Problem 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.

def p55(C = 10000):
	res = 0
	for n in range(1, C):
		term = n
		term_rev = int(str(term)[::-1])
		for k in range(50):
			term += term_rev
			term_rev = int(str(term)[::-1])
			if term_rev == term:
				break
		else:  # loop fell through
			res += 1
	return res