#726 - Falling Bottles
Consider a stack of bottles of wine. There are \(n\) layers in the stack with the top layer containing only one bottle and the bottom layer containing \(n\) bottles. For \(n=4\) the stack looks like the picture below.
The collapsing process happens every time a bottle is taken. A space is created in the stack and that space is filled according to the following recursive steps:
- No bottle touching from above: nothing happens. For example, taking \(F\).
- One bottle touching from above: that will drop down to fill the space creating another space. For example, taking \(D\).
- Two bottles touching from above: one will drop down to fill the space creating another space. For example, taking \(C\).
This process happens recursively; for example, taking bottle \(A\) in the diagram above. Its place can be filled with either \(B\) or \(C\). If it is filled with \(C\) then the space that \(C\) creates can be filled with \(D\) or \(E\). So there are 3 different collapsing processes that can happen if \(A\) is taken, although the final shape (in this case) is the same.
Define \(f(n)\) to be the number of ways that we can take all the bottles from a stack with \(n\) layers. Two ways are considered different if at any step we took a different bottle or the collapsing process went differently.
You are given \(f(1) = 1\), \(f(2) = 6\) and \(f(3) = 1008\).
Also define \[S(n) = \sum_{k=1}^n f(k).\]
Find \(S(10^4)\) and give your answer modulo \(1\,000\,000\,033\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=726. Published Saturday, 19th September 2020, 08:00 pm. Solved by 235 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.2 Primes, Sieves, and Integer Factorization · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #746 · #960 · #997
Concepts: combinatorics number-theory brute-force-reduction
Likely techniques: memoization
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 726? Is it a bound (1000000033), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single residue (the answer is reduced modulo a given number, so keep everything in modular arithmetic from the start). Required format: give your answer modulo 1000000033.
- 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 f(n), f(1) 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 1000000033?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1000000033 and the cost of testing one.
- Which combinatorics 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 combinatorics / memoization - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1000000033, 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, taking F.") - 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 combinatorics structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the memoization 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 1000000033 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 'memoization' 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: #746 · #960 · #997
- 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.