# Lychrel numbers (Euler Problem 55)¶

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https://projecteuler.net/problem=55

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.

In [1]:

    def get_digits(n):
        d = []
        while n:
            d.append(n % 10)
            n //= 10
        return d

    def is_pilandrome(n):
        ds = get_digits(n)
        len_ds = len(ds)
        if len_ds < 2:
            return True
        for i in range(0, len_ds // 2):
            if ds[i] != ds[len_ds - i - 1]:
                return False
        return True

    assert(is_pilandrome(1337) == False)
    assert(is_pilandrome(1331))
    assert(is_pilandrome(131))
    assert(is_pilandrome(132) == False)

    def get_digit_inverse(n):
        ds = get_digits(n)
        base = 1
        i = 0
        for d in ds[::-1]:
            i += (base * d)
            base *= 10
        return i

    assert(get_digit_inverse(47) == 74)
    assert(get_digit_inverse(47) == 74)

In [2]:

    def is_not_lychrel(n, iterations=50):
        for i in range(0, iterations):
            n = n + get_digit_inverse(n)
            if is_pilandrome(n):
                return (i + 1)
        return 0

    assert(is_not_lychrel(47) == 1)
    assert(is_not_lychrel(349) == 3)
    assert(is_not_lychrel(10677, 100) == 53)

In [3]:

    lychrels = [n for n in range(1, 10000) if is_not_lychrel(n) == 0]
    s = len(lychrels)
    print(s)
    assert(s == 249)

    249

In [ ]: