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There was a problem which asked to connect some resistors in some order(parallel and series) to achieve an equivalent resistor with a specified ohms. I solved that problem but I think we can argue about the generalized form of it:

$\bigstar$ Suppose we have $n$ inputs(Consider them to be number and equal for now) and $m$ unique functions.Each function takes two inputs and outputs a number which is always acceptable.
$Example:$ In here we have $n$ elements(resistors)with value of $R$ and two functions Parallel and Series namely: $$P(i_1,i_2) = \frac{i_1 i_2}{i_1+i_2} , \ S(i_1,i_2)=i_1+i_2$$
$\bigstar$ Question1: How many unique numbers can the combination of $n$ inputs and $m$ functions generate?
$Example:$ How many unique resistors can we obtain by different combination of resistors in series and parallel?
$My \ Strategy: for \ n=3 \implies R\square R\square R $
Convention:Every function looks at its left and if right is filled too,immediately outputs the $f(left,right)$ and we replace the whole beginning number to the right of the function with that output.(This should prevent manipulating parenthesis and precedence I guess)
We fill the blanks with either P or S. So we will have
$PP \implies R_{eq}=R/3$
$PS \implies R_{eq}=3R/2$
$SP \implies R_{eq}=2R/3$
$SS \implies R_{eq}=3R$
flaw:I'm not sure whether this approach produces unique outputs or not

$\bigstar$ Question2:Can we predict a specific number definitely could be generated? Can we predict a specific number can never be generated?
$Example:$ Can we say that a 10 ohms can never be generated by any configuration(series or parallel) of 6 3 ohms resistors?

$\bigstar$ Question3: Now consider $n$ inputs to be arbitrary(may be different) and rest of the constraints be just as presented. How can we answer the questions above? Is this argument related to a certain topic of mathematics(I think the first two questions are combinatorics questions but I'm skeptical about the last question...)

Sorry if this post is long but these questions came into my mind and I think there should be interesting approaches in solving them

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If the answer to some questions is easy or you think I can work it out myself, please provide a hint –  Zeta.Investigator Sep 13 '12 at 11:17
    
Must all $n$ resistors be used? (That is, is leaving out a resistor an option?) Your example says "yes"; I just want to double check. –  Willie Wong Sep 13 '12 at 12:35
    
@WillieWong : Yes, All of the resistors must be used –  Zeta.Investigator Sep 13 '12 at 13:25
    
Hum, the question is likely to involve not just combinatorics but also some amount of number theory. Notice that in the case of $2k$ resistors ($k > 1$) there are at least two ways to get a total resistance of $k/2$. You can either put $k$ in series, another $k$ in series, and the two sets in parallel overall, or pair up the resistors to have $k$ pairs, each pair has the two resistors in parallel, and the pairs are placed in series. Something similar can be done with $nk$ resistors to get $k/n$ overall resistance more than one way. –  Willie Wong Sep 13 '12 at 14:44
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