#585 - Nested Square Roots
Consider the term \(\small \sqrt{x+\sqrt{y}+\sqrt{z}}\) that is representing a nested square root. \(x\), \(y\) and \(z\) are positive integers and \(y\) and \(z\) are not allowed to be perfect squares, so the number below the outer square root is irrational. Still it can be shown that for some combinations of \(x\), \(y\) and \(z\) the given term can be simplified into a sum and/or difference of simple square roots of integers, actually denesting the square roots in the initial expression.
Here are some examples of this denesting:
\(\small \sqrt{3+\sqrt{2}+\sqrt{2}}=\sqrt{2}+\sqrt{1}=\sqrt{2}+1\)
\(\small \sqrt{8+\sqrt{15}+\sqrt{15}}=\sqrt{5}+\sqrt{3}\)
\(\small \sqrt{20+\sqrt{96}+\sqrt{12}}=\sqrt{9}+\sqrt{6}+\sqrt{3}-\sqrt{2}=3+\sqrt{6}+\sqrt{3}-\sqrt{2}\)
\(\small \sqrt{28+\sqrt{160}+\sqrt{108}}=\sqrt{15}+\sqrt{6}+\sqrt{5}-\sqrt{2}\)
As you can see the integers used in the denested expression may also be perfect squares resulting in further simplification.
Let F(\(n\)) be the number of different terms \(\small \sqrt{x+\sqrt{y}+\sqrt{z}}\), that can be denested into the sum and/or difference of a finite number of square roots, given the additional condition that \(0<x \le n\). That is,
\(\small \displaystyle \sqrt{x+\sqrt{y}+\sqrt{z}}=\sum_{i=1}^k s_i\sqrt{a_i}\)
with \(k\), \(x\), \(y\), \(z\) and all \(a_i\) being positive integers, all \(s_i =\pm 1\) and \(x\le n\).
Furthermore \(y\) and \(z\) are not allowed to be perfect squares.
Nested roots with the same value are not considered different, for example \(\small \sqrt{7+\sqrt{3}+\sqrt{27}}\), \(\small \sqrt{7+\sqrt{12}+\sqrt{12}}\) and \(\small \sqrt{7+\sqrt{27}+\sqrt{3}}\), that can all three be denested into \(\small 2+\sqrt{3}\), would only be counted once.
You are given that \(F(10)=17\), \(F(15)=46\), \(F(20)=86\), \(F(30)=213\) and \(F(100)=2918\) and \(F(5000)=11134074\).
Find \(F(5000000)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=585. Published Saturday, 7th January 2017, 07:00 pm. Solved by 222 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact counting underlies discrete pricing lattices, scenario enumeration, and combinatorial probability (Phase 7).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
1.1 Sets, Functions, and Relations · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.11 Integer Partitions and Counting Structures · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 2.1 Functions, Limits, and Continuity · 3.1 Vectors, Multivariable Functions, and Level Sets · 4.2 Linear Maps, Matrices, Rank, and the Null Space · 7.1 Probability Spaces, Random Variables, and Distributions
Recommended stepping-stone problems: #269 · #438 · #674
Concepts: algebra combinatorics geometry brute-force-reduction
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Understand the problem
- What exactly is the input to problem 585? Is it a bound (5000000), 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 F(n), F(10) 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 5000000?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 5000000 and the cost of testing one.
- Which algebra 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 algebra - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 5000000, 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: "You are given that F(10)=17, F(15)=46, F(20)=86, F(30)=213 and F(100)=2918 and F(5000)=11134074.") - 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?
- 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 algebra structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the algebra 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 5000000 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 'algebra' 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?
- Constraints: if the limit were 10× larger, which step fails, and what would you replace it with?
- Related problem: #269 · #438 · #674
- 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.