Answer
The total moment of inertia of all particles is $0.06kg.m^2$
Work Step by Step
The figure below shows one face of the cube. On this face, the moment of inertia for each of 3 particles is: $$I_1=I_3=mr^2$$ $$I_2=mr_2^2=m(r^2+r^2)=2mr^2$$
The total moment of inertia of 3 particles is $I=4mr^2$
The opposite face of this cube has the same 3 particles with the same moment of inertia, so its total $I=4mr^2$.
The total moment of inertia of all particles is $$\sum I=2I=8mr^2=8(0.12kg)(0.25m)^2=0.06kg.m^2$$
