2015-04-28

# Problem 055: Lychrel numbers

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.

Solution:
v                           v-1\+g02<
0          v/+55\+g01\p01%+<        \$  v/+55\+g01\p01%+<
>"}P"*>::0\>:!#v_\55+*\:55 ^>:!#v_\::0\>:!#v_\55+*\:55 ^
>\$ .@      >\$+38*   #v  ^#   <  |-p02:\$<
^_^#!:-1             <\+1\\$\$<^\$\$<
Start
??
Pause
Reset
Output:
Stack:   (0)

Explanation:

Not much to say about this one. I reuse the isPalindrome code from P-36 and bruteforce through all the numbers.
I think it would be really useful to cache the intermediate results - but our befunge space is too small for such data structures :/

 Interpreter steps: 10 470 329 Execution time (BefunExec): 2.22s (4.73 MHz) Program size: 56 x 5 (fully conform befunge-93) Solution: 249 Solved at: 2015-04-28

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