A few nights ago I couldn't sleep and so started doing this:
I would take a number and add up all of its digits to get a new number and then add up all of those digits and so on until there was only one digit left.
For example, I would say $543\mapsto 12\mapsto 3$ or $94286\mapsto 29\mapsto 11\mapsto 2$.
I noticed a few patterns and then after explaining the rules of this game to some friends was able to impress them by adding up the digits of huge numbers in a couple seconds! The only pattern really worth mentioning is this:
Adding $9$ to a digit doesn't change the final answer (i.e. $12391234$ has the same final answer as $1231234$. Similarily, $158$ has the same final answer as $15899999999999$) So when adding, you can just ignore the $9$. This helped a lot in trying to impress my friends because instead of adding $136271845$ you can just recognize this as $19999$ which will sum up to $1$ since you can ignore $9$.
There are a couple other patterns I noticed (like what happens if you keep doubling a number), but I want to know why the above happens, or how to prove why this happens? I have some rough ideas of why it happens but was hoping someone out there would have a slick proof of this?