## Question

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

Not all numbers produce palindromes so quickly. For example,

\displaystyle \begin{aligned} 349 + 943 & = 1292 \\ 1292 + 2921 & = 4213 \\ 4213 + 3124 & = 7337 \end{aligned}

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.

palindrome :: String -> Bool
lychrel n = not $any palindrome$ take 50 $tail (iterate (\x -> show$ (read x) + (read $reverse x)) (show n)) main :: IO () main = print$ length $filter lychrel [1..10000] $ ghc -O2 -o lychrel lychrel.hs
$time ./lychrel real 0m0.295s user 0m0.288s sys 0m0.000s ## Python #!/usr/bin/env python def is_palindrome(s): return s == ''.join(reversed(s)) def is_lychrel(n): s = str(n) i = 0 done = False while not done: if i > 50: return True s = str(int(s) + int(''.join(reversed(s)))) i += 1 if is_palindrome(s): done = True return False def main(): count = 0 for n in range(10000): if is_lychrel(n): count += 1 print(count) if __name__ == "__main__": main() $ time python3 lychrel.py
sys    0m0.004s