Pentacubes (Planar ones mostly)

 A while ago when I made my set of tetracubes I mentioned the possibility of making the next set up, the pentacubes, if I ever felt in the mood for a lot of gluing and sanding. Well at some point between then and now (probably last year when I wasn't really doing a lot of blog post or website writing) I was obviously up for it because now I have a set of pentacubes to call my own. Very rough ones, admittedly - I glued but didn't bother to sand - but they work and can be fitted together without issue so they'll do.

The plan was (and still is, I suppose) to colour the edges of each piece just like I did for the unit cubes so the boundaries are a bit more distinct. But to do that I'd need to sand them, and to sand them I'd need to bring them to somewhere without a bit of outside space. So it's not happened yet.

The total volume of the pentacubes is 5x29 = 145 which clearly can't be factorised into three numbers greater than 1 to be the length, width and girth of a solid cuboid. The closest thing I could immediately find was adding two monocubes (or, you know, cubes) to the set to bump up the volume to 147 which can be factored as 3x7x7. And it turns out that packing them into this shape (as in the above picture) is a fun and satisfying challenge. And it can be made more challenging by specifying the position of the unit cubes before hand to yield a symmetrical configuration.

Schematic for a symmetrical solution to the 3x7x7. A square indicates the piece extends up into the layer above, and a dot indicates it extends down into the layer below.

Isometric diagram of the above solution.

Planar Pentacubes

These are the twelve pentacubes whose cubes all lie in the same plane - essentially, just chunky pentominoes. As well as solving into cuboids of sizes 1x6x10, 1x5x12 etc., these can also be solved into 2x3x10, 2x5x6 and 3x4x5 cuboids. And these, from my experience, are way harder than the polyomino rectangle counterparts. There are apparently 3,940 solutions to the 3x4x5 but you wouldn't think it trying to find one manually. I had tried on and off for several months and never really even got that close until a few nights ago when I found the below solution.

The 2x3x10 cuboid is even more of a nightmare, and it's mainly because of the I-pentomino (or rather, I-pentacube). Wherever you stick it it creates a narrow little space that seems to severely limit the pieces it can accommodate. I managed to stumbled across one of the 12 solutions, and it took several hours I'll never get back.

The 2x5x6 cuboid is interesting. It has 264 solutions, of which one is special in that none of the pieces extend into both 'layers' of the shape. Meaning that it's essentially the pentominoes solved into two congruent 5x6 rectangles then stacked up on top of each other. Like this:

I've noticed the shading on these images seems to subconsciously reflect where I am relative to the light source in the room. Here in the day time I have a window to my right, so I drew the surfaces facing it the lightest, but when I drew these ones a while back I must have been working at night with artificial light.

Another configuration possible with the planar pentacubes is the following sort of 'ring' shape. This can be made trivially in a few ways by adapting the solutions to the 3x20 pentomino rectangle, effectively folding it around on itself and bringing the ends together.

The ring shape. That 1x7 void in the top goes right through and out the bottom.

A better diagram of the assembly.

There's clearly a lot of possibilities with this set and with the full set of pentacubes that I've only just started to dip a toe into here. So I imagine there'll probably be further posts on here as I keep playing about and maybe gaining a little bit of competence with them. Right now it feels very trial and error.

In the mean time, check out David Goodger's page which has a way more in-depth exploration of what these sets can do.