#985 - Telescoping Triangles
Given a triangle \(T_k\), it is sometimes possible to construct a triangle \(T_{k+1}\) inside \(T_k\) such that
- The three vertices of \(T_{k+1}\) lie one on each side of \(T_k\).
- For each side of \(T_k\), the angles formed between it and the two sides of \(T_{k+1}\) it touches are equal to each other.
Illustrated above is such a sequence of three triangles starting with \(T_0\) (in blue) having side lengths \((8,9,10)\). Then \(T_1\) is shown in green and \(T_2\) in red. However, no triangle can be drawn inside \(T_2\) that satisfies the requirements. In other words, \(T_3\) does not exist.
Amongst all integer-sided triangles \(T_0\) such that \(T_2\) exists but \(T_3\) does not exist, the smallest possible perimeter is \(10\) when \(T_0\) has side lengths \((3, 3, 4)\).
Suppose another triangle \(T_0\) has integer side lengths, and \(T_{20}\) exists, but \(T_{21}\) does not exist. What is the smallest possible perimeter of \(T_0\)?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=985. Published Saturday, 21st February 2026, 10:00 pm. Solved by 358 members at time of mirroring.
Why this is useful
Algorithmic Development. Graph/state-space search transfers to routing, dependency resolution, and execution-path optimisation (Phases 16, 17).
We classify relevance honestly - not every Euler problem is a trading application.
Prerequisites
Lessons that prepare you:
17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.9 Graph Algorithms: BFS, DFS, Dijkstra, and Minimum Spanning Trees · 19.6 Recurrence Relations and Generating Functions · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 2.7 Sequences, Series, Convergence, and Power Series · 3.1 Vectors, Multivariable Functions, and Level Sets
Recommended stepping-stone problems: #577 · #349 · #745
Concepts: geometry graph-theory sequences-series string-processing
Learning mode
Pick how much scaffolding you want. Your choice is remembered per problem.
Understand the problem
- What exactly is the input to problem 985? Is it a bound (10), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: one extremal object (or the value attached to it), so a search-with-pruning shape is natural.
- 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 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?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by 10 and the cost of testing one.
- Which geometry 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 geometry - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = 10, 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 - 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
Real Python (Pyodide) in a sandboxed Web Worker - no network, no filesystem, no DOM access. Ctrl/Cmd+Enter runs. Escape leaves the editor. Stop terminates the worker.
Check your answer
Answers are checked against a salted hash held in a separate file - not printed in this page. This prevents accidental spoilers; it is not cryptographic protection (see the build notes).
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 geometry structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the geometry 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 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 'geometry' 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: #577 · #349 · #745
- 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.