Squares Part 2: This time it's Octominoes

(Part 1 is here, I'd recommend reading it first.)

After making the heptomino constructions in the above post, octominoes seemed like the logical next step. And we are spoiled for choice when it comes to which size squares we can make from them.

Firstly for solid squares with no centre hole, n² for when n is a multiple of 4, so we can construct 4x4, 8x8, 12x12 and so on. Great stuff. But it gets even better..! Through some happy coincidence of mathematics it turns out that for all even n, n²-1 is divisible by 8. This means we can fill any odd edge-length square with octominoes as long as we give it a one-cell central hole. In fact, the only sizes we can't theoretically do are the squares with edge lengths even but not a multiple of 4: 2x2, 6x6, 10x10 etc. (Oh, and really tiny squares like 1x1 which are too small to do anything with. I guess in a strange way, the 1x1 square is just zero octominoes around a central hole so it sort of counts, but it's a weird edge case.)

Next step - pick a bunch of these squares whose total area is 2952. There are many ways of doing this; I picked one that included as many different sizes of square as possible. One of each possible size from 3x3 all the way up to 21x21, and then a second 12x12 and 13x13 in order to make the numbers work.

The 3x3 was easy enough to solve, but it was all downhill from there...

This took all afternoon and long into the evening to solve by hand. Early on the most challenging part was digging through the box of octominoes to find the specific bit I needed - my set are made of fluorescent acrylic so I can shine a UV torch into the box to make the edges stand out a bit better, but it's still infuriating digging through three-hundred-and-odd near identical pieces looking for a specific one. Especially when it turns out the piece I'm after has already been used up.

But by the time I was down to the last three squares (the three largest; I did them in ascending size order because doing one of the tiny ones last just sounded utterly horrifying) I was faced with a different problem. The thing with solving lots of separate squares is that there's lots of edges - way more than just solving one large rectangle. And the boundary between solution and empty space demands a higher proportion of straight-edged pieces. And pieces capable of filling 2, 3 and 4 cell deep indentations right at the edge of the construction.

Stuff like this.

Early on this wasn't much of a problem, but the further I progressed the less pieces I had remaining capable of plugging a hole like this. On the very last square I totally ran out and had to just build really carefully so as to not create a hole like this at the edge. And I managed - just.

The other tough bit was each time I finished off a square. Even with all the forward planning in the world, you don't seem to get a lot of say in what the space left for the last 2 or 3 pieces in a square looks like. And although sometimes you can partition it up without using an already-used piece or a nice piece with a 2x2 block in it, most times you've just got to take one for the team and sacrifice a potentially useful late game piece. It hurts, but there's no other way. And with octominoes, there's so many pieces with 2x2 blocks you can afford to use up one or two (or ten) prematurely along the way. Having to spend the big P-shaped piece in the top-right of the 19x19 square though... that wasn't fun.

So yeah, other related challenges. For the reader of course, because on completing this I swore off touching another octomino for at least a week.

• My construction is 17 squares total. Can anyone improve on that? (As in: more, smaller squares.)
• My construction has 15 distinct sizes of square. Is there a way to increase that, to take those duplicated 12x12 and 13x13 squares and somehow use that area to introduce another variety in there?
  

Squares

This has bound to have been done before, it seems too obvious a thing not to. But rediscovery or not I'm going to write about it at length anyway and nobody can stop me.

Imagine this: squares, right, but made out of polyominoes. Gripping stuff, eh?

Pentominoes

With tetrominoes and below you can't really do a lot, for all the usual reasons. Tetrominoes have got parity imbalance, triominoes and dominoes are just tiny sets you can't do a lot with, and while I suppose you can make a 1x1 square out of the set of monominoes (all one of them) it's not the most interesting thing in the world so we'll jump ahead to where it starts to get interesting.

One set of pentominoes can pack a 5x5 square and a 6x6 square simultaneously, if you don't mind an off-centre hole in the 6x6 to bring its total area down to 35. Shown above is one way of doing this, but there are at least 2 more, not counting rotations and reflections as distinct solution. Sadly there is no combination of squares without holes that the pentominoes can fill, just because 60 can't be partitioned into square numbers divisible by 5.

Hexominoes

This lot are frustrating due to the usual parity constraints but nevertheless we'll give it a go. Square sizes permitted are 6² = 36 and 12² = 144 for solid squares, and 5² = 24+1, 7² = 48+1, 11² = 120+1, 13² = 168+1 if you want to allow a square with one solitary central hole. Which feels like bending the rules but alleviates parity issues somewhat, and will be needed for heptominoes and above where we've got pieces with holes whether we like it or not.

