#653 - Frictionless Tube
Consider a horizontal frictionless tube with length \(L\) millimetres, and a diameter of 20 millimetres. The east end of the tube is open, while the west end is sealed. The tube contains \(N\) marbles of diameter 20 millimetres at designated starting locations, each one initially moving either westward or eastward with common speed \(v\).
Since there are marbles moving in opposite directions, there are bound to be some collisions. We assume that the collisions are perfectly elastic, so both marbles involved instantly change direction and continue with speed \(v\) away from the collision site. Similarly, if the west-most marble collides with the sealed end of the tube, it instantly changes direction and continues eastward at speed \(v\). On the other hand, once a marble reaches the unsealed east end, it exits the tube and has no further interaction with the remaining marbles.
To obtain the starting positions and initial directions, we use the pseudo-random sequence \(r_j\) defined by:
\(r_1 = 6\,563\,116\)
\(r_{j+1} = r_j^2 \bmod 32\,745\,673\)
The west-most marble is initially positioned with a gap of \((r_1 \bmod 1000) + 1\) millimetres between it and the sealed end of the tube, measured from the west-most point of the surface of the marble. Then, for \(2\le j\le N\), counting from the west, the gap between the \((j-1)\)th and \(j\)th marbles, as measured from their closest points, is given by \((r_j \bmod 1000) + 1\) millimetres.
Furthermore, the \(j\)th marble is initially moving eastward if \(r_j \le 10\,000\,000\), and westward if \(r_j > 10\,000\,000\).
For example, with \(N=3\), the sequence specifies gaps of 117, 432, and 173 millimetres. The marbles' centres are therefore 127, 579, and 772 millimetres from the sealed west end of the tube. The west-most marble initially moves eastward, while the other two initially move westward.
Under this setup, and with a five metre tube (\(L=5000\)), it turns out that the middle (second) marble travels 5519 millimetres before its centre reaches the east-most end of the tube.
Let \(d(L, N, j)\) be the distance in millimetres that the \(j\)th marble travels before its centre reaches the eastern end of the tube. So \(d(5000, 3, 2) = 5519\). You are also given that \(d(10\,000, 11, 6) = 11\,780\) and \(d(100\,000, 101, 51) = 114\,101\).
Find \(d(1\,000\,000\,000, 1\,000\,001, 500\,001)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=653. Published Sunday, 27th January 2019, 01:00 am. Solved by 373 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:
13.2 Monte Carlo Estimation and Error Analysis · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.7 Dynamic Programming: Memoization and Tabulation · 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 · 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: #856 · #375 · #869
Concepts: probability sequences-series brute-force-reduction
Likely techniques: bfs-dfs memoization
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- What exactly is the input to problem 653? Is it a bound (1000000000), 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 d(L,N,j) 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 1000000000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000000000 and the cost of testing one.
- Which probability 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 probability / bfs-dfs - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000000000, 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, with N=3, the sequence specifies gaps of 117, 432, and 173 millimetres.") - then run it. A surprise here is worth more than an hour of debugging later.
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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.
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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 probability 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 1000000000 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: #856 · #375 · #869
- 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.