#55 - Lychrel Numbers
If we take \(47\), reverse and add, \(47 + 74 = 121\), which is palindromic.
Not all numbers produce palindromes so quickly. For example,
\[\begin{align} 349 + 943 &= 1292\\ 1292 + 2921 &= 4213\\ 4213 + 3124 &= 7337 \end{align}\]That is, \(349\) took three iterations to arrive at a palindrome.
Although no one has proved it yet, it is thought that some numbers, like \(196\), never produce a palindrome. A number that never forms a palindrome through the reverse and add process is called a Lychrel number. Due to the theoretical nature of these numbers, and for the purpose of this problem, we shall assume that a number is Lychrel until proven otherwise. In addition you are given that for every number below ten-thousand, it will either (i) become a palindrome in less than fifty iterations, or, (ii) no one, with all the computing power that exists, has managed so far to map it to a palindrome. In fact, \(10677\) is the first number to be shown to require over fifty iterations before producing a palindrome: \(4668731596684224866951378664\) (\(53\) iterations, \(28\)-digits).
Surprisingly, there are palindromic numbers that are themselves Lychrel numbers; the first example is \(4994\).
How many Lychrel numbers are there below ten-thousand?
NOTE: Wording was modified slightly on 24 April 2007 to emphasise the theoretical nature of Lychrel numbers.
Problem text © Project Euler, licensed under CC BY-NC-SA 4.0. Original: projecteuler.net/problem=55. Published Friday, 24th October 2003, 06:00 pm. Solved by 59,809 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.
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Prerequisites
Lessons that prepare you:
1.3 Proof Techniques: Direct, Contrapositive, and Contradiction · 19.14 Computational Complexity, Feasibility Estimation, and Proving Algorithms Correct · 19.1 Divisibility, GCD, and the Euclidean Algorithm · 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
Concepts: number-theory sequences-series brute-force-reduction
Likely techniques: modular-exponentiation
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- What exactly is the input to problem 55? Is it a bound (ten-thousand), a supplied dataset, or a definition you must generate from?
- What is the required output - restate it precisely: a single count.
- Write out, in your own words, the definition of lychrel number as the statement gives it. Which integers/objects are excluded by that definition?
- What constraint does the bound ten-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 ten-thousand?
- Why is brute force hard HERE specifically? Estimate the number of candidates implied by ten-thousand 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?
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- Predict the strategy: in one sentence, what will your solution do? (The classification says number-theory / modular-exponentiation - do you agree, and why?)
- Predict the complexity of your intended method in terms of N = ten-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, 349 + 943 = 1292 1292 + 2921 = 4213 4213 + 3124 = 7337 That is, 349 took three iterations to arrive at a palindrome.") - then run it. A surprise here is worth more than an hour of debugging later.
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