4½-ominoes (sort of)

An interesting set of pieces brought to my attention in some discussions on the Puzzle Fun Facebook group a while ago. Basically, you take a monomino that's been sliced in half diagonally and add it to a tetromino, creating a shape with an area of 4.5 square units. There are 44 ways of doing this, yielding a total area of 198 units² which divides up nicely in various ways.

The full set in (almost) an 11x18 rectangle. Laser cut from MDF then coloured in a way that looks like a six year old did it.

These pieces aren't quite a continuation of ideas from the 3½-ominoes I discussed here; there's a subtle difference. I think those were created by taking a tetromino and removing a half-square, meaning there are pieces like the ones below that are essentially a monomino and domino bridged together by the triangle, which can't happen if we're determining the piece set additively like we are doing here.

Fig. 1: The original set of '3½-ominoes, with the two pieces that can't be thought of as augmented triominoes shown in dark orange.

So twelve of the 14 3½-ominoes are of the same sort as these tetromino+tan shapes. And actually that's nice because combining those twelve with our 44 here gives a total of 198 + 42 = 240 square units. Which means rectangles ahoy. And the fact 12 and 44 both divide by 4 means we should be able to solve four small congruent rectangles, each with 11 big pieces and 3 small pieces. In theory...

But first, things you can do with just the 4½'ers on their own.

Rectangles

198 divides up nicely: 2x3x3x11. Which means rectangles of 6x33, 9x22 and 11x18 are all possible (I'm not actually sure if 3x66 is possible - it seems too thin and the pieces that look like mutilated X and Z pentominoes might cause issues since they'll touch two sides of the rectangle no matter how they're oriented. Might need further investigation). I've so far managed to solve the 11x18 (the shape of the tray I made to hold the pieces) and the 9x22, but I haven't yet had a crack at the 6x33. Call it an exercise for the reader.

Fig. 2: The pieces in a 9x22 rectangle. I didn't bother digitising it because the piece boundaries are nice and clear with these.

Notice that each piece has one unit of diagonal edge with which it touches another piece, and that this pair of pieces combines to make (effectively) an enneomino - see the image below, where I've drawn on a few of these pairs. This is how I solved the latter half of the above rectangle (and is in fact my go-to technique for solving with these shapes). Once you've burned through all the really scary tricky pieces, pair the rest up to make nice chunky likely-looking enneominoes and tile the remaining shape with those, so as to not be left with any diagonal edges that are too awkward or concave to fill with any of the remaining pieces. Sometimes you find yourself with one of those pieces that's a P-pentomino with a slice taken out of it, and nothing that'll reach inside and fill it.

Fig. 3: Enneominoes made up of pairs of pieces marked a bit more clearly. (The other pieces make pairs too, but these were the deliberate ones.)

Doing this technique reduces the puzzle to something a bit more rectilinear which is perhaps easier for the human brain to deal with then all those 45° angles. And additionally, the choice of enneominoes you end up with isn't set in stone, quite often the pieces are interchangeable (e.g. two pieces can fit together in two different ways to make a sort of adaptable Swiss-army enneomino) which makes things easier still.

Easier, but not easy. These things are still way trickier than the likes of polyominoes and polyhexes by my reckoning.

Non-rectangular Shapes

198's proximity to 200 means that the following 'rounded' rectangle with clipped corners is possible:

A little trickier than the plain rectangles before, just because those corners demand certain pieces placed certain ways right off the bat. I get the feeling that two 10x10 squares with two corners clipped each ought to be possible too. And might be a nice way to store a set of these, if one were to make a two-layer high box for them.

In fact, I think I'm only really just scratching the surface of what can be done with this set, so expect a 'Part 2' post in the future. (In about, what... 2029, if the recent blog posting schedule here is anything to go by...)

Combining the 3½- and 4½-ominoes

As mentioned before, the 44 pieces of size 4.5 and the 12 of size 3.5 give a total area of 240 square units. And even better, since there's a sprinkling of smaller pieces in there, the solves should be (fingers crossed) slightly easier than a solve with just the 4½-ominoes alone.

Fig. 4 or 5, I lost count: Two 11x11 squares with centre holes, filled with the 3½- and 4½-ominoes.

Here's one such construction, using the 44 4½-ominoes and the 12 3½-ominoes (six in each square). After having solved this I take back what I said just before about them potentially being easier - there's really not much in it. Though I do think the pieces I'm saving for late in the game are not the best. I perhaps need to focus more on saving useful pairs of pieces than individual pieces, which I'm not really doing.

