#555 - McCarthy 91 Function
The McCarthy 91 function is defined as follows: \[ M_{91}(n) = \begin{cases} n - 10 & \text{if } n > 100 \\ M_{91}(M_{91}(n+11)) & \text{if } 0 \leq n \leq 100 \end{cases} \]
We can generalize this definition by abstracting away the constants into new variables: \[ M_{m,k,s}(n) = \begin{cases} n - s & \text{if } n > m \\ M_{m,k,s}(M_{m,k,s}(n+k)) & \text{if } 0 \leq n \leq m \end{cases} \]
This way, we have \(M_{91} = M_{100,11,10}\).
Let \(F_{m,k,s}\) be the set of fixed points of \(M_{m,k,s}\). That is, \[F_{m,k,s}= \left\{ n \in \mathbb{N} \, | \, M_{m,k,s}(n) = n \right\}\]
For example, the only fixed point of \(M_{91}\) is \(n = 91\). In other words, \(F_{100,11,10}= \{91\}\).
Now, define \(SF(m,k,s)\) as the sum of the elements in \(F_{m,k,s}\) and let \(S(p,m) = \displaystyle \sum_{1 \leq s < k \leq p}{SF(m,k,s)}\).
For example, \(S(10, 10) = 225\) and \(S(1000, 1000)=208724467\).
Find \(S(10^6, 10^6)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=555. Published Sunday, 10th April 2016, 04:00 am. Solved by 856 members at time of mirroring.
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Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization
Recommended stepping-stone problems: #191 · #346 · #679
Concepts: string-processing
Likely techniques: hashing
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Understand the problem
- What exactly is the input to problem 555? Is it a bound (10^6), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single exact integer.
- 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(n), S(p,m) 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 10^6?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^6 and the cost of testing one.
- Which string-processing 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 string-processing / hashing - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10^6, 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, the only fixed point of M_91 is n = 91.") - then run it. A surprise here is worth more than an hour of debugging later.
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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 string-processing structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the hashing idea faster? Which words in the statement were pointing at it, and did you notice them?
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How would your solution change if the bound 10^6 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 'hashing' underneath?
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- 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: #191 · #346 · #679
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