#327 - Rooms of Doom
A series of three rooms are connected to each other by automatic doors.
Each door is operated by a security card. Once you enter a room the door automatically closes and that security card cannot be used again. A machine at the start will dispense an unlimited number of cards, but each room (including the starting room) contains scanners and if they detect that you are holding more than three security cards or if they detect an unattended security card on the floor, then all the doors will become permanently locked. However, each room contains a box where you may safely store any number of security cards for use at a later stage.
If you simply tried to travel through the rooms one at a time then as you entered room 3 you would have used all three cards and would be trapped in that room forever!
However, if you make use of the storage boxes, then escape is possible. For example, you could enter room 1 using your first card, place one card in the storage box, and use your third card to exit the room back to the start. Then after collecting three more cards from the dispensing machine you could use one to enter room 1 and collect the card you placed in the box a moment ago. You now have three cards again and will be able to travel through the remaining three doors. This method allows you to travel through all three rooms using six security cards in total.
It is possible to travel through six rooms using a total of \(123\) security cards while carrying a maximum of \(3\) cards.
Let \(C\) be the maximum number of cards which can be carried at any time.
Let \(R\) be the number of rooms to travel through.
Let \(M(C,R)\) be the minimum number of cards required from the dispensing machine to travel through \(R\) rooms carrying up to a maximum of \(C\) cards at any time.
For example, \(M(3,6)=123\) and \(M(4,6)=23\).
And, \(\sum M(C, 6) = 146\) for \(3 \le C \le 4\).
You are given that \(\sum M(C,10)=10382\) for \(3 \le C \le 10\).
Find \(\sum M(C,30)\) for \(3 \le C \le 40\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=327. Published Saturday, 5th March 2011, 07:00 pm. Solved by 1,316 members at time of mirroring.
Why this is useful
Algorithmic Development. Graph/state-space search transfers to routing, dependency resolution, and execution-path optimisation (Phases 16, 17).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.6 Recurrence Relations and Generating Functions · 2.7 Sequences, Series, Convergence, and Power Series
Recommended stepping-stone problems: #51 · #67 · #107
Concepts: graph-theory sequences-series
Likely techniques: bfs-dfs
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 327? Is it a bound (40), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single sum.
- 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 M(C,R), M(3,6) 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 40?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 40 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 / bfs-dfs - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 40, 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 (the statement says: "For example, you could enter room 1 using your first card, place one card in the storage box, and use your third card to exit the room back to the start.") - 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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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 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 40 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: #51 · #67 · #107
- 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.