#57 - Square Root Convergents
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
\(\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}\)
By expanding this for the first four iterations, we get:
\(1 + \frac 1 2 = \frac 32 = 1.5\)
\(1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4\)
\(1 + \frac 1 {2 + \frac 1 {2+\frac 1 2}} = \frac {17}{12} = 1.41666 \dots\)
\(1 + \frac 1 {2 + \frac 1 {2+\frac 1 {2+\frac 1 2}}} = \frac {41}{29} = 1.41379 \dots\)
The next three expansions are \(\frac {99}{70}\), \(\frac {239}{169}\), and \(\frac {577}{408}\), but the eighth expansion, \(\frac {1393}{985}\), is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
In the first one-thousand expansions, how many fractions contain a numerator with more digits than the denominator?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=57. Published Friday, 21st November 2003, 06:00 pm. Solved by 46,858 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 1.1 Sets, Functions, and Relations · 19.12 Continued Fractions, Pell Equations, and Diophantine Approximation · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 2.1 Functions, Limits, and Continuity · 2.7 Sequences, Series, Convergence, and Power Series · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.2 Linear Maps, Matrices, Rank, and the Null Space
Concepts: algebra geometry number-theory sequences-series
Likely techniques: binary-search continued-fractions exact-rational
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Understand the problem
- What exactly is the input to problem 57? Is it a bound (1393985), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- 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 1393985 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 1393985?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 1393985 and the cost of testing one.
- Which sequences-series 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 sequences-series / continued-fractions - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 1393985, 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: "It is possible to show that the square root of two can be expressed as an infinite continued fraction.") - then run it. A surprise here is worth more than an hour of debugging later.
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- Reduce the time complexity. What is the bottleneck, and what mathematical fact removes it?
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- Replace brute force with a closed form, a sieve, a recurrence, or a symmetry argument.
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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 sequences-series structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the continued-fractions 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 1393985 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 'continued-fractions' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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- Variation: change the bound (or a rule) in the statement. Does your method still work? What breaks first?
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- Related problem: explore the same concept filter in the Lab
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