## Colored Solid Tetrominoes

Let's take the five tetrominoes with thickness one. There are many different orientations in space: 3 for the square and the straight tetromino, 12 for the skew and T tetromino and 24 for the L tetromino. Regarding all pieces with different orientations as distinct we have 54 fixed solid tetrominoes with a total volume of 216. Due to the fixed orientation there are some restrictions for constructions. For the sum S of the x-coordinates of all cubes the equation S mod 4 = 2 must hold. The same applies for the y and z-coordinates. The 6x6x6 cube doesn't match these conditions and can't be constructed.

Therefore I decided to color the faces of the pieces with 6 colors according to the 6 orientations +x, -x, +y, -y, +z, -z. It's allowed to turn the pieces around but in the final construction faces with same orientation must have the same color. Now you can get a cube, because the colors of the pieces inside can't be seen and you are free to rotate them. A manual solution should be possible within a couple of hours.

A lot of other constructions can be made as well: boxes, open boxes, 3-fold replicas of octacubes and other special figures. In the following applet some of them are shown. Click the pieces to remove them from the construction and click the removed pieces to put them back. A click on the little tripod rotates the whole figure. Can you guess which examples are made without any rotation?

For each construction a drawing of the layers is provided, where the pieces are labeled. Together with the picture of the set each piece can easily be identified.

 Boxes 6x6x6, 3x3x24, 3x6x12 3-fold replicas of octacubes crosstower, 9x9x3 ring, 2 tripods, 2 boxes, replica 1, replica 2 Open boxes 6x6x10 with wall thickness 1 8x6x5 with wall thickness 2 8x6x8 with wall thickness 1 is possible, too. I found the solution manually, but it's rather difficult. Others 3 pyramids

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