#314 - The Mouse on the Moon
The moon has been opened up, and land can be obtained for free, but there is a catch. You have to build a wall around the land that you stake out, and building a wall on the moon is expensive. Every country has been allotted a \(\pu{500 m}\) by \(\pu{500 m}\) square area, but they will possess only that area which they wall in. \(251001\) posts have been placed in a rectangular grid with \(1\) meter spacing. The wall must be a closed series of straight lines, each line running from post to post.
The bigger countries of course have built a \(\pu{2000 m}\) wall enclosing the entire \(\pu{250 000 m^2}\) area. The Duchy of Grand Fenwick, has a tighter budget, and has asked you (their Royal Programmer) to compute what shape would get best maximum enclosed-area/wall-length ratio.
You have done some preliminary calculations on a sheet of paper.
For a \(2000\) meter wall enclosing the \(\pu{250 000 m^2}\) area the
enclosed-area/wall-length ratio is \(125\).
Although not allowed , but to get an idea if this is anything better: if you place a circle inside the square area touching the four sides the area will be equal to \(\pi \times \pu{250^2 m^2}\) and the perimeter will be \(\pi \times \pu{500 m}\), so the enclosed-area/wall-length ratio will also be \(125\).
However, if you cut off from the square four triangles with sides \(\pu{75 m}\), \(\pu{75 m}\) and \(75\pu{\sqrt 2 m}\) the total area becomes \(\pu{238750 m^2}\) and the perimeter becomes \(1400+300\pu{\sqrt 2 m}\). So this gives an enclosed-area/wall-length ratio of \(130.87\), which is significantly better.
Find the maximum enclosed-area/wall-length ratio.
Give your answer rounded to \(8\) places behind the decimal point in the form abc.defghijk.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=314. Published Sunday, 12th December 2010, 07:00 am. Solved by 620 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:
19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.6 Recurrence Relations and Generating Functions · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 3.1 Vectors, Multivariable Functions, and Level Sets · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Recommended stepping-stone problems: #210 · #456 · #904
Concepts: computational-geometry geometry numerical-methods sequences-series brute-force-reduction
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 314? Is it a bound (251001), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: one extremal object (or the value attached to it), so a search-with-pruning shape is natural. Required format: Give your answer rounded to 8 places behind the decimal point in the form abc.
- 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 251001 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 251001?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 251001 and the cost of testing one.
- Which geometry 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 geometry - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 251001, 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.
Python workbench
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Check your answer
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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 geometry structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the geometry 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 251001 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 'geometry' 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: #210 · #456 · #904
- 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.