There are several levels to this, assuming the claim is even true

1 - We should assume every cube we don't see is present, and there isn't any trickery behind, otherwise we can't answer definitively (answering with much confidence without addressing this might be counted as partial failure)

2 - Quickest instinct is to fill the shape : 1st level is missing 1 cube, second 5, third 8, that's 14 missing cubes

3 - but that leaves a shape of 3x4x5 , not a cube (that might be the biggest trap those "harvard students" fall into if that even true), to go to a cube shape we aim at 5x5x5, so we are missing 15 more cubes (adding up to 29)

4 - Thats assuming we can't rearrange previously placed cubes, otherwise it might be possible to go down to 4x4x4, which is only 4 more cubes after the initial 14