A palindrome is a number that is the same way forward as it is backwards.

For example, 22 is a palindrome.

Clearly, however, 23 is not.

But by adding 23 reversed on to itself (32), you get 55, which is a palindrome.

That took 1 repetition.

With 64, 1 repetition gives you 110, and since that isn't a palindrome, you must do another repetition. You'll end up with 121, which is a palindrome.

That took 2 repetitions.

So my question is this:

What is the lowest number to require more than 25 repetitions to become a palindrome? (Yes, brute force will work)

## Palindromic Repetitions

### Re: Palindromic Repetitions

Should we assume that all numbers will create a palindrome eventually? What if the number is already a palindrome to begin with?

### Re: Palindromic Repetitions

In the case of this problem, it would not matter if the theory that all numbers eventually arrive at a palindrome is later proven to be false, for we are only looking for the first one that takes more than 25 repetitions. Also, if a number already is a palindrome, then did it not take 0 repetitions to arrive at a palindrome? This is simply my thought process, at least...

### Re: Palindromic Repetitions

See also:

http://mathworld.wolfram.com/Palindromi ... cture.html

(Note: contains spoiler for the problem in the OP)

http://mathworld.wolfram.com/Palindromi ... cture.html

(Note: contains spoiler for the problem in the OP)

_{Jaap's Puzzle Page}

### Re: Palindromic Repetitions

I'm not sure it does; it contains a number which isn't known to ever become a palindrome, but we're looking for a number which does become a palindrome, but in more than 25 steps.jaap wrote:(Note: contains spoiler for the problem in the OP)

There is a smallest initial value *known* to take more than 25 steps, though it can't be proven it is the smallest. (I believe that's what pj6444 was getting at.)