Problem 55, in short, says “How many Lychrel numbers are there below ten-thousand?”

I didn’t know what a Lychrel number was until I read the explanation. Here it is:

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?

At first I made the mistake of thinking that a number was Lychrel if it does produce a palindromic number within 50 iterations. Then I realized my mistake: a number is Lychrel if it doesn’t produce a palindromic number within 50 iterations.

Here is what I came up with in Clojure to determine the number of Lychrel numbers less than 10,000:

(import '(java.math BigInteger))

(defn reverse-str [s]
  (apply str (reverse s)))

(defn reverse-int [n]
  (BigInteger. (reverse-str (str n))))

(defn add-reverse [n]
  (+ n (reverse-int n)))

(defn lychrel? [n iterations]
  (if (< iterations 1)
    true
    (let [i (add-reverse n)
          s (str i)]
      (if (= (reverse-str s) s)
        false
        (recur i (- iterations 1))))))

(println (count (filter #(lychrel? % 49) (range 10000))))