Eleven equal wires of each of resistance r from the edges of an incomplete cube. Find the total resistance from one end of vacant edge of the cube to the other.

Dear student,



Let A and B be the two vacant edges of the incomplete cube.Let current 2i enter at the point A and as AC and AG are symmetrical the current gets divided as i₁ along AC and AG.At C a part of current i₂ flows along CE and remaining current along CD.G is in similar position as C,current i₂ flows along GE and remaining along GH.At point E current along CE and GE combine to give current 2i₂ along EF.At F current along FD and Fh are equal to i_{2 .}The current along FD and Cd combine to give current i₁ along DB.Suppose V is potential difference across A and B.
Applying kirchoff second law to loop ABCD

 $$\begin{gathered} i_{1}R+\left(i_1-i_2\right)R+i_{1}R=V \\ 3i_{1}R-i_{2}R =V \to eqn 1 \end{gathered}$$
Applying Kirchoffs law to loop ABEFDB,

$$\begin{gathered} i_{1}R+i_{2}R+2i_{2}R+i_{2}R+i_{1}R=V \\ 2i_{1}R+4i_{2}R=V \to eqn 2 \end{gathered}$$

From the above two eqn 

$$\begin{gathered} 3i_{1}R-i_{2}R=2i_{1}R+4i_{2}R \\ {i}_1=5i_{2 }\to eqn 3 \end{gathered}$$
From eqn 1 and 3 

$$\begin{gathered} 3R\times 5i_2-i_{2}R=V \\ 14i_{2}R=V \to eqn 4 \end{gathered}$$

If R_AB represents resistance between A and B,then 

$V=R_{AB}\times 2i_{2 \to eqn 5}$

From eqn 4 and 5

$$\begin{gathered} R_{AB }\times 2i_1=14i_{2}R \\ {R}_{AB }\times 2\times 5i_2=14i_{2}R \\ {R}_{AB }=\frac{14}{10}R=1.4R \end{gathered}$$

Regards.