#42 - Coded Triangle Numbers
The \(n\)th term of the sequence of triangle numbers is given by, \(t_n = \frac12n(n+1)\); so the first ten triangle numbers are: \[1, 3, 6, 10, 15, 21, 28, 36, 45, 55, \dots\]
By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is \(19 + 11 + 25 = 55 = t_{10}\). If the word value is a triangle number then we shall call the word a triangle word.
Using words.txt (right click and 'Save Link/Target As...'), a 16K text file containing nearly two-thousand common English words, how many are triangle words?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=42. Published Friday, 25th April 2003, 06:00 pm. Solved by 82,101 members at time of mirroring.
Why this is useful
Mathematical Foundation. Exact integer reasoning and algorithmic efficiency. Foundational rigour and computational discipline rather than a direct trading application.
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 · 17.1 Python for Quants: NumPy, pandas, and Vectorization · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.10 Search: Backtracking, Branch-and-Bound, Binary Search, Meet-in-the-Middle · 19.2 Primes, Sieves, and Integer Factorization · 2.7 Sequences, Series, Convergence, and Power Series · 3.1 Vectors, Multivariable Functions, and Level Sets
Concepts: geometry number-theory sequences-series string-processing
Learning mode
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Understand the problem
- What exactly is the input to problem 42? Is it a bound (two-thousand), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- 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 two-thousand impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly two-thousand?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by two-thousand and the cost of testing one.
- Which sequences-series 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 sequences-series - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = two-thousand, 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, the word value for SKY is 19 + 11 + 25 = 55 = t_10.") - then run it. A surprise here is worth more than an hour of debugging later.
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- 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 sequences-series structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the sequences-series 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 two-thousand 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 'sequences-series' 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: explore the same concept filter in the Lab
- 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.