (\(K\) represents the king):
Notice that the sums \(4+1\), \(1+7\), \(7+6\), \(6+2\), \(2+3\), and \(3+5\) are all Fibonacci numbers, as required. It should also be mentioned that the king always prefers an arrangement where the knight to the his left has a smaller number than the knight to his right. With this additional rule, the above arrangement is unique for \(n=7\), and the knight sitting in the 3rd chair from the king’s left is knight number \(7\).
Later, several new knights are appointed to the Order, giving \(34\) knights and chairs in addition to the king's throne. The knights eventually determine that there is a unique seating arrangement for \(n=34\) satisfying the above rules, and this time knight number \(30\) is sitting in the 3rd chair from the king's left.
Now suppose there are \(n=99\,194\,853\,094\,755\,497\) knights and the same number of chairs at the roundtable (not including the king’s throne). After great trials and tribulations, they are finally able to find the unique seating arrangement for this value of \(n\) that satisfies the above rules.
Find the number of the knight sitting in the \(10\,000\,000\,000\,000\,000\)th chair from the king’s left.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=669. Published Saturday, 11th May 2019, 10:00 pm. Solved by 364 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).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 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 · 19.13 Matrix Exponentiation and Linear Recurrence Acceleration · 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: #324 · #762 · #746
Concepts: combinatorics number-theory brute-force-reduction
Likely techniques: hashing matrix-exponentiation
Learning mode
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Understand the problem
- What exactly is the input to problem 669? Is it a bound (99194853094755497), 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 constraint does the bound 99194853094755497 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 99194853094755497?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 99194853094755497 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?
Predict & plan (before you code)
- Predict the strategy: in one sentence, what will your solution do? (The classification says combinatorics / matrix-exponentiation - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 99194853094755497, 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 - then run it. A surprise here is worth more than an hour of debugging later.
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Optimization
You have a correct answer. That is the start of the learning, not the end.
- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
- Reduce memory. Can you stream, or keep only the last k states?
- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
- Prove the optimized version computes the same thing.
- Compare two implementations and time them.
Explain it
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 combinatorics structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the matrix-exponentiation idea faster? Which words in the statement were pointing at it, and did you notice them?
What was the bug that cost you the most time, and what CLASS of bug was it (off-by-one, definition misread, precision, state under-specified)?
How would your solution change if the bound 99194853094755497 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?
Which problem you have already solved is this most similar to, and what is the shared skeleton - is it really 'matrix-exponentiation' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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Low confidence schedules this problem for spaced review, even if you solved it.
Mastery check
- 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: #324 · #762 · #746
- Transfer: where else does this technique appear? Name a lesson and a real computational setting.
- Spaced re-attempt: come back after the review interval and re-solve it with no hints.