Blocked Half-Cube Box

The Blocked Half-Cube Box is a new design that uses pieces made of pairs of diagonal half-cubes.
The first known design of a packing puzzle using this type of piece is the Fujimoto Cube designed by Shinichi Fujimoto in 1982. His cube was made up of eight pairs of diagonal half-cubes that pack into a 2x2x2 cube, with nine solutions. Since that time, many other puzzle designs using diagonal half-cubes have been produced.
Fujimoto’s eight pieces make up a complete set when connections are allowed only between the square-to-square and square-to-diagonal faces of pairs of half-cubes. If diagonal-to-diagonal face connections are also allowed, the new complete set consists of 12 pieces that pack into a 2x2x3 solid. However, one of the new pieces has an edge-to-edge connection that is physically impossible in theory, although possible to make using pin-type connections. A more elegant scheme is to simply glue the two half-cubes into the box into which the pieces are to be packed. Gluing them into the upper center part of the box, as in the Blocked Half-Cube Box, has the additional benefit of complicating the process of packing the remaining pieces into the box.
Four solutions are known. One is provided and the other three are left as an exercise for the solver.