Finding Optimal Resistor Combination for Desired Equivalent Resistance

Resistor series values are designed such that you can select for ratios with an accuracy on par with the tolerance. For example, the E96 series (https://en.wikipedia.org/wiki/Preferred_number) is used for 1% resistors. Pick two values from the table and calculate the divider ratio. If you switch one of them out for an adjacent value in the table, the difference in the ratio will be less than 1%. This works for any resistor series. If you need to do better than that, you can use a trimmer, expensive laser-trimmed resistor arrays, or possibly some form of calibration to compensate for an offset divider value. If you really need high precision, don't forget about the temperature coefficients.

One way to solve it would be to connect a 1 volt source between X and Y, then use nodal analysis to determine the source current. There are six nodes (not counting X and Y), so six equations would be needed. The equivalent resistance is the voltage divided by the current.

Another way is to take advantage of symmetry, as in your previous question. Remove \$R_{PQ}\$ and \$R_{RS}\$. Now look at nodes P, Q, R, and S. Each one is halfway between X and Y in terms of resistance along their branch. This means that they all should have the same voltage, which is \$\frac {V_X - V_Y} 2\$. Since they have the same voltage, a resistor connected between any two of the points won't draw any current. Thus, \$R_{PQ}\$ and \$R_{RS}\$ can be ignored. Now the circuit is a straightforward series and parallel resistor problem.