I-Tetracubes with Pegs and Notches
Here are the 39 I-Tetracubes with exactly one peg and one notch.
You cannot construct a 2x6x13 box with these pieces, because the pieces can't be placed parallel to
the edges of size 2 and the 6x13 rectangle can't be filled because the area isn't a multiple of 4.
The layers of the 3x4x13 box are shown below.
The total volume of the pieces with exactly two male or two female faces is 200 and a lot of boxes can be made.
The 2x4x25 box was split into a 2x4x13 and a 2x4x12 box.
The 2x5x20 box was split into a 2x5x8 and a 2x5x12 box.
The 4x5x10 box was split into a 4x2x10 and a 4x3x10 box.
The 5x5x8 box was split into a 5x3x8 and a 5x2x8 box.
You can't get a 2x10x10 box, because the pieces can't be placed parallel to the edges of size 2 and the
10x10 area can't be filled. If the 25 2x2 sub squares are checkerboard colored there are without loss of generality
12*4 white and 13*4 black unit squares. But since each piece covers two white and two black unit squares
the numbers of white and black squares must be equal.
The 2x2x50 box is impossible, because the pieces must be parallel to the edges of size 50,
but 50 isn't a multiple of 4.