I had several tiles left over from my chess set, and was pondering what to do with them. I decided to try making a checkered Lego® Soma cube. Lego® bricks have the wonderful property that 2 studs is exactly the same length as 5 plates height. I decided to double this, making each of the 27 smaller cubes measure 4 studs on a side. I connected the cubes together internally, giving no evidence on the outside.

After completing the first cube I decided to try making a smaller one. Since each of the component cubes measured 2 studs on a side, there would be no room to put hidden bricks on the inside. I figured that by using 2 x 2 corner bricks, I could connect any place where 3 cubes came together at a right angle. All the pieces in the Soma cube could be oriented in a way to make this possible except the large 'L'. For this piece I used both the corner bricks and 1 x 2 technic bricks with 2 holes. Here is the piece with the top tiles removed to see the internal structure. As it happens this second strategy would work for every piece without the corner pieces, except for the 'Y' piece, which requires the corner brick.

The smaller cube has the same color pattern as the larger one except for the large 'L' which is inversed. Both cubes are solvable with a chequered pattern. The large cube has 219 solutions, while the smaller cube has only 21, making it much more difficult. The larger cube is built to have a solution with all studs oriented the same way; the smaller cube can be solved such that all faces are smooth (no undersides of bricks visible.)

Ldraw models: (If you don't have a viewer, it can be downloaded for free here.)

Soma_s_corner.ldr

Soma_s_ll.ldr

Soma_s_ltwist.ldr

Soma_s_rtwist.ldr

Soma_s_s.ldr

Soma_s_sl.ldr

Soma_s_t.ldr

Have fun building and solving your own; I only ask that if you publish it anywhere to give a link to me.