An Egyptian pharaoh wants to build a monument to the Sun God, Ra. He wants this to be a solid stone cube, which is $20$ hectocubits tall, long and wide.
His engineers get right to work planning this. The only stones the Egyptians have access to are $2000$ enormous rectangular blocks leftover from a failed project, whose dimensions are $2\times 2\times1 $ (all units in hectocubits), and which cannot be broken into smaller usable blocks. They figure that the easiest way to build this monument is in horizontal layers, each layer being a 10 by 10 array of blocks which lie flat. They present this plan to the pharaoh.
However, after seeing the plan, the pharaoh is furious! Since the edges of the blocks are lightly rounded, the cracks between blocks can let some light through. This is of course a problem, since the cube is supposed to be an homage to the generosity of the Sun God, so it should graciously accept the gift of the Sun's rays instead of letting them pass by.
The engineers get confused at this point, since they aren't as religious, and don't understand the importance of the pharaoh's request. Totally losing his cool, the pharaoh yells, "You fools! All I ask is that you
build a $20\times20\times20$ cube out of $2\times2\times 1$ blocks so no light can shine in one face though the cracks between the blocks and exit the opposite face!
IS THAT SO DIFFICULT!?"
Can the engineers succeed?
Note that the pharaoh wants the cube to block all possible beams of light, not just vertical ones. In other words, every line which enters one face and exits the opposite face must also pass through the interior of one of the $2\times2\times1$ blocks. The solution does not involve lateral-thinking, so the answer is not "just fill the cracks with mortar."
Edit: I changed it so that the cube is only required to block light which enters and leaves opposite faces. As @frodoskywalker pointed out, it was trivially impossible without this addition. Furthermore, I clarified other confusing points (breaking blocks is not allowed, and the cube must be solid, implying all the blocks are axis aligned).