Lychrel numbers

Author: Shlomi Fish

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.

Source code: prob055-shlomif.p6

#!/usr/bin/raku

use v6;

sub rsum($x)
{
    return $x + Int($x.flip());
}

sub is_palindrome($int)
{
    my $s = Str($int);
    return $s.flip eq $s;
}

sub is_lycherel($start)
{
    my $n = rsum($start);
    for 1 .. 50 -> $i
    {
        return False if is_palindrome($n);
        $n = rsum($n);
    }
    return True;
}

if (False)
{
    say is_palindrome(11);
    say rsum(13);
}
say +((1..10000).grep( { is_lycherel($_) }));