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For most integers, if you reverse the digits and add it to itself, it quickly becomes a palindromic number, that is, the digits are the same forwards and backward. For example, start with 79:
79 + 97 = 176
176 + 671 = 847
847 + 748 = 1585
1585 + 5851 = 7546
7546 + 6457 = 14003
14003 + 30041 = 44044

Until you get to 196, all the integers become palindromes quickly.
Except for the number 196. In 1990, John Walker tested it by adding itself until it had become a number with 1,000,000 digits, and it still wasn't palindromic. In 1995 someone tested out to 2,000,000 with no success.

It seems strange that 196 should be such a special number. Of course, this is all in the totally arbitrary base 10. According to mathworld, the Palindromic Number Conjecture has been proven false in base 2, since the number 10110 never produces a palindrome.

These numbers can be viewed as Clues from the keeper of the vat.

These non-palindromics numbers are called "Lychrel Numbers", although no number (in base 10 anyway) can be proven to be ever non-palindromic; they're just non-palindromic in the first few, ah, million iterations. Of course, they also get less likely to become a Lychrel Number after each iteration*.

As of August 14, 2002, someone has calculated 196's iterations up to 41 Million digits; that's around a hundred million iterations.

*: According to Dan Hoey, the probability can be calculated as such:
You suggested that 196 will yield eventually. If you mean it will become a palindrome, I don't think that is very likely. The reason is that the number will only yield a palindrome if there is no carry out of any digit when the addition is performed. If the digits are randomly distributed, the probability of this is about 2^(-n/2) for an n-digit number. The number of digits is about 2/5 of the number of steps taken, and will certainly increase at least once every five steps. So the probability that the number will eventually reach a palindrome on the k'th step should be less than 2^(-k/10), and the sum of that for k=3,000,000 to infinity is something like 10^-90,000.

There are some numbers in other bases for which the process can be shown not to terminate (http://www.seanet.com/~ksbrown/kmath004.htm, for example) but that is because a particularly orderly kind of progression is followed. None is now known in base 10, though I wouldn't be surprised if one is found. But I doubt that 196 will turn out this way either. It looks to me like it will probably continue to grow chaotically, and I don't think our mathematics is up to proving anything about it.

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