#988 - Non-attacking Frogs
Frogs can be placed on the real number line at integer locations. Given coprime positive integers \((a,b)\), each frog has the ability to make jumps of distances \(a\) or \(b\) in the positive direction.
Two frogs placed at \(m\) and \(n\), \(m<n\), are attacking if the frog at \(m\) can hop to \(n\) with some series of jumps. For example if \((a,b)=(3,5)\), frogs placed at \(0\) and \(11\) are attacking as the former can make two jumps of \(3\) and one jump of \(5\) to reach \(11\). However, frogs placed at \(4\) and \(11\) are non-attacking.
A non-attacking configuration is a placement of any number of frogs such that:
- one frog is placed at \(0\);
- all other frogs are placed at distinct positive integers;
- no two frogs are attacking.
Define \(F(a,b)\) to be sum of the integer locations of every frog, summing over all non-attacking configurations. For example if \((a,b)=(3,5)\) there are seven non-attacking configurations: \[\{0\}\quad\quad\{0,1\}\quad\quad\{0,2\}\quad\quad\{0,4\}\quad\quad\{0,7\}\quad\quad\{0,1,2\}\quad\quad\{0,2,4\} \]giving \(F(3,5)=23\).
You are also given \(F(5,13)=16336\).
Find \(F(19,53)\).
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=988. Published Sunday, 15th March 2026, 07:00 am. Solved by 169 members at time of mirroring.
Why this is useful
Algorithmic Development. Optimal substructure and state-space reasoning are exactly how American-option pricing and optimal execution are solved (Phases 13, 16).
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.8 Bit Manipulation and State Compression · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.2 Primes, Sieves, and Integer Factorization · 2.7 Sequences, Series, Convergence, and Power Series
Recommended stepping-stone problems: #934 · #734 · #754
Concepts: number-theory sequences-series brute-force-reduction
Likely techniques: bfs-dfs bitmask-dp gcd-euclid hashing
Learning mode
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Understand the problem
- What exactly is the input to problem 988? Is it a bound (19), 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(a,b), F(3,5) 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 19?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 19 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 / bitmask-dp - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 19, 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: "For example if (a,b)=(3,5), frogs placed at 0 and 11 are attacking as the former can make two jumps of 3 and one jump of 5 to reach 11.") - 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.
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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 number-theory structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the bitmask-dp 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 19 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 'bitmask-dp' 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: #934 · #734 · #754
- 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.