 Solution

Let $a$ be the cube's side and $z$ a small upward displacement. Its displaced water volume decreases by $a^{2}z$, so the change in buoyancy is $-{\rho }_{w}g{a}^{2}z$. With cube mass $m=0.9{\rho }_w{a}^3$,The displacement must be small enough that the upper face remains above water and the bottom remains submerged. This buoyancy estimate neglects fluid added mass, damping and melting.