Tool for "designing" a resistor network?
I don't know if you've considered continued fractions, or if they'd do anything you care about, but here's something that may be worth considering.

Let's say you want \$R_x=12.5\:\Omega\$ between nodes \$N_A=N_0\$ and \$N_B\$. Set \$i=0\$:


Select the next higher value above \$R_x\$. In E12, that's \$R_{2i+1}\$ (for \$i=0\$, \$R_1=15\:\Omega\$.) If satisfied with the results, use \$R_{2i+1}\$ to connect \$N_i\$ to \$N_B\$ and you are done!
Otherwise, compute \$R_y=\frac{R_{2i+1}\cdot R_x}{R_{2i+1}-R_x}\$ (for \$i=0\$, \$R_y=75\:\Omega\$) and select the next lower value of \$R_{2i+2}\$ (for \$i=0\$, \$R_2=68\:\Omega\$.) Connect one end of \$R_{2i+2}\$ to \$N_i\$. The other end is left open and labeled \$N_{i+1}\$. If satisfied with the results of jumpering \$N_{i+1}\$ to \$N_B\$, then jumper those two nodes and you are done!
Otherwise, compute \$R_x=R_y-R_{2i+2}\$ (for \$i=0\$, \$R_x=7\:\Omega\$), increment \$i\$ (for \$i=0\$, \$i\$ becomes \$i=1\$) and then go to step 1.


In the above starting case of \$12.5\:\Omega\$ and using E12 resistor values, the following possible circuits develop:







simulate this circuit &ndash; Schematic created using CircuitLab

And so on. You can see that it rapidly approaches the desired value. Most of the dissipation is usually carried in the first resistor. Whether or not this is good or bad, depends. The gist of this is that you are almost always very, very close with no more than three or four resistors with E12.

Here's the same approach, but used with E24 resistor values:







simulate this circuit

Note that at the even steps of \$i\ge 2\$, you can also consider summing the two series resistor values and picking the next higher value that closely matches as an alternative to step \$i-1\$, too. For example, in step \$i=2\$ of the above E24 resistor selections, \$R_2\$ and \$R_3\$ could be summed and then the choice of \$R_2=330\:\Omega\$ could be used instead of what was used in step \$i=1\$. In the above case, this would have been a better choice for \$i=1\$ than just strictly following the algorithm I gave earlier.

This points up a slight improvement to the algorithm. The value of \$R_x\$ should be considered as either one resistor of next higher value, or else two resistors composed of the next lower value, plus the difference rounded up to its next (or similarly convenient) higher value.

Also, the algorithm doesn't always find optimal choices. For example, this could have been \$12\:\Omega\$ in series with two, parallel \$1\:\Omega\$ resistors. And that would have been exact. But what it does do is to continually diminish the residual error (assuming exact resistor values.)

(Offered in the case it matters to you to have a codified, step-wise algorithm you can follow without just doing "hit and miss.")