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.

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

20x20 Diagonal Square with Heptominoes

It turns out the heptominoes can do four congruent right-angled triangles, each with a single-square hole at the right angle:

Fig. 1: Four 19x19 triangles

I solved the first three quadrants of this by hand (the red, blue and green ones) then utterly despaired at the thought of having to do the fourth one. As a result of using my normal solving technique, I'd found myself left with a selection of mostly quite blocky, squarish pieces then realised that these pieces don't generally lend themselves well to building wiggly edges. For finishing off relatively square shapes like the last corner of a rectangle or something they're fine, but for this I wasn't sure. So I wrote it off as a bad job.
Then a few days later, I drew the pieces into FlatPoly2 just for the sheer hell of it and it found a solution in about a minute.

These four triangles can then be put together in various different ways, including this 19x19 (Edit: it's 20x20, I can't count) diagonal square:

Fig. 2: The holes don't quite match the symmetry of the outer perimeter but whatever.

'Measure twice, construct once'

Building something only to discover that I've somehow put the central holes in the wrong place seems to be a common theme for me. I need to implement some 'measure twice, construct once' philosophy maybe.
A short while after I wrote that big long post about combining pentominoes and hexominoes into one set, I thought about how just possible it would be to knock out a 4xn rectangle with those pieces. I mean, each piece would physically fit into a 4 cell high rectangle, and if it's possible to squeeze all 1- to 7-ominoes into a 5x211* then this could well be possible too.

The whole 4-cell-high thing made it tricky but not as big a challenge as I had initially suspected it was going to be. After 45 minutes with my hexomino set (and a shoddy set of pentominoes cut out of graph paper to supplement them) I found the pattern below. With the two holes off-centre by one...

Fig. 1a: Side note, I'm not mad on this arrangement of holes but it's the most symmetrical one I can think of for two holes in an even x even rectangle.

Thankfully, as I was sketching down this solution, I notices it wasn't as bad as I'd feared - there was a central section (marked in light gray above) which could be flipped over and had the effect of transposing the two dots over by one cell - into the middle!

Fig. 1b: The finished construction.

Of course, there's rarely a nice quick fix like that, as I discovered with a heptomino construction from a few weeks ago. The harbour heptomino and its little central hole must have been accidentally knocked during the solve process and I hadn't realised. Only upon drawing the solution down onto graph paper did I notice it was slightly off. There was no quick little rearrangement of pieces that would fix it this time. I just had to solve the entire thing again in a few days' time.

Fig. 2: The heptominoes in an octagon with central hole. Coloured by how far each heptomino is from the edge of the construction, just to see how it would look really.

Notice the way the diagonally-symmetric pieces are all grouped nicely along a diagonal running from near the south-west edge up through the centre. Basically, I noticed in one of David Bird's nonomino constructions (typically, the one I can't find on the internet anywhere to link to; it's in S.W. Golomb's 'Polyominoes', page 116) that all (or at least a good chunk) of the rotationally symmetric pieces were grouped together in the central section surrounded by holes**. And somehow I'd never noticed that before. I just like the idea of little things like that, 'hidden' in plain sight within the construction that reveal themselves on closer inspection. (Another one I like is in a heptomino construction by Nick Maeder, third one from the bottom on this page, which has the crucifix-shaped heptomino in the centre of a triplicated version of itself, all positioned centrally within the pattern.)

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* I've seen this done but can't find it anywhere on the internet, but can't find in anywhere or I'd link it. Maybe it was in the middle of an Internet Archive binge of polyomino-related sites that are no longer up.

** Actually, there's something similar going on in this nonomino construction, also by David Bird. A lot of symmetrical pieces are grouped around the line of holes running upwards from the middle.

Heptominoes Miscellany

A few more heptominoes type things I've done recently, since lots of small posts is easier than one big post. Again, nothing particularly groundbreaking here.

Here's a 11x71 with 25 holes arranged in a central grille:

I solved the middle bit first seeing as it put the most restriction on pieces that could be used. Then built the left side (using up a handful of nice useful 2x2-blocky pieces in the process), then the right hand side. I don't know whether it's possible to do a 23x35 with a grid of 49 internal holes. The maths says yes but the pieces themselves might not allow it.

And here's an 11x69 rectangle with 3 holes.

I think I'm getting the hang of rectangles now; the biggest challenge here seemed to be finding a surface long enough to construct it on. Next time I think I'm going to have to try my hand at putting together some more involved figures, and confront my fear of building things with diagonal edges. (I mean, it can't be that hard can it? Look at how many heptominoes have got those wiggly zig-zag edges anyway.)

Congruent Rectangles with Heptominoes - Part 2

Part 1 is here. It's dead disappointing, mind, there's only one set of rectangles and it's the 4x 10x19s. But in that post I mentioned a couple of other possibilities for sets of rectangles that use the entire set of heptominoes between them. Here are some (coincidentally none of these are the ones mentioned in the above post...)

