#597 - Torpids
- A division consists of \(n\) boats (typically 13), placed in order based on past performance.
- All boats within a division start at 40 metre intervals along the river, in order with the highest-placed boat starting furthest upstream.
- The boats all start rowing simultaneously, upstream, trying to catch the boat in front while avoiding being caught by boats behind.
- Each boat continues rowing until either it reaches the finish line or it catches up with ("bumps") a boat in front.
- The finish line is a distance \(L\) metres (the course length, in reality about 1800 metres) upstream from the starting position of the lowest-placed boat. (Because of the staggered starting positions, higher-placed boats row a slightly shorter course than lower-placed boats.)
- When a "bump" occurs, the "bumping" boat takes no further part in the race. The "bumped" boat must continue, however, and may even be "bumped" again by boats that started two or more places behind it.
-
After the race, boats are assigned new places within the division, based on the bumps that occurred. Specifically, for any boat \(A\) that started in a lower place than \(B\), then \(A\) will be placed higher than \(B\) in the new order if and only if one of the following occurred:
- \(A\) bumped \(B\) directly
- \(A\) bumped another boat that went on to bump \(B\)
- \(A\) bumped another boat, that bumped yet another boat, that bumped \(B\)
- etc
Suppose that, in a particular race, each boat \(B_j\) rows at a steady speed \(v_j = -\)log\(X_j\) metres per second, where the \(X_j\) are chosen randomly (with uniform distribution) between 0 and 1, independently from one another. These speeds are relative to the riverbank: you may disregard the flow of the river.
Let \(p(n,L)\) be the probability that the new order is an even permutation of the starting order, when there are \(n\) boats in the division and \(L\) is the course length.
For example, with \(n=3\) and \(L=160\), labelling the boats as \(A\),\(B\),\(C\) in starting order with \(C\) highest, the different possible outcomes of the race are as follows:
| Bumps occurring | New order | Permutation | Probability |
|---|---|---|---|
| none | \(A\), \(B\), \(C\) | even | \(4/15\) |
| \(B\) bumps \(C\) | \(A\), \(C\), \(B\) | odd | \(8/45\) |
| \(A\) bumps \(B\) | \(B\), \(A\), \(C\) | odd | \(1/3\) |
| \(B\) bumps \(C\), then \(A\) bumps \(C\) | \(C\), \(A\), \(B\) | even | \(4/27\) |
| \(A\) bumps \(B\), then \(B\) bumps \(C\) | \(C\), \(B\), \(A\) | odd | \(2/27\) |
Therefore, \(p(3,160) = 4/15 + 4/27 = 56/135\).
You are also given that \(p(4,400)=0.5107843137\), rounded to 10 digits after the decimal point.
Find \(p(13,1800)\) rounded to 10 digits after the decimal point.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=597. Published Sunday, 2nd April 2017, 07:00 am. Solved by 206 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:
1.1 Sets, Functions, and Relations · 13.2 Monte Carlo Estimation and Error Analysis · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.11 Integer Partitions and Counting Structures · 2.1 Functions, Limits, and Continuity · 2.4 Taylor Series and Local Approximation · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 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: #901 · #367 · #973
Concepts: algebra combinatorics numerical-methods probability brute-force-reduction
Likely techniques: bfs-dfs
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- What exactly is the input to problem 597? Is it a bound (13), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a real number quoted to a stated precision, so the whole computation must control rounding error.
- 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,L), p(3,160) 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 13?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 13 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?
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- 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 = 13, 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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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 13 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?
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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: #901 · #367 · #973
- 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.