#255 - Rounded Square Roots
We define the rounded-square-root of a positive integer \(n\) as the square root of \(n\) rounded to the nearest integer.
The following procedure (essentially Heron's method adapted to integer arithmetic) finds the rounded-square-root of \(n\):
Let \(d\) be the number of digits of the number \(n\).
If \(d\) is odd, set \(x_0 = 2 \times 10^{(d-1)/2}\).
If \(d\) is even, set \(x_0 = 7 \times 10^{(d-2)/2}\).
Repeat:
until \(x_{k+1} = x_k\).
As an example, let us find the rounded-square-root of \(n = 4321\).
\(n\) has \(4\) digits, so \(x_0 = 7 \times 10^{(4-2)/2} = 70\).
\[x_1 = \Biggl\lfloor{\dfrac{70 + \lceil{4321 / 70}\rceil}{2}}\Biggr\rfloor = 66\]
\[x_2 = \Biggl\lfloor{\dfrac{66 + \lceil{4321 / 66}\rceil}{2}}\Biggr\rfloor = 66\]
Since \(x_2 = x_1\), we stop here.
So, after just two iterations, we have found that the rounded-square-root of \(4321\) is \(66\) (the actual square root is \(65.7343137\cdots\)).
The number of iterations required when using this method is surprisingly low.
For example, we can find the rounded-square-root of a \(5\)-digit integer (\(10\,000 \le n \le 99\,999\)) with an average of \(3.2102888889\) iterations (the average value was rounded to \(10\) decimal places).
Using the procedure described above, what is the average number of iterations required to find the rounded-square-root of a \(14\)-digit number (\(10^{13} \le n \lt 10^{14}\))?
Give your answer rounded to \(10\) decimal places.
Note: The symbols \(\lfloor x \rfloor\) and \(\lceil x \rceil\) represent the floor functionthe largest integer not greater than \(x\) and ceiling functionthe smallest integer not less than \(x\) respectively.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=255. Published Friday, 11th September 2009, 09:00 pm. Solved by 1,053 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).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.1 Sets, Functions, and Relations · 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 · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 2.1 Functions, Limits, and Continuity · 2.7 Sequences, Series, Convergence, and Power Series · 2.4 Taylor Series and Local Approximation · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Recommended stepping-stone problems: #714 · #469 · #794
Concepts: algebra geometry numerical-methods sequences-series brute-force-reduction
Likely techniques: precision-control
Learning mode
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Understand the problem
- What exactly is the input to problem 255? Is it a bound (10^14), 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 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 10^14 impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly 10^14?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10^14 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 / precision-control - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10^14, 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: "As an example, let us find the rounded-square-root of n = 4321.") - 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 precision-control 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 10^14 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 'precision-control' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
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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: #714 · #469 · #794
- 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.