#236 - Luxury Hampers
Suppliers 'A' and 'B' provided the following numbers of products for the luxury hamper market:
| Product | 'A' | 'B' |
|---|---|---|
| Beluga Caviar | 5248 | 640 |
| Christmas Cake | 1312 | 1888 |
| Gammon Joint | 2624 | 3776 |
| Vintage Port | 5760 | 3776 |
| Champagne Truffles | 3936 | 5664 |
Although the suppliers try very hard to ship their goods in perfect condition, there is inevitably some spoilage - i.e. products gone bad.
The suppliers compare their performance using two types of statistic:
- The five per-product spoilage rates for each supplier are equal to the number of products gone bad divided by the number of products supplied, for each of the five products in turn.
- The overall spoilage rate for each supplier is equal to the total number of products gone bad divided by the total number of products provided by that supplier.
To their surprise, the suppliers found that each of the five per-product spoilage rates was worse (higher) for 'B' than for 'A' by the same factor (ratio of spoilage rates), m>1; and yet, paradoxically, the overall spoilage rate was worse for 'A' than for 'B', also by a factor of m.
There are thirty-five m>1 for which this surprising result could have occurred, the smallest of which is 1476/1475.
What's the largest possible value of m?
Give your answer as a fraction reduced to its lowest terms, in the form u/v.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=236. Published Saturday, 14th March 2009, 09:00 am. Solved by 1,133 members at time of mirroring.
Why this is useful
Optimization. The transferable skill is replacing infeasible enumeration with a mathematical reduction - the core move in calibration and large-scale computation (Phases 10, 13).
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 · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps
Recommended stepping-stone problems: #783 · #933 · #898
Concepts: game-theory brute-force-reduction
Likely techniques: exact-rational
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 236? Is it a bound (1476), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer. Required format: Give your answer as a fraction reduced to its lowest terms, in the form u / v.
- 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 1476 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1476?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1476 and the cost of testing one.
- Which game-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 game-theory / exact-rational - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1476, 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 game-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the exact-rational 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 1476 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 'exact-rational' 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: #783 · #933 · #898
- 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.