First there's this set of three 15x17s using 36 pieces each which I found ages ago, but didn't feel like it deserved an entire blog post to itself at the time

Fig. 1: Three 15x17 rectangles. You can almost tell by the pieces used in each which order I built them in.

In the first post or Part 1 or whatever we're calling it, I had somehow overlooked the fact that 108 divides by 6, and that a set of 6 rectangles should therefore be possible. The minimum number of holes that works is 12 (I think), 2 per rectangle, and that gives each an area of 128 cells which can be done as 8x16 (I can't rule out 4x32, but at the same time I'm utterly terrified of the prospect of trying to fit heptominoes into a 4-cell-wide anything.)

Fig. 2: Six 8x16 rectangles.

There wasn't really any pleasing ways of distributing the holes, this is about as good as it gets. Also, just look at those four hideous stretched-F-pentomino-looking pieces clustered in the middle of the top-right rectangle. I had accidentally forgotten to use those up sooner, and wound up stuck with them at the very end. That last rectangle (top-right, the one with all the supposedly easy to work with pieces) was partially a computer search job too. I think it was getting late and I needed the table space for something else so I had to speed up the solving process somehow.

Still not touching the nine 5x17s with centered holes though. It's gonna take one hell of a rainy day to drive me to attempt that.

Octominoes!

Well, this was an utter ordeal. I used the same method as my last little failed attempt at an octomino construction, the whole two MS Paint windows deal, but with a slightly differently shaped rectangle (20x148 this time, with eight holes, six of which are due to the holey octominoes.) And this time I ran into a different set of problems. I got about 90% of the way through the construction and did the usual sanity check, counting the amount of free area left and praying it divides by eight and I haven't drawn a heptomino (or a nonomino* for that matter) in there anywhere by mistake.

I counted 168 unit squares remaining. So far so good. But then I counted the remaining octominoes. And got 21. And 23x8 is not 168. Something was afoot, but it being fairly late on Friday night (because I lead such an exciting life...) I was far too tired to work out exactly what was up. so I hit the hay and resolved to see what was up tomorrow.

Oddly enough, as I was falling asleep I had some kind of major tetris effect going on, seeing endless visions of octominoes fitting together in various ways all wiggly like; a veritable kama sutra for tetris blocks. And, as I'd been reading a fair bit on organic chemistry recently, in my bizarre sleep-deprived state I was also fruitlessly trying to assign the 'systematic name' to each octomino based on the way it branched and twisted, not quite awake enough to notice that the octomino was not in fact a molecule.

Well, the next morning and with a fresh pair of eyes, I took another look at the almost-complete rectangle and couldn't immediately spot any foul play just by eyeballing it. My guess was that I'd somehow used a piece twice, forgot to cross it off the 'used bits' list the first time round. And so began the laborious task of verifying this - getting a fresh image of all the octominoes up, then crossing each one off as I highlighted it in my construction. And if I found one that had been previously used... well, I'd cross that bridge when I came to it.

And so it transpired I had inadvertently duplicated two pieces. Thankfully, they were both right near the bottom of the construction; I only had to backtrack about 15 pieces to be back in a state where the rest was solvable. I guess I had started to get a bit careless just as I was becoming too tired to think properly and in hindsight it was lucky I called it a night when I did on the Friday. So I continued (being just a little more careful this time) and managed to get the rest of the pieces in without incident.

Of course I used FlatPoly2 as a further sanity check when I had about 12 pieces left, just to make sure I hadn't solved myself into an impossible endgame. And in running that check I might have accidentally glimpsed the position of two or three pieces that allowed a solution. But I did the rest of it. All by hand! And it only took me, what, six hours or so? (Actually, putting it that way, it feels like a colossal waste of time I could have spent doing something useful, but...)

Anyway, here it is, in all its glory:

Fig. 1: All 369 octominoes in a 20x148 rectangle with symmetrically placed holes. Not pictured: enough blood, sweat and tears to fill an Olympic swimming pool.

All of this points to one thing - I need to get myself a physical set of octominoes by any means necessary. It's a little bit harder to use a shape twice when you've only got one of 'em to hand.

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* Or is it 'enneomino'? I think I've seen both but I can't decide which I prefer.

Combining Pentominoes and Hexominoes (into one nice big happy family)

In an earlier post, (this one in fact) I mentioned the possibility of combining the sets of hexominoes and pentominoes together, creating a nice (if mathematically incomplete) set of 47 pieces covering a total area of 270 square units and not having any of the irritating parity issues that make hexominoes on their own such a ball-ache to build shapes with.

And as an extra added bonus, you can hoard some of the nicer pentominoes until the end (ones like the P, L and V) which makes solving things with these generally fairly easy too. Nice practice before tackling pure hexomino things, if nothing else.

Rectangles

So... the rectangles. 270 divides by nearly anything so we've got a lot of choice here. 1x270 and 2x135 can be ruled out straight of the bat, because the pieces are too big. 3x90 I don't know, I mean, every piece fits within a 3xn box but I could see that getting ugly fast - look at how few 3xn solutions there are for just the pentominoes alone! Maybe it has a solution, like the thin solution with the pentahexes, but I don't have the balls to go looking for it.

