Pentominoes
A pentomino is a plane figure composed of five congruent squares joined along edges. The diagram below shows three pentominoes. Allowing for rotations and reflections, there are exactly twelve possible pentominoes.
My project was to build a set of twelve pentominoes out of LEGO elements. First of all, note that it is possible to construct a perfect cube out of LEGO as shown below.
If you have some LEGO bricks around, try it yourself to see that this is practically a perfect cube! Five such cubes together form a sort of three-dimensional pentomino. (Of course, to make the pentomino stay together in one piece, it is necessary to build it out of larger bricks and plates than those depicted above.) I built each pentomino using a different color. My finished set of pentominoes appears below.
Pentominoes are commonly found in mathematical puzzles. For example, one such puzzle involves arranging all twelve pentominoes in a 5 × 12 rectangle. Though hundreds of solutions exist, finding one usually involves much trial-and-error. One solution appears here. Other puzzles involve fitting the pentominoes in rectangles measuring 3 × 20, 4 × 15, or 6 × 10, or in a variety of other shapes with area 60.
Since my pentominoes are three-dimensional, they also have a combined volume of 60 cubic units, so it is possible that they might fit together to form a rectangular solid measuring, say, 3 × 4 × 5 units. However, I have yet to figure out how to arrange the pentominoes in such a way.
For more information on pentominoes, see the Pentomino article in Wikipedia.
