A cube is constructed from 12 identical wires. Current enters one corner of the cube and it leaves the opposite corner. If the resistance of each wire is r, then equivalent resistance will be

The distribution of current in the cube is shown in fig. Let V be the potential difference between A and G.

Consider any path between, A and G say ABCG, we have 
$\mathrm{V}=2\mathrm{i}\times \mathrm{r}+\mathrm{i}\times \mathrm{r}+2\mathrm{i}\times \mathrm{r}=5\mathrm{ir}$
Let -R be the equivalent resistance between A and G. Then
$\mathrm{V}=\left(6\mathrm{i}\right)\times \mathrm{R}$
or $5\mathrm{ir}=6\mathrm{iR}\text{ or }\mathrm{R}=(5/6)\mathrm{r}$