# Definition:Reverse-and-Add

## Definition

Let $m \in \Z_{>0}$ be a positive integer expressed in decimal notation.

Let $r: \Z_{>0} \to \Z_{>0}$ be the mapping defined as:

The mapping $r$ is the **reverse-and-add operation**.

## Also known as

Some sources give this as **reverse-then-add**.

## Historical Note

There has been a lot of work in recent years on the results of iterating the reverse-and-add process, in several number bases.

The usual aim of the exercise is to determine how long it takes until a palindromic number appears.

Most of the research has, as can be expected, been done on numbers expressed in base $10$ notation.

Of the $900$ numbers with $3$ digits, $90$ are already palindromic and $735$ require between $1$ and $5$ iterations.

David Wells includes the following quote in his *Curious and Interesting Numbers, 2nd ed.* of $1997$:

*The remaining $75$ numbers can be classified into just a few groups, the members of which after one or two reversals each produce the same number and are therefore essentially the same. One of these groups consists of the numbers $187$, $286$, $385$, $583$, $682$, $781$, $869$, $880$ and $968$, each of which when reversed once or twice forms $1837$ and eventually forms the palindromic number $8,813,200,023,188$ after $23$ reversals.*

He attributes this quote to **Richard Hamilton**, but no reference to it can be found on the internet, apart from its appearance in the above source work, and it has not been possible to corroborate it.

Technically speaking, of course, $869$ and $968$ take just $22$ iterations.

The first $10 \, 000$ numbers contain $5996$ which have not yielded a palindrome.

The longer a number gets, the lower the probability that it will form a palindrome when added to its reversal, so for large numbers it appears that it becomes less and less likely for the reverse-and-add process to end in such a way.

## Sources

- Weisstein, Eric W. "Reverse-Then-Add Sequence." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/Reverse-Then-AddSequence.html