Problem Description

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.

Analysis

Our bounds are specified by the problem’s description; specifically to search a range from 10 to 9999 and only check a depth to 49 iterations.

The function is_lychrel() takes the candidate and adds itself to its reverse. If this sum is a palindrome then the it’s not a Lychrel number and we return a false result. This process is repeated up to 49 times.

Solution

Runs < 1 second in Python.

from Euler import is_palindromic
 
def is_lychrel(n):
  for i in xrange(0, 50):
    n = n + int(str(n)[::-1])
    if is_palindromic(n): return False
  return True
 
print sum(1 for n in xrange(10, 10000) if is_lychrel(n))

Comments

More information on the Euler module can be found on the tools page.
for N<100,000 and up to 500 iterations produces 6091 Lychrel numbers