#607 - Marsh Crossing
Frodo and Sam need to travel 100 leagues due East from point A to point B. On normal terrain, they can cover 10 leagues per day, and so the journey would take 10 days. However, their path is crossed by a long marsh which runs exactly South-West to North-East, and walking through the marsh will slow them down. The marsh is 50 leagues wide at all points, and the mid-point of AB is located in the middle of the marsh. A map of the region is shown in the diagram below:
The marsh consists of 5 distinct regions, each 10 leagues across, as shown by the shading in the map. The strip closest to point A is relatively light marsh, and can be crossed at a speed of 9 leagues per day. However, each strip becomes progressively harder to navigate, the speeds going down to 8, 7, 6 and finally 5 leagues per day for the final region of marsh, before it ends and the terrain becomes easier again, with the speed going back to 10 leagues per day.
If Frodo and Sam were to head directly East for point B, they would travel exactly 100 leagues, and the journey would take approximately 13.4738 days. However, this time can be shortened if they deviate from the direct path.
Find the shortest possible time required to travel from point A to B, and give your answer in days, rounded to 10 decimal places.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=607. Published Saturday, 10th June 2017, 01:00 pm. Solved by 1,980 members at time of mirroring.
Why this is useful
Numerical Computing. Precision, conditioning, and numerical iteration transfer directly to pricing engines and model calibration (Phases 5, 13, 15).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.6 Recurrence Relations and Generating Functions · 2.4 Taylor Series and Local Approximation · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Concepts: dynamic-programming numerical-methods
Likely techniques: bfs-dfs hashing precision-control
Learning mode
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Understand the problem
- What exactly is the input to problem 607? Is it a bound (10), 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. Required format: give your answer in days, rounded to 10 decimal places.
- 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 10 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10 and the cost of testing one.
- Which numerical-methods 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 numerical-methods / bfs-dfs - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10, 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
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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 numerical-methods 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 10 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.
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: explore the same concept filter in the Lab
- 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.