#74 - Digit Factorial Chains
The number \(145\) is well known for the property that the sum of the factorial of its digits is equal to \(145\): \[1! + 4! + 5! = 1 + 24 + 120 = 145.\]
Perhaps less well known is \(169\), in that it produces the longest chain of numbers that link back to \(169\); it turns out that there are only three such loops that exist:
\[\begin{align} &169 \to 363601 \to 1454 \to 169\\ &871 \to 45361 \to 871\\ &872 \to 45362 \to 872 \end{align}\]It is not difficult to prove that EVERY starting number will eventually get stuck in a loop. For example,
\[\begin{align} &69 \to 363600 \to 1454 \to 169 \to 363601 (\to 1454)\\ &78 \to 45360 \to 871 \to 45361 (\to 871)\\ &540 \to 145 (\to 145) \end{align}\]Starting with \(69\) produces a chain of five non-repeating terms, but the longest non-repeating chain with a starting number below one million is sixty terms.
How many chains, with a starting number below one million, contain exactly sixty non-repeating terms?
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=74. Published Friday, 16th July 2004, 06:00 pm. Solved by 30,375 members at time of mirroring.
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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 · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 19.7 Dynamic Programming: Memoization and Tabulation · 19.4 Exact Arithmetic: Big Integers, Rationals, and Floating-Point Traps · 19.3 Modular Arithmetic, Inverses, and Fast Exponentiation · 19.6 Recurrence Relations and Generating Functions · 19.2 Primes, Sieves, and Integer Factorization
Concepts: dynamic-programming number-theory
Likely techniques: big-integer
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Understand the problem
- What exactly is the input to problem 74? Is it a bound (one million), 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 one million impose, and is it inclusive or exclusive?
- What are the edge cases: the smallest legal object, zero/one, ties, and the boundary at exactly one million?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by one million and the cost of testing one.
- Which dynamic-programming 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 dynamic-programming / big-integer - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = one million, 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.
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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 dynamic-programming structure do the real work? Name the single observation that collapsed the search space.
Could you have reached the big-integer 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 one million 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 'big-integer' 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
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