Solving equations where the solution is an operator Ok, so here's some context.
Solving regular equations we might have something like this:
$2 + x = 5$, solving for $x$ we get 3. We might even have an equation like $x + y = 5$ where there are multiple solutions.
But what's in common with all these equations is that the process, or the algorithm, we follow to solve them is determined by the operators that show the relations between variables.
Now if you think back to very early elementary school, I'm sure you solved stuff like
$5 \_ 2 = 3$ where you would fill in the blank with a minus. I'm wondering if there's a branch or mathematics that studies actual systematical ways to solve "equations" like that. It might seem trivial from this example but it obviously would grow in complexity. Perhaps the solutions would be operators AND numbers.
The concrete problem I thought up that led me to this was trying to find a way to "map" any $\frac{1}{n}$ to $\frac{1}{n+1}$ by just adding/multiplying/something the first fraction by a constant. The "equation" would look something like this:
$\frac{1}{n}\_C=\frac{1}{n+1}$ 
I can solve similar problems in my head, like:
$n\_C=n+1$ where the obvious solution is $+$ and 1. Or for example, $n\_C=2n$ where the solution is $\times$ and 2. The last one can also have $+$ as a solution but then $C$ would have to be $n$ and would no longer be a constant.
(I didn't know what to put as the tag)

Equations where "operations" (really just a funny notation for functions) are the unknowns are called functional equations. Solving them can range from the trivial (like the examples you give) to very complex. Some techiques are given in the Wikipedia link above.
