#575 - Wandering Robots
It was quite an ordinary day when a mysterious alien vessel appeared as if from nowhere. After waiting several hours and receiving no response it is decided to send a team to investigate, of which you are included. Upon entering the vessel you are met by a friendly holographic figure, Katharina, who explains the purpose of the vessel, Eulertopia.
She claims that Eulertopia is almost older than time itself. Its mission was to take advantage of a combination of incredible computational power and vast periods of time to discover the answer to life, the universe, and everything. Hence the resident cleaning robot, Leonhard, along with his housekeeping responsibilities, was built with a powerful computational matrix to ponder the meaning of life as he wanders through a massive 1000 by 1000 square grid of rooms. She goes on to explain that the rooms are numbered sequentially from left to right, row by row. So, for example, if Leonhard was wandering around a 5 by 5 grid then the rooms would be numbered in the following way.
Many millenia ago Leonhard reported to Katharina to have found the answer and he is willing to share it with any life form who proves to be worthy of such knowledge.
Katharina further explains that the designers of Leonhard were given instructions to program him with equal probability of remaining in the same room or travelling to an adjacent room. However, it was not clear to them if this meant (i) an equal probability being split equally between remaining in the room and the number of available routes, or, (ii) an equal probability (50%) of remaining in the same room and then the other 50% was to be split equally between the number of available routes.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=575. Published Saturday, 22nd October 2016, 01:00 pm. Solved by 704 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 · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.13 Matrix Exponentiation and Linear Recurrence Acceleration · 19.11 Integer Partitions and Counting Structures · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.3 Determinants, Eigenvalues, and Eigenvectors · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 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: #425 · #215 · #393
Concepts: combinatorics geometry graph-theory linear-algebra probability 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 575? Is it a bound (1000), 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 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 graph-theory 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 graph-theory - 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 graph-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the graph-theory 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 'graph-theory' 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: #425 · #215 · #393
- 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.