Reverse and add until palindrome
Take a number in base 10: if it is a palindrome, stop: otherwise reverse its digits, and add them. If the result is a palindrome, stop.
Otherwise, continue.
121 121 |
847 847 |
1595 1595 |
7546 7546 |
14003 14003 |
44044 44044 |
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261 261 |
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491 491 |
887 1675 7436 13783 52514 94039 187088 1067869 10755470 18211171 887 1675 7436 13783 52514 94039 187088 1067869 10755470 18211171 |
Questions to investigate:
\begin{enumerate}
\item Starting at $n$, do we eventually reach a palindrome?
\item Starting at $n$, how many iterations of reverse and add does it take to reach a palindrome?
\item For those numbers that do reach a palindrome, can we spot patterns?
\item For those numbers that don't reach a palindrome, how fast do they grow?
\item If we just iterate reverse and add, without worrying about palindromes, how big is the Nth iterate?
\item If we just iterate reverse and add, how many of the iterates we see are palindromes?
\item Can we extend this problem to numbers with decimal expansions?
\item If all of the digits in the number are less than 5 will it become a palindrome in a single step?
\item If a number reaches a palindrome, must it reach another palindrome?
\item In a given range, for the numbers which do reach a palindrome, what is the average number of steps?
\item In a given range, what proportion of numbers reach palindromes?
\item Do number theoretic properties of $n$ influence whether it reaches a palindrome?
\item For numbers that do reach palindromes multiple times, what is the distance between palindromes?
\item What happens if we change the operation to reverse and multiply?
\item If all digits are less than or equal to 4, we get a palindome. Are there any other $n$ which give a palindrome in one step?
\end{enumerate}
2 4 8 16 77 154 605 1111 2222 4444 8888 17776 85547 160105 661166 1322332 3654563 7309126 13528163 49710694 2 4 8 16 77 154 605 1111 2222 4444 8888 17776 85547 160105 661166 1322332 3654563 7309126 13528163 49710694 |
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