#532 - Nanobots on Geodesics
Bob is a manufacturer of nanobots and wants to impress his customers by giving them a ball coloured by his new nanobots as a present.
His nanobots can be programmed to select and locate exactly one other bot precisely and, after activation, move towards this bot along the shortest possible path and draw a coloured line onto the surface while moving. Placed on a plane, the bots will start to move towards their selected bots in a straight line. In contrast, being placed on a ball, they will start to move along a geodesic as the shortest possible path. However, in both cases, whenever their target moves they will adjust their direction instantaneously to the new shortest possible path. All bots will move at the same speed after their simultaneous activation until each bot reaches its goal.
Now Bob places \(n\) bots on the ball (with radius \(1\)) equidistantly on a small circle with radius \(0.999\) and programs each of them to move toward the next nanobot sitting counterclockwise on that small circle. After activation, the bots move in a sort of spiral until they finally meet at one point on the ball.
Using three bots, Bob finds that every bot will draw a line of length \(2.84\), resulting in a total length of \(8.52\) for all three bots, each time rounded to two decimal places. The coloured ball looks like this:
In order to show off a little with his presents, Bob decides to use just enough bots to make sure that the line each bot draws is longer than \(1000\). What is the total length of all lines drawn with this number of bots, rounded to two decimal places?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=532. Published Sunday, 1st November 2015, 07:00 am. Solved by 367 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:
1.1 Sets, Functions, and Relations · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 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 · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.6 Recurrence Relations and Generating Functions · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 2.1 Functions, Limits, and Continuity · 2.4 Taylor Series and Local Approximation · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Recommended stepping-stone problems: #897 · #416 · #994
Concepts: algebra dynamic-programming geometry numerical-methods brute-force-reduction
Likely techniques: bfs-dfs inclusion-exclusion 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 532? Is it a bound (1000), 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 constraint does the bound 1000 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000 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 / inclusion-exclusion - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000, 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
Real Python (Pyodide) in a sandboxed Web Worker - no network, no filesystem, no DOM access. Ctrl/Cmd+Enter runs. Escape leaves the editor. Stop terminates the worker.
Check your answer
Answers are checked against a salted hash held in a separate file - not printed in this page. This prevents accidental spoilers; it is not cryptographic protection (see the build notes).
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 inclusion-exclusion 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 1000 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 'inclusion-exclusion' 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: #897 · #416 · #994
- 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.