Project Euler Solution 55: Lychrel numbers

In Project Euler Problem 55: Lychrel numbers we find another curious number property builds on palindromes.

If we take 47, reverse and add, 47 + 74 = 121, which is palindromic.

Not all numbers produce palindromes so quickly. For example,

349 + 943 = 1292,
1292 + 2921 = 4213
4213 + 3124 = 7337

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.

We can write a test from the examples:

def test_is_lychrel() -> None:
    assert is_lychrel(47)
    assert is_lychrel(349)
    assert not is_lychrel(196)

In order to implement this, we need a function which does this reversing and adding. We can have another test:

def test_reverse_and_add() -> None:
    assert reverse_and_add(47) == 121

Implementing this function is easy with the string manipulations of Python:

def reverse_and_add(number: int) -> int:
    return number + int(str(number)[::-1])

Then we can write a predicate:

def is_lychrel(number: int) -> int:
    for iteration in range(50):
        number = reverse_and_add(number)
        if is_palindrome(number):
            return False
    else:
        return True

I have used function is_palindrome from Solution 36: Double-base palindromes.

With that predicate it is easy to just go through all the numbers and see how many we find.

def solution() -> None:
    lychrel_numbers = list(filter(is_lychrel, range(1, 10_000)))
    return len(lychrel_numbers)

That runs in 31 ms, so it seems fast enough. No deep insight needed in order to solve this one.