#975 - A Winding Path
Given a pair \((a,b)\) of coprime odd positive integers, define the function \[H_{a,b}(x)=\frac{1}{2}-\frac{1}{2(a+b)}\Bigl(b\cos(a\pi x)+a\cos(b\pi x)\Bigr) \]It can be seen that \(H_{a,b}(0)=0\), \(H_{a,b}(1)=1\), and \(0 < H_{a,b}(x) < 1\) for all \(x\) strictly between \(0\) and \(1\).
Given two such pairs \((a,b)\) and \((c,d)\), paths of infinitesimal width traverse the unit cube internally through every point \((x,y,z)\in [0,1]^3\) such that \(z=H_{a,b}(x)=H_{c,d}(y)\). Remarkably, it can be shown that the point \((0,0,0)\) is always connected to the opposite corner \((1,1,1)\). Furthermore, with the additional condition \(\gcd(a+b,c+d)\in\{2,4\}\), it can be shown that there is exactly one path connecting the two points.
Shown above are two examples, as viewed from above the cube. That is, we see the paths projected onto the \(xy\)-plane, with corresponding \(z\) values indicated with varying colour. In the second example some paths are coloured grey to indicate that, while they exist, they do not form part of the path from \((0,0,0)\) to \((1,1,1)\).
Define \(F(a, b, c, d)\) to be the sum of the absolute changes in height (or \(z\)-coordinate) over all uphill and downhill sections of the path from \((0,0,0)\) to \((1,1,1)\). In the first example above, the path climbs \(\approx4.00886\) over eleven uphill sections, and descends \(\approx3.00886\) over ten downhill sections, giving \(F(3,5,3,7)\approx7.01772\). You are also given \(F(7,17,9,19)\approx 26.79578\).
Let \(G(m, n)\) be the sum of \(F(p,q,p,2q-p)\) over all pairs \((p,q)\) of primes, \(m\leq p < q\leq n\). You are given \(G(3, 20)\approx463.80866\).
Find \(G(500,1000)\) giving your answer rounded to five digits after the decimal point.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=975. Published Saturday, 20th December 2025, 07:00 pm. Solved by 135 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:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.5 Combinatorics: Counting, Binomials, and Inclusion–Exclusion · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 2.4 Taylor Series and Local Approximation · 3.1 Vectors, Multivariable Functions, and Level Sets · 5.1 Floating-Point Arithmetic, Conditioning, and Stability
Recommended stepping-stone problems: #875 · #886 · #929
Concepts: geometry graph-theory number-theory numerical-methods brute-force-reduction
Likely techniques: bfs-dfs gcd-euclid inclusion-exclusion sieve
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 975? Is it a bound (500), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a real number quoted to a stated precision, so the whole computation must control rounding error.
- 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 b(x), b(0) 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 500?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 500 and the cost of testing one.
- Which number-theory 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 number-theory / inclusion-exclusion - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 500, 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 can be seen that H_a,b(0)=0, H_a,b(1)=1, and 0 < H_a,b(x) < 1 for all x strictly between 0 and 1.") - then run it. A surprise here is worth more than an hour of debugging later.
Scratchpad
Mathematical notes, formulas, pseudocode, hypotheses, complexity notes. Saved automatically with your progress.
Python workbench
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Check your answer
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Progressive hints
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 number-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the inclusion-exclusion 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 500 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 'inclusion-exclusion' underneath?
State the transferable technique in one sentence, without mentioning this problem's story at all.
Self-assess (mastery is not a correct number)
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: #875 · #886 · #929
- 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.