# 2021-03-12 MATH 3600 Reverse and Add

## 434 days ago by calkin

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 # is a palindrome so we stop
 121 121
572 + 275
 847 847
847+748
 1595 1595
1595+5951
 7546 7546
7546+6457
 14003 14003
14003+30041
 44044 44044
n=162
digit_list=n.digits()
m=0 for i in digit_list: m=m*10+i print m
 261 261
def reverse_n(n): digit_list=n.digits() m=0 for i in digit_list: m=m*10+i return(m)
reverse_n(194)
 491 491
n=196 for j in srange(10): n=n+reverse_n(n) print(n)
 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}

n=1 for i in srange(20): n=n+reverse_n(n) print n
 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