Enneiamonds Again

A breakthrough, of sorts. This was the first time I managed to solve something with the full set of enneiamonds without having to resort to any kind of help from computer solvers or anything to knock in the last few pieces. The last few solves I did with them there was always a little bit of 'cheating' going on, not necessarily using the computer so fully solve the end of the puzzle but to check that a solution existed with the pieces I had remaining. But none of that this time. The training wheels are off.

Solve time took a while, but it was lots of short bursts spread out over a week or so. The first 80% of it (solving left to right) came in one sitting, probably an hour or so, then the end was just twenty minutes or so here and there, between university and other things, often getting all but one piece in there. And then today, by fluke more than design I'm sure, I hit upon a solution.

(That one little green unit triangle is there to mark 50 units along the top edge - I'd started off with more there, marking out every ten rows so I didn't build to the wrong dimensions like I have a habit of doing - but as the region between 50 and 60 units was the hardest to complete that marker stuck around and I guess I forgot to tidy it up.)

Here's a nicer image of the full thing, because the photo isn't exactly clear when there's two or more pieces of the same colour bordering each other. Clear acrylic would have solved this, but hindsight and all that.

Fig. 1: The completed trapezium.

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.

Solving Technique: 'Piece Substitution'

By sheer chance when solving a hexomino thing a few days ago I was presented with a really nice clear example of a technique that I'm sure I always just referred to in blog posts and things as 'piece substitution' but never actually ever bothered to clarify. So here it is. Consider yonder picture:

The two pieces in red to the side are the two I'm left with, and the hole remaining just won't accommodate them in any way short of physically snapping the pieces apart. The best we can do is getting the more irregular piece in there in the obvious place, leaving a longer thinner 'L' shaped gap than we're capable of filling. Like this.

The trick here is to look at the two pieces in light blue, one of which is the long skinny 'L' piece we need. Notice that we can do this:

which uses up our unusable P-shaped piece, and at the same time frees up out long piece, allowing us to fill the other hole and complete the puzzle.

Of course, there's no guarantee that it'll fall into place as nicely as that. Sometimes it's two pairs of pieces that can make the same shape that need to be swapped, or sometimes it's even uglier, like a chain of substitutions that free up one particular piece then use that piece to free up another. But it's a viable technique surprisingly often given how much of an utter fluke it looks.