#481 - Chef Showdown
A group of chefs (numbered #\(1\), #\(2\), etc) participate in a turn-based strategic cooking competition. On each chef's turn, he/she cooks up a dish to the best of his/her ability and gives it to a separate panel of judges for taste-testing. Let \(S(k)\) represent chef #\(k\)'s skill level (which is publicly known). More specifically, \(S(k)\) is the probability that chef #\(k\)'s dish will be assessed favorably by the judges (on any/all turns). If the dish receives a favorable rating, then the chef must choose one other chef to be eliminated from the competition. The last chef remaining in the competition is the winner.
The game always begins with chef #\(1\), with the turn order iterating sequentially over the rest of the chefs still in play. Then the cycle repeats from the lowest-numbered chef. All chefs aim to optimize their chances of winning within the rules as stated, assuming that the other chefs behave in the same manner. In the event that a chef has more than one equally-optimal elimination choice, assume that the chosen chef is always the one with the next-closest turn.
Define \(W_n(k)\) as the probability that chef #\(k\) wins in a competition with \(n\) chefs. If we have \(S(1) = 0.25\), \(S(2) = 0.5\), and \(S(3) = 1\), then \(W_3(1) = 0.29375\).
Going forward, we assign \(S(k) = F_k/F_{n+1}\) over all \(1 \le k \le n\), where \(F_k\) is a Fibonacci number: \(F_k = F_{k-1} + F_{k-2}\) with base cases \(F_1 = F_2 = 1\). Then, for example, when considering a competition with \(n = 7\) chefs, we have \(W_7(1) = 0.08965042\), \(W_7(2) = 0.20775702\), \(W_7(3) = 0.15291406\), \(W_7(4) = 0.14554098\), \(W_7(5) = 0.15905291\), \(W_7(6) = 0.10261412\), and \(W_7(7) = 0.14247050\), rounded to \(8\) decimal places each.
Let \(E(n)\) represent the expected number of dishes cooked in a competition with \(n\) chefs. For instance, \(E(7) = 42.28176050\).
Find \(E(14)\) rounded to \(8\) decimal places.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=481. Published Sunday, 21st September 2014, 04:00 am. Solved by 316 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:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 10.4 Algorithms: Gradient Descent and Newton's Method · 10.1 Convex Sets and Convex Functions · 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 · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.13 Matrix Exponentiation and Linear Recurrence Acceleration · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.11 Integer Partitions and Counting Structures · 19.6 Recurrence Relations and Generating Functions · 19.2 Primes, Sieves, and Integer Factorization · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 5.1 Floating-Point Arithmetic, Conditioning, and Stability · 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: #310 · #509 · #569
Concepts: combinatorics dynamic-programming game-theory number-theory numerical-methods optimization probability sequences-series brute-force-reduction
Likely techniques: matrix-exponentiation 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 481? Is it a bound (14), 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 do the arguments of S(k), S(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 14?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 14 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 / matrix-exponentiation - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 14, 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 instance, E(7) = 42.") - 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.
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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 game-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the matrix-exponentiation 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 14 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 'matrix-exponentiation' 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: #310 · #509 · #569
- 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.