#300 - Protein Folding
In a very simplified form, we can consider proteins as strings consisting of hydrophobic (H) and polar (P) elements, e.g. HHPPHHHPHHPH.
For this problem, the orientation of a protein is important; e.g. HPP is considered distinct from PPH. Thus, there are \(2^n\) distinct proteins consisting of \(n\) elements.
When one encounters these strings in nature, they are always folded in such a way that the number of H-H contact points is as large as possible, since this is energetically advantageous.
As a result, the H-elements tend to accumulate in the inner part, with the P-elements on the outside.
Natural proteins are folded in three dimensions of course, but we will only consider protein folding in two dimensions.
The figure below shows two possible ways that our example protein could be folded (H-H contact points are shown with red dots).
The folding on the left has only six H-H contact points, thus it would never occur naturally.
On the other hand, the folding on the right has nine H-H contact points, which is optimal for this string.
Assuming that H and P elements are equally likely to occur in any position along the string, the average number of H-H contact points in an optimal folding of a random protein string of length \(8\) turns out to be \(850 / 2^8 = 3.3203125\).
What is the average number of H-H contact points in an optimal folding of a random protein string of length \(15\)?
Give your answer using as many decimal places as necessary for an exact result.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=300. Published Saturday, 4th September 2010, 01:00 pm. Solved by 1,310 members at time of mirroring.
Why this is useful
Probability Statistics. Expectation and state-based probability reasoning underpin pricing, risk, and statistical inference (Phases 7, 11, 13).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
10.4 Algorithms: Gradient Descent and Newton's Method · 10.1 Convex Sets and Convex Functions · 13.2 Monte Carlo Estimation and Error Analysis · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.6 Recurrence Relations and Generating Functions · 2.4 Taylor Series and Local Approximation · 5.1 Floating-Point Arithmetic, Conditioning, and Stability · 7.3 Independence, Conditional Probability, and Conditional Expectation · 7.2 Expectation, Moments, and Key Inequalities · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #103 · #190 · #183
Concepts: dynamic-programming numerical-methods optimization probability string-processing brute-force-reduction
Likely techniques: hashing precision-control
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 300? Is it a bound (15), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a string (a concatenation), so its exact length and character order are part of the answer. Required format: Give your answer using as many decimal places as necessary for an exact result.
- 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 15 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 15?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 15 and the cost of testing one.
- Which optimization 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 optimization / precision-control - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 15, 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.
Scratchpad
Mathematical notes, formulas, pseudocode, hypotheses, complexity notes. Saved automatically with your progress.
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Progressive hints
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 optimization structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the precision-control 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 15 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 'precision-control' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
Self-assess (mastery is not a correct number)
You reach Mastered only when you have solved it, rated yourself at least Solid across the dimensions, and written a real explanation.
Confidence
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: #103 · #190 · #183
- 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.