I have one more final thought that occurred to me while I was drawing up the image for that last solution - these sets of pieces ought to allow eight pieces to meet at a vertex. Is it possible in practice to find solutions (to any fairly regular shape) that include this mythical 8-way corner? Looking through the solutions in my notes there's a few that have corners where five pieces meet, but that's the maximum so far. I imagine you'd need to start with the eight touching pieces first, and build outwards.

Miscellaneous Polyform-related Puzzles, Part 1

It's been ages since I last wrote anything on here (so much for that promise I'd update this with a little more frequency) so have some first-draft-quality rambling about polyform-ish puzzles that are sort of just close enough to fit the theme of the blog.

3x3x3 Serially Interlocking Cube

I built this one after seeing a diagram of the pieces in Stewart T. Coffin's The Puzzling World of Polyhedral Dissections and not being able to visualise how on earth they fit together. It's a four-piece puzzle with the unusual property that it must be assembled and disassembled in a certain order (hence 'serially interlocking'); the pieces all kind of hold each other in place. It's not too hard a solve; the very first assembly takes a little bit of thinking but any assemblies and disassemblies after that aren't hard at all because you end up just sort of memorising what goes where, which spoils the fun a bit.

The two visible red bits are two parts of the same piece. Likewise with the blue bits. It's a weird puzzle.

Building this was something of a trial run for other polycube puzzles. I wanted a full set of pentacubes for the longest time (and still do), and I decided the cheapest way to make polycubes was probably to buy the cubes and attach 'em together myself. So I went out (okay, stayed in and fired up the internet) and ordered in enough wooden cubes to make Mio Naganohara seethe with envy, then set about making the tetracubes you can see half way down this page as well as these pieces. The problem is, the combination of my own shoddy handiwork and the cubes not being exactly cubic meant that the resulting pieces didn't quite fit together as snugly as I'd hoped. Not so bad with the relatively simple tetracube shapes, but with these pieces there's C-pentomino-esque indentations that need to be able to fit round another cube and those require a degree of exactness. So I spent a good couple of hours sanding down the glued-up pieces just to make them actually fit together and then come apart afterwards without getting stuck.

And then I scribbled on them with felt tips because I couldn't be arsed to look for paint. It's eye-catching from a distance, but when you actually hold the pieces and examine them up close it's like something a child would make in their first woodworking lesson. But it's the best I can do without resorting to things like putting effort in...

Fig. 2: The pieces in all their terrifying glory. Between this photo and the one above there's probably enough info for readers to make their own set if desired.

Polyarcs

Another one I made myself. It's made of the same materials as my heptiamonds and their tray, which makes me think I just chucked the SVGs for these in the margins then sent the lot to the laser cutting place to get more puzzle for my money.

Polyarcs are the polyforms where the base units are the two shapes you get drawing a quarter circle inside a unit square. There's a bunch of information and constructions involving the 1- through 3-arcs over on Henri Picciotto's site.

Fig. 3: These bad boys were spared the humiliation of a felt tip paint job.

This is the set of two 1-arcs and seven 2-arcs, and they fit in that 2x4 tray in dozens of ways; it's not particularly hard to get them back in there after tipping them out. Putting them in so that the wood grain lines up works as a marginally harder challenge. I kind of hoped I'd be able to make some other shapes with these too, but they're kind of limiting in that there's such a small total area to work with. They do the rounded shape from that linked site in a few ways too.

Sadly, I don't think they can make the 4-way rotationally symmetrical shape on the right - I can get all but one piece in so I'm wondering if there's some parity-like constraint preventing this from working, or if it's just a case of 'the pieces won't go'.

Unnamed 8x8 polyomino puzzle

This one's a doozy. At least it is if you take your dictionary, scribble out the definition for the word 'doozy' and write 'pain in the ass' there instead.

It solves into an 8x8 tray, not unlike the pentominoes + O-tetromino set I probably thought this was when I first bought it (for the grand sum of £2). But the pieces are bizarre. Four of the most uncooperative pentominoes (U, X, T and W), the T- and U-hexominoes (!) and a handful of larger 'ominoes seemingly picked at random. And the worst of it is, it's a really infuriatingly tricky puzzle. There's not a time I've picked this up and not put it down feeling like uppercutting a nun. Even when I manage to solve the thing, the rag-tag assortment of pieces is so illogical and un-mathematical it winds me up anyway.

