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Fitting Blocks in a Cube

Mark Jason Dominus was one of the original subjects in Julian Stanley's Study of Mathematically Precocious Youth (aka, SMPY). He posted the following problem to Mattababy, a listserv for math instructors at the Johns Hopkins Center for Talented Youth summer program. In his post to the listserv, he claimed that the problem came to him in a dream; when he later awoke, he still thought it was a good problem, so he decided to share it.

The volume of a 3 × 3 × 3 cube is 27 cubic units, and the volume of a 2 × 2 × 1 prism is 4 cubic units. Theoretically, six prisms should be able to fit inside the cube, with three cubic units empty. But can you arrange six 2 × 2 × 1 prisms so they fit inside a 3 × 3 × 3 cube?  
Blocks in a Box

The following problem is related to the main problem on this page. The main difference is that this problem uses a box that is not a cube.

How many 2 × 2 × 6 blocks will fit into a box that measures 10 × 10 × 20?

The main similarity is that both will tax your spatial visualization skills!