Pentominoes

Today I turned 25900 days old and one of the OEIS entries for this number states:

A248102

Number of tilings of a 5 X 2n rectangle using 2n pentominoes of shapes N, Y.

So 25900 is a member of this sequence, specifically for the case where n=8 and so, when there is a 5 x 16 rectangle (with area 80 square units), it can be tiled with 16 pentominoes (each with an area of 5 square units) in 25900 different ways. The pentominoes must be of type N or Y. Of course, I then had to look up what was meant by a pentomino of type N and type Y. Figure 1 shows the twelve types of free pentominoes, lettered according to the shapes that approximate those letters.

Figure 1: the 12 types of free pentominoes (source)

By free pentominoes, it is meant that chiral versions are not considered distinct. For example, the F pentomino can be lifted out of the plane and turned into its mirror image. If chiral versions are considered distinct, then there are 18 different types. See Figure 2.

Figure 2: The 12 pentominoes can form 18 different shapes,
with 6 of them (the chiral pentominoes) being mirrored. (source)

To quote from Wikipedia:

A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20. 

Figure 3 shows some of the possible configurations:

Figure 3

There are many websites where puzzles of this sort can be solved. Figure 4 is a screenshot of one of these sites. Link

Figure 4: link to website

In the screenshot in Figure 4, the 6 x 10 rectangle is to be tiled. There are 2339 ways in which this can be done. For the 5 x 12 case, there are 1010 ways. For the 4 x 15 case, there are 368 ways and for the 3 x 20 case there are just 2 ways. 

There's a lot more that could be said about pentominoes but I'll go on and make some comments about pentacubes which are polycubes made out of five cubes. There are 29 possible pentacubes of which twelve are flat (corresponding to the twelve pentominoes extruded to a depth of 1 unit). Figure 5 shows one of the non-flat pentacubes:

Figure 5: source 

To quote again from Wikipedia:

A standard pentomino puzzle is to tile a rectangular box with the pentominoes, i.e. cover it without overlap and without gaps. Each of the 12 pentominoes has an area of 5 unit squares, so the box must have an area of 60 units. Possible sizes are 6×10, 5×12, 4×15 and 3×20.

Figure 6

The diagrams in Figure 6 do require a little scrutiny but the shapes of the various polycubes do reveal themselves. Obviously, the topics of pentominoes and pentacubes form a rich vein of recreational Mathematics and I'm just grazing the surface in this post. There are other puzzles that involve pentacubes. I'll mention one such puzzle and that is the Bedlam Cube, named after its inventor Bruce Bedlam. To quote from Wikipedia:

The puzzle consists of thirteen polycubic pieces: twelve pentacubes and one tetracube. The objective is to assemble these pieces into a 4 x 4 x 4 cube. There are 19,186 distinct ways of doing so, up to rotations and reflections. The Bedlam cube is one unit per side larger than the 3 x 3 x 3 Soma cube, and is much more difficult to solve. See Figures 7 and 8.

Figure 7: an assembled Bedlam Cube

Figure 8: a disassembled Bedlam Cube 

There are many more puzzles and much more to say about polyominoes and polycubes but I'll leave it at that for the moment. Meanwhile, I've not been idle. My trophy cabinet is shown in Figure 9 with my trophy for attaining Level 4 in Pentominoes.

Figure 9