(This was a question asked by my dear little 10 year old brother.) Let's define some kind of algorithm, where we take a number, reverse its digits, and add it to the original, and iterate until we obtain a palindrome/mirror number. For example:
start with $213$; $213+312=525$, which is a palindrome. We can define this as a "function" $f(213)=535=f(312)$, $f(12)=f(21)=33$, etc.
My brother asked the following two questions:
If we begin with $196$, and follow this algorithm, can we ever obtain such a palindromic number? If yes, what's that number? I.e., does $f(196)$ (which is equal to $f(691)$) exist?
How about if we begin with $9988$? Is it possible to obtain such a palindromic number? I.e., does $f(9988)$ exist? If yes, what is $f(9988)$?
I couldn't answer him, and so I've resorted to the great minds of MSE.
P.S.: I was curious to see whether, if we start with a general number $n=\sum^n_{i=1}a_i10^i$, what will the resulting palindromic number be, if it exists? But this is just a "bonus"! 🙂
Best Answer
As the comments have already informed you, this is an open problem several amateur and professional mathematicians have been working on for decades now. According to a p196.org page, calculations had advanced past 400 million decimal digits as of 2011.
To give you some perspective, consider that scientists say there are $10^{80}$ atoms in the entire universe. If an odd perfect number exists, it is at least $10^{300}$. It might be the case that the smallest odd perfect number is less than the first palindrome reached from 196 by the reverse-and-add function. You have a better chance of winning a million dollar lottery jackpot than you do of finding a palindrome in 196's trajectory.
As for 4994, most people consider it trivial because it's already a palindrome. The question you need to be asking then is whether 9988 reaches a palindrome by this algorithm. According to a German Wikipedia page, if I've understood the language correctly and if the information is correct (two big ifs), 9988's trajectory might never reach a palindrome. But I doubt calculations have gone as far for it as for 196.