5x54 is where things start to get definitely possible, and the rest of the possible rectangles (6x45, 9x30, 10x27 and 15 by whatever 270 divided by 15 is) are all easy enough if you don't mind a little trial and error.

Seriously, get yourself a set of these. Buy them, make them, hack them out of the back of a cereal box, whatever it takes. You won't regret it.*

Fig. 1: Some rectangles. Looking at the notebook I transcribed these from, some of these were done back in 2015 or so, before I'd really refined what little technique I have.

Now, moving onwards to some more tricky stuff...

Rectangles 2: The Revenge

Since 270 divides up so well, not only can you make rectangles but you can do sets of congruent smaller rectangles too! Check it out:

Fig. 2: Three 5x18s, two 9x15s, three 9x10s and five 6x9s.

The challenge here is making sure you have the right balance of pent- and hexominoes in each rectangle. For example, for the bottom row (the five pink 6x9s) each rectangle is 54 units, and the only way to get 54 by adding 5's and 6's is either nine hexominoes (9x6) or six pentominoes and four hexominoes (6x5 + 4x6 = 54).

For the same sort of reason, we can rule out six 5x9s. Each rectangle would have 45 units, which can only be filled by 3 pentominoes and 5 hexominoes (since 3x5 + 5x6 = 45) but that would require 15 pentominoes total so it can't be done :(

Nine 5x6s is out too. A 30-cell rectangle must have either 0 or 6 pentominoes so that the remainder can be filled with hexominoes - that is, it must be either all pent or all hex. This would mean that all the hexominoes would end up together in seven 5x6 blocks... but then you could just push those together and make a 6x21 rectangle of pure hexominoes, which ain't possible because of parity constraints.

Parallelograms

Moving on to some less rectangular shapes...
Again, we've got a lot of variety possible here -  the base lengths 6, 9, 10, 15, 18, 27, 30, 45 and 54 are solvable in theory, but in reality it's a little bit of a different story.

Of these, I've only done a selection of these by hand, mainly because they're not massively challenging or interesting (then again, if I didn't bang on about uninteresting things here this blog would have like no posts.) The longer the diagonal sides, the trickier it is, just because you run out of wiggly pieces building one side then have to use the nice easy pieces on the second side.

Fig. 3: Two example parallelograms. You can solve the rest yourself (with your newly-acquired set of pieces!) because it takes so long to transfer them from notebook sketches into pretty diagrams and it's really not worth it for not-that-interesting solutions like these.

Of course, you don't need to stop there. With the aid of computer search ('cause I'm lazy like that) I found the following two even taller skinnier parallelograms

Fig. 4: Just look at these freaks of nature. And I'm willing to bet the even thinner 6x45 is possible too, but so far FlatPoly2 has failed me.

Rectangles with little bits missing from inside 'em

...because not only does 270 factorise well, but some of the numbers just above it factorise fairly nicely too!

Fig. 5: Holey rectangles. (I fought the urge to follow that with 'Batman')

Getting all Fancy

Depending on how much you like solving wiggly diagonal edges there are all sorts of other possibilities too, so long as you don't mind the odd hole slap-bang in the middle of everything. Check out the following bad lads, sorted in order of how many tears were shed before finding the solution:

Fig. 6: How many images have I produced for this blog? And I still haven't settled on a colour scheme... sheesh!

That central heptomino is completely arbitrary really, it's just the most fitting one that has a few axes of symmetry. The same patterns could be done with a hole the shape of the H-looking heptomino (for example) but it just doesn't look as neat.

Another option (which to be honest I haven't really looked at myself on account of being proper lazy) is putting additional restrictions on the way the pentominoes are distributed, for example, making sure that each pentomino isn't touching any others, or that they are all bunched together in particular ways. The solution below has the pentominoes grouped into two symmetrical end bits. Actually, I didn't solve this one thinking about it as a pent+hex construction, I did the middle hexomino bit and only realised the two pentomino end caps were possible a while later.

Fig. 7: A nice five-cell high pattern with the pentominoes bunched together at each end all nice and symmetrical.

Another Possibility

A slightly more complete-feeling set would be the entire range of 1 through 6-ominoes, but sadly, their total area is 299 units, which you can't really do a lot with. There's a 13x23 rectangle, and by extension two 13x23 parallelograms, and adding a unit cell hole allows things like the 24x24 right-angled triangle.
But then you've got monominoes and monomino-sized holes together in the same construction, which just isn't very aesthetically pleasing for some reason. And since you've got all the tiny little triomino, domino and monomino pieces to work with, these aren't that much of a challenge either. So it's not such a good set after all.

Heptominoes next time. I promise.

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* Polyominoes the Blog accepts no responsibility for any regret caused by the acquisition of any polyform sets.