Going through my cupboard of polyomino stuff and failed laser-cutting experiments, there's probably enough stuff there to write a second one of these posts some time, so consider this a Part 1.

Wide Tetrominoes, and a Different Kind of Parity

Note: This post was written ages ago (like, February or something) but for whatever reason I didn't publish it at the time, I think I'd decided it wasn't up to the standards of other blog posts. But today I've looked at the shambles that passes for other posts and this isn't really noticeably worse so here goes...

Introduction

Recently, it dawned on me that being able to get things laser cut has opened up a lot of possibilities for polyomino/polyform-related mischief. For example, for smaller sets of pieces if I was to cut the pieces in transparent acrylic then cut a tray for them from two opaque white bits, it would make a nice cute little self-contained puzzle. I decided to do a trial run, drawing up the .SVG files one evening for the hexiamonds set below:

Some day I'll invest in a proper camera. Today is not that day.

Which worked fantastically (read: I managed to glue the two parts of the tray together without attaching any of my fingers to the table.) It's a cute, if fiendishly tricky, little puzzle.

Wide Tetrominoes

So the obvious question was, what next, seeing as little puzzles like this are completely feasible? While brainstorming ideas I stumbled across a set of pieces I hadn't really seen mentioned or investigated anywhere else. These ones:

Like tetrominoes. But made of rectangles. Preferably ones where the aspect ratio means you have to join long side to long side, short side to short side, and can't have any pieces in at a 90° angle.
There are nine of these possible, and with an area of 4 square (well, rectangular) units each, giving a total of 36 units to play with. Promising! (If playing with units is your thing.)

I did the check for checkerboard parity, just to rule that out. Tetrominoes have this issue, as do one-sided tetrominoes (a.k.a. the Tetris pieces) so I wanted to be absolutely sure. And they looked fine. the presence of two variations of the T-tetromino meant overall parity was balanced. 6x6 and 4x9 rectangles, here we come!

But after several minutes of playing on with a set of these (well, simulating a set with pen and paper) I couldn't get all 9 in. A few near misses but no cigar. Maybe this is just one of those puzzles where there are no solutions and no good reason for it.

A Good Reason for It

At around the same time as all this I was balls deep in an octomino construction and was concerned that parity restrictions could rule out a solution for this shape and render the hours I'd sunk into it a bit of a waste. Sure, it would still have been valuable practice for other polyomino constructions in the future, but it'd still be a shame. Anyway, while worrying about this I found a page* detailing other restrictions on the set of 363 unholey octominoes, and various subsets of those. One of these was to consider not a checkerboard colouring of the pieces and solution shape, but to colour every second column, like this:

The table below shows the total imbalance each piece can contribute towards the total number of white and black squares in the construction. Five have no overall effect, three imbalance the amounts by two, and one (the vertical I-tetromino) either contributes four white squares or four black squares.

Very professional-looking MSPaint table here. LaTeX eat your heart out.

And for these to all fit the 6x6 box, the overall difference between the amounts of black squares and white squares must be zero. That is, ± 2 ± 2 ± 2 ± 4 = 0. Each ± can either be a plus or a minus depending on whether the unbalanced piece contains more white squares than black or vice versa.

In fact, no matter which combination of + and - you pick to stick in there, the equation never gives zero, meaning that there's no way the set of pieces can cover the same number of black and white squares.

Similarly, the difference in counts of each colour square for the 4x9 and 3x12 cases are 4 and 12 respectively**, meaning that the set of pieces will never fill them either. In fact, the 3x12 cases are even easier, it's just now dawned on me when writing this that the length-4 I tetrominoes wouldn't fit inside.

What next?

Well, maybe there are nice things to be done with the tetromino set but I'll leave that as an exercise for the reader (because I can't be arsed to do it myself, as per usual). The next thing worth looking into seems to be pentomino-analogs made of rectangles. In fact I'd be highly surprised if this hasn't already been independently thought of and subsequently done to death by others.
But I'll have a crack at it anyway. In another blog post, another time.

---

* It's on The Poly Pages somewhere under 'More Octomino constructions', but I can't make a link to it work properly for some reason. Linking to Poly Pages gets weird sometimes.

** For the rotations of the rectangles shown in the diagram. the 9x4 and 12x3 cases, having an even number of columns each, both have parity 0, same as the 6x6.

