## Monday, June 20, 2011

### The 196 Problem

Here's a weird but interesting problem I just learned about: the 196 problem.

Its about what are called "Lychrel numbers" -- numbers which do not form palindromes when their digits are added iteratively.

It's easiest to show some examples. Consider the number 23. Reverse its digits (to get 32) and add these two numbers: 23 + 32 = 55. "55" is palindromic -- it reads back-to-front the same as front-to-back -- so 23 is not a Lychrel number.

How about 431? 431 + 134 = 565, a palindrome.

Or 59: 59 + 95 = 154. 154 + 451 = 605. 605 + 506 = 1111, a palindrome (after 3 iterations).

10,911 reaches the palindrome 4668731596684224866951378664 after 55 steps.

Question: What numbers are Lychrel numbers?

Answer: No one knows. 196 is suspected to be the lowest possible Lychrel number, but there is no proof.

196 has been iterated to a number with 300 million digits without reaching a palindrome.

But here's an even stranger thing about 196: it's been proven to be a Lychrel number in all bases between 2 and 18 (inclusive) except base 10. Weird. Why would 10 be special?

--
UPDATE: 196 is not a Lychrel number in base 19: 196 base 19 = A6; A6+6A base 19 = GG, a palindrome.

Or in base 20: 196 base 20 = 9G; 9G+G9 base 20 = 44, a palindrome.

#### 1 comment: Anonymous said...

This is why I like this blog, it's full of intriguing stuff that I don't read about elsewhere and that I'm glad I now do know about.