Electronic – Current through each resistor

I'm solving this high-school level problem, where I have to find the current through each resistor.

^{simulate this circuit – Schematic created using CircuitLab}

There aren't any resistors in parallel or series, which could be combined, so I tried using Kirchoff's law for junctions and loops. The resulting system of linear equations doesn't have any solution according to Mathematica. \$I_0\$ is the current through the source, \$I_{15}\$ is an auxiliary current through the empty wire in the middle of the circuit. My resulting system of linear equations:

My questions are:

• Is my approach to this problem right? Maybe I chose wrong loops. Is there any rule for which loops I should choose?
• There are 16 unknown currents, I have 10 junction equations. Does it
  mean I have to create at least 6 loops, to find all the unknowns?
• If my approach is wrong, what is the first step or direction I should use to solve this problem?

EDIT: There are a few mistakes in the equations:

• equation 3 is \$I_3=I_5+I_7\$
• equation 15 and 16 don't include \$I_{15}\$ (there is no voltage change over the wire without any resistor)
• With these changes, the system can be solved by satisfying \$\mathbf{A}^\intercal \mathbf{A}\vec{x}=\mathbf{A}^\intercal\vec{b}\$. However, the accepted answer shows a much better approach to this problem (the results are the same).