#666 - Polymorphic Bacteria
- Every minute, each individual will simultaneously undergo some kind of transformation.
- Each individual \(A\) of type \(\alpha\) will, independently, do one of the following (at random with equal probability):
- clone itself, resulting in a new bacterium of type \(\alpha\) (alongside \(A\) who remains)
- split into 3 new bacteria of type \(\beta\) (replacing \(A\))
- Each individual \(B\) of type \(\beta\) will, independently, do one of the following (at random with equal probability):
- spawn a new bacterium of type \(\alpha\) (alongside \(B\) who remains)
- die
If a population starts with a single bacterium of type \(\alpha\), then it can be shown that there is a 0.07243802 probability that the population will eventually die out, and a 0.92756198 probability that the population will last forever. These probabilities are given rounded to 8 decimal places.
Now consider another species of bacteria, \(S_{k,m}\) (where \(k\) and \(m\) are positive integers), which occurs in \(k\) different types \(\alpha_i\) for \(0\le i< k\). The rules governing this species' lifecycle involve the sequence \(r_n\) defined by:
- \(r_0 = 306\)
- \(r_{n+1} = r_n^2 \bmod 10\,007\)
Every minute, for each \(i\), each bacterium \(A\) of type \(\alpha_i\) will independently choose an integer \(j\) uniformly at random in the range \(0\le j<m\). What it then does depends on \(q = r_{im+j} \bmod 5\):
- If \(q=0\), \(A\) dies.
- If \(q=1\), \(A\) clones itself, resulting in a new bacterium of type \(\alpha_i\) (alongside \(A\) who remains).
- If \(q=2\), \(A\) mutates, changing into type \(\alpha_{(2i) \bmod k}\).
- If \(q=3\), \(A\) splits into 3 new bacteria of type \(\alpha_{(i^2+1) \bmod k}\) (replacing \(A\)).
- If \(q=4\), \(A\) spawns a new bacterium of type \(\alpha_{(i+1) \bmod k}\) (alongside \(A\) who remains).
In fact, our original species was none other than \(S_{2,2}\), with \(\alpha=\alpha_0\) and \(\beta=\alpha_1\).
Let \(P_{k,m}\) be the probability that a population of species \(S_{k,m}\), starting with a single bacterium of type \(\alpha_0\), will eventually die out. So \(P_{2,2} = 0.07243802\). You are also given that \(P_{4,3} = 0.18554021\) and \(P_{10,5} = 0.53466253\), all rounded to 8 decimal places.
Find \(P_{500,10}\), and give your answer rounded to 8 decimal places.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=666. Published Saturday, 20th April 2019, 01:00 pm. Solved by 359 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:
1.1 Sets, Functions, and Relations · 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 · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.11 Integer Partitions and Counting Structures · 19.6 Recurrence Relations and Generating Functions · 2.1 Functions, Limits, and Continuity · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 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: #286 · #461 · #307
Concepts: algebra combinatorics numerical-methods probability sequences-series brute-force-reduction
Likely techniques: memoization 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 666? Is it a bound (10007), 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. Required format: give your answer rounded to 8 decimal places.
- 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 10007 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10007?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10007 and the cost of testing one.
- Which numerical-methods 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 numerical-methods / memoization - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10007, 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 numerical-methods 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 10007 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: #286 · #461 · #307
- 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.