In Love with Resistance!

Due to symmetry, there is no difference in voltage between points $C$ , $D$ , and $E$ , whether or not there is any connection in the vertical direction. The system on the left is therefore electrically equivalent to the one on the right. (Sorry about the strange resistors, but I don’t have the software for better ones.)

$\dfrac1R=4\times\dfrac1{2r}=\dfrac2r$

$R=\boxed{0.5r}$

This by no means a rigorous solution but it relies on the concept of "In a circuit, the overall resistance must be less than the smallest resistance chain". If you imagine the circuit as only existing of the top and bottom chain (no links connecting the center are extant), the overall resistance of that circuit would be 1/(1/(1+1) + 1/(1+1)) = 1/(½ + ½) = 1/1 = 1r . Using the concept mentioned above, the overall resistance of the circuit has to be less than 1r, since you can imagine all the central resistors as having infinite resistance in order to get to the 1r, lowering those infinities to a finite value will decrease the resistance below 1r. The only option less than 1r is .5r.