Polytans, polyaboloes, whatever you want to call them

Polytans (also polyaboloes, depending on which website you're looking at) are the shapes made from joining isoceles right-angled triangles (45°-45°-90° triangles*) together edge-to-edge. Theres one 1-tan, which is just the triangle on its own, then three 2-tans, four 3-tans (tritans? triaboloes?) and 14 tetratans/tetraboloes. The numbers grow colossally fast compared to the numbers of polyominoes, polyiamonds or just about anything else; probably partly because there's often more than way to append a triangle to an edge.

Here's some pretty pictures of the 1- through 4-tans:

Fig. 1: Here they are, courtesy of a tedious as balls half-hour in Microsoft Paint. Upon completing this I realised I could have just generated them using Peter Esser's solver and took a screen shot. You live and learn.

Above this, there are 30 penta-tans, 107 hexa-tans, 318 hepta-tans and none of those look like real words, I can kinda see why the 'aboloes suffix gets used. Yeah, pentaboloes and hexaboloes rolls off the tongue a lot better.

I have made little acrylic sets of the 1- to 4-aboloes to play with. The larger sets I haven't gotten around to doing yet, partially because coronavirus and lockdown and all that, and also because the place I usually get them made have upped their prices and I've only got so much annual budget for polyomino-related spending. But tetraboloes are more than enough of a fiendish challenge in the mean time.

Surprise, surprise, I still can't take a decent photo for toffee.

The combined area covered by the triaboloes and tetraboloes is 34 which is a bit of an ugly number but it's still workable. For a start, we can do rectangles of area 36 with the corners snipped off, as in the image below. this woks for 6x6, 4x9 and 3x12 rectangles.

Fig. 2: The triaboloes are highlighted in a slightly lighter shade of yellow.

 Difficulty-wise, the thin rectangle doesn't seem noticeably easier or harder than the square, but then again they're all infuriatingly tricky for something so deceptively simple-looking. Best technique seems to be to try and use up pieces with lots of diagonal edges first. But that only gets you so far. Prepare for lots of trial and error.

And when you turn over the tray I made for them there's the following configuration, which is just unfairly difficult. The centre requires the square shaped bit, leaving the remaining 13 pieces to fill the square doughnut around it.

Fig. 3: The design on the other side of the tray. Finding a solution to this is left as an exercise for the reader.

Apparently there are 45 solutions to this. I've sunk literally hours into it by hand and found only one so far.
---

* Fun fact: It's insanely hard to describe specific triangles without a diagram.

3½-ominoes?

Not strictly polyominoes but close enough really. I have no idea how I first found out about this set of pieces - I always had a hunch it was the set used in Martin Watson's puzzle 'DemiTri' (which doesn't look like it's on the site any more) but that says 12 pieces. And attempts to create similar sets in Peter Esser's program by slicing tetrominoes or adding half-squares to triominoes yields 12 and 13 piece sets respectively.

Fig. 1: The set, crudely rendered in Microsoft Paint.

But this looks like a complete set to me, all the ways of putting together three squares and a triangular half-square (if there's a fifteenth one and I've missed it let me know) and it's got a total area of 14 x 3.5 = 49 unit squares, which suggests (among other things) a 7x7 square:

Fig. 2: Here's one I made earlier.

Technique for solving these is a tad unusual. Since they each have one diagonal side, if the outer perimeter of the shape you're filling has no diagonal sides than the pieces are effectively 'paired up' by joining two at the diagonal edge. This results in any shape like this being split into seven heptominoes which can be in turn split in half to give two pieces. So my technique was to first put together a couple of promising looking heptominoes (i.e. ones containing 2x2 or 2x3 rectangles) then trying to fit those together. I used this method to cobble together the shapes below.

Fig 3. 10x5 with a bite taken out of it.

Fig. 4: These. Which can be put together to make the shape in Fig. 3.

Fig. 5: More shapes!

...and this nightmare shape that I found with a solver because there's no way I'd have the patience to do it by hand.

Sadly, these seem to be more limited with what you can do with them compared to, say, pentominoes or hexiamonds, both of which have a similar number of pieces (that, or I'm just really uncreative. I have a hunch it may be the latter.)
And there's also a scary bonus thought - this set of pieces is just one in a family. There's scope for doing things with the sets of pieces which are four squares and a triangle*, or two squares and two triangles, and so on, and at that point we're approaching just regular sets of polyaboloes or polytans or whatever people generally call them.
But that's going to have to be a post for another time.

---

* If I counted correctly there's an odd number of these which might further limit what can be done with them.