#494 - Collatz Prefix Families
The Collatz sequence is defined as: \(a_{i+1} = \left\{ \large{\frac {a_i} 2 \atop 3 a_i+1} {\text{if }a_i\text{ is even} \atop \text{if }a_i\text{ is odd}} \right.\).
The Collatz conjecture states that starting from any positive integer, the sequence eventually reaches the cycle \(1,4,2,1, \dots\).
We shall define the sequence prefix \(p(n)\) for the Collatz sequence starting with \(a_1 = n\) as the sub-sequence of all numbers not a power of \(2\) (\(2^0=1\) is considered a power of \(2\) for this problem). For example:
\(p(13) = \{13, 40, 20, 10, 5\}\)
\(p(8) = \{\}\)
Any number invalidating the conjecture would have an infinite length sequence prefix.
Let \(S_m\) be the set of all sequence prefixes of length \(m\). Two sequences \(\{a_1, a_2, \dots, a_m\}\) and \(\{b_1, b_2, \dots, b_m\}\) in \(S_m\) are said to belong to the same prefix family if \(a_i \lt a_j\) if and only if \(b_i \lt b_j\) for all \(1 \le i,j \le m\).
For example, in \(S_4\), \(\{6, 3, 10, 5\}\) is in the same family as \(\{454, 227, 682, 341\}\), but not \(\{113, 340, 170, 85\}\).
Let \(f(m)\) be the number of distinct prefix families in \(S_m\).
You are given \(f(5) = 5\), \(f(10) = 55\), \(f(20) = 6771\).
Find \(f(90)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=494. Published Saturday, 20th December 2014, 07:00 pm. Solved by 269 members at time of mirroring.
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Algorithmic Development. Graph/state-space search transfers to routing, dependency resolution, and execution-path optimisation (Phases 16, 17).
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Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.6 Recurrence Relations and Generating Functions · 2.7 Sequences, Series, Convergence, and Power Series
Recommended stepping-stone problems: #908 · #308 · #977
Concepts: sequences-series brute-force-reduction
Likely techniques: bfs-dfs hashing
Learning mode
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Understand the problem
- What exactly is the input to problem 494? Is it a bound (6771), 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 p(n), p(13) 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 6771?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 6771 and the cost of testing one.
- Which sequences-series 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 sequences-series / bfs-dfs - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 6771, 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: "For example: p(13) = \13, 40, 20, 10, 5\ p(8) = Any number invalidating the conjecture would have an infinite length sequence prefix.") - 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 sequences-series structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bfs-dfs 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 6771 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 'bfs-dfs' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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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: #908 · #308 · #977
- 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.