Make a circuit so that it heats water as rapidly as possible To do this I understand that we need to create the greatest power.
What I don't understand is why would we need to use a parallel circuit instead of a series circuit to create the greater power?
 A: It depends on your power supply.
The power $P$ through a circuit element is always given by
$$ P_\text{thing} = I_\text{thing}V_\text{thing} $$
where $I$ is the current through and $V$ the voltage drop across the thing that you're interested in.  If your circuit element obeys Ohm's Law
$$
V_\text{thing} = I_\text{thing} R_\text{thing}
$$
then you can write the power is
$$
P_\text{thing} = I_\text{thing}^2R_\text{thing} \quad\text{or}\quad P_\text{thing} = V_\text{thing}^2/R_\text{thing}.
$$
The thing that we're interested in here is our power supply: we want to build a circuit where as much of the power as possible flows into the water.  There are two possibilities:


*

*Our power supply might output a constant current, $I$.  In that case we'll get more power $P=I^2R$ if we connect the resistors in series, so that the full current from the supply flows through each resistor. We'd get less total power if we connect the resistors in parallel, so that only a small part of the current flows through each resistor. A set of resistors connected in series has an equivalent resistance $$R_\text{series} = R_1 + R_2 + \cdots$$ which is larger than any of the individual $R_i$, and a constant-current circuit uses more power $P=I^2 R$ for larger $R$.

*Our power supply might output a constant voltage, $V$. In that case we'll get more power $P=V^2/R$ if we connect the resistors in parallel, so that each resistor is across the full voltage of the power supply. We'd get less total power if we connect the resistors in series, so that each resistor sees only a small fraction of the voltage drop. A set of resistors in parallel has an equivalent resistance $$R_\text{parallel} = \left(\frac1{R_1} + \frac1{R_2} + \cdots\right)^{-1}$$ which is smaller than any of the individual resistances $R_i$, and a constant-voltage circuit uses more power $P=V^2/R$ for smaller $R$.
We happen to live in a world where constant-voltage power supplies (like batteries) are more common that constant-current power supplies (like those driven by photocurrents), so a parallel array of resistors is more likely to be a better heater than a series array.
