#368 - A Kempner-like Series
The harmonic series \(1 + \frac 1 2 + \frac 1 3 + \frac 1 4 + \cdots\) is well known to be divergent.
If we however omit from this series every term where the denominator has a \(9\) in it, the series remarkably enough converges to approximately \(22.9206766193\).
This modified harmonic series is called the Kempner series.
Let us now consider another modified harmonic series by omitting from the harmonic series every term where the denominator has \(3\) or more equal consecutive digits.
One can verify that out of the first \(1200\) terms of the harmonic series, only \(20\) terms will be omitted.
These \(20\) omitted terms are:
\[\frac 1 {111}, \frac 1 {222}, \frac 1 {333}, \frac 1 {444}, \frac 1 {555}, \frac 1 {666}, \frac 1 {777}, \frac 1 {888}, \frac 1 {999}, \frac 1 {1000}, \frac 1 {1110},\] \[\frac 1 {1111}, \frac 1 {1112}, \frac 1 {1113}, \frac 1 {1114}, \frac 1 {1115}, \frac 1 {1116}, \frac 1 {1117}, \frac 1 {1118}, \frac 1 {1119}.\]
This series converges as well.
Find the value the series converges to.
Give your answer rounded to \(10\) digits behind the decimal point.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=368. Published Sunday, 22nd January 2012, 01:00 am. Solved by 633 members at time of mirroring.
Why this is useful
Numerical Computing. Precision, conditioning, and numerical iteration transfer directly to pricing engines and model calibration (Phases 5, 13, 15).
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Prerequisites
Lessons that prepare you:
19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.6 Recurrence Relations and Generating Functions · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Recommended stepping-stone problems: #863 · #398 · #796
Concepts: numerical-methods sequences-series brute-force-reduction
Likely techniques: exact-rational
Learning mode
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Understand the problem
- What exactly is the input to problem 368? Is it a bound (1200), 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 rounded to 10 digits behind the decimal point.
- Write out, in your own words, the definition of kempner series as the statement gives it. Which integers/objects are excluded by that definition?
- What constraint does the bound 1200 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1200?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1200 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 / exact-rational - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1200, 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.
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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 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 1200 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.
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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: #863 · #398 · #796
- 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.