Can You Solve this Ice Diamond Riddle? ft. Vsauce’s Michael Stevens

The level of water will go down, here's why:
-> If it would be only the ice cube, the level of water wouldn't move since the law of archimedes states that the upwards force is equal to the volume of displaced water. Since the forces are in equilibrium, the volume displaced is equal to the volume of water needed to weigh the same amount as the ice cube. Ice has a density of 917kg/m3, so a bloc of 917kg would take up 1m3 and displace 0.917m3 of water. Because of the conservation of mass, if the cube melts, it would still weigh 917kg of liquid water (same amount of atoms than before), now taking up 0.917m3 of volume since the density of water is 1000kg/m3. The level pf water didn't change. (This is also the reason why floating icebergs don't raise the level of the oceans when they melt, only ice supported by ground will).
-> Now if we add a diamond to the cube, it will displace more water. Let's imagine it weighs 3.5kg (nice size right?) : 3.5L more of water will be displaced by the floating ice cube. Now when it melts, the diamond will be submerged and displace a volume of 1L (since its density equals 3.5kg/L). Less water is displaced, thus the level of water will go down.

BONUS: You can also notice that in the compound ice cube + diamond, the mass of the ice has to be at least 7.9 times the mass of the diamond for the compound not to drown immediately. So for 1cm3 of diamond, you will need over 30cm3 of ice.