Making 210 with a combination of the above squares is our next problem. Each of the above squares would contain an even number of hexominoes, meaning that no combination of them would ever contain exactly 35 hexominoes. Damn.

One avenue to explore is to allow duplication of one of the hexominoes, bringing the set size up to 36 and total area to 216. This can be broken into squares, the most obvious partition being six 6x6 squares. This is... difficult. I've tried it by hand, I've tried running various software to find a solution, no luck so far. All I know is that for this to work the duplicated hexomino must be one of the ones with 2-4 parity imbalance. And that it's possible to solve six 6x6s for the set of hexominoes plus triominoes. So if solutions do exist, they look to be pretty few and far between.

But there are other ways of making 216 from the above numbers. 6² + 6² + 12² works, as does 7² + 13² with a central hole in each. The duplicated hexomino in each case is shown in a different colour (again, the duplicate must be one of the 11 pieces with unbalanced parity.)

6² + 6² + 12² = 216

 

7² + 13² = 216 + 2

As a general rule, the more squares and the smaller the squares are, the harder it is to find a solution.

5² + 5² + 7² + 11² = 216 + 4

Another fun challenge would be to place the two duplicate pieces in a symmetrical or otherwise aesthetically pleasing way. I've just been letting pure chance dictate which piece is duplicated and where it ends up, i.e. solving the 35 unique pieces and hoping the 6 remaining squares are all joined together. It works but the solutions aren't always the prettiest.

Heptominoes

This is where it starts getting fun. For the first time we've got a real choice of how many squares we can do and what size they are. For unholey squares the sizes possible are 7², 14² and 21² (28² > 757 so we don't need to worry about that) and for squares with a centre hole (which we're gonna need for the harbour heptomino) the available sizes are 13² and 15². Actually, a 6² or 8² would be possible but the hole would be off-centre so I didn't consider these. We can afford to be picky here.)

I didn't do an exhaustive list of what was possible, I just found a bunch of squares whose area added up to 756 and jumped right in. The first solution I found was the picture below.

Three 7x7s, a 13x13 and 21x21 with heptominoes.

I did the little squares first because they're the most restrictive in terms of which pieces can be used then did the great big square at the very end. I then realised that instead of having a 21x21 square, I could create four 7x7s, two 14x14s and a 13x13 which would yield a little more challenging of a solve.

 

'A little more challenging' is putting it lightly.

For a start, it took me two attempts. Admittedly this was down to my own stupidity - on the first attempt I'd done the four small squares first then moved onto the larger three, only to discover near the end that one of my 7x7 squares was actually a 6x7 and I had one leftover piece too many. The second attempt (the one pictured above) just took ages to do. Same solve order, but when I got to the last square (the bottom right 14x14) I was left with some really difficult uncooperative pieces and completing it took probably about six hours spread out over the space of the weekend.

The obvious next step here is this: One 14x14 is equivalent to four 7x7s. So It should be possible to solve a full set of heptominoes into the following:

• Eight 7x7s, a 13x13 and a 14x14.
• Twelve 7x7s and a 13x13.
  

Whether either of these is possible (or feasible solving by hand) is another matter entirely. I noticed with the construction with four 7x7s I was running out of pieces that would fit comfortably in corners or along edges by the end of the last square, and introducing even more edge is only going to make that worse. I might tackle those other possibilities at some point soon. But that's a pretty big 'might'.

Little stopgap post while I think of something decent to write about

It's been a little while since I last posted anything on here. Here's a nice 4-way symmetrical heptomino construction, does that make it better?

Fig. 1: Those corners are surprisingly restricting of which heptominoes can make them.

 

In the past I used to crack open the box of polyform stuff in evenings as a way of relaxing and calming down, but recently I stumbled upon this monstrosity which seems to have the opposite effect on me:

Fig. 2: Eww gross.

Called the 'sawblade' in my little notebook where I sketch out possible constructable shapes, and although its just a pure hexomino puzzle it's way harder than it has any right to be. I found the above sort of near-miss early on, which fit all the pieces but had an asymmetrical clump for a central hole which just looks wrong. I mean, even the 'proper' solution has a 2-fold symmetrical hole in the middle of an otherwise 4-fold construction so it's never going to be perfect, but this was just too imperfect to live with.

Fig. 3: Better (marginally...)

It took a further three days of trying on and off, just whenever I had half an hour or so spare until I finally stumbled on a solution that had the central hole looking some way presentable. The real nightmare pieces in solving this were the long straight bits, the I-hexomino and the various pieces with a 5xn bounding box. And the big L-shape piece. Usually these can be sat against the flat walls of rectangular constructions but not in this case... Learning to use these up early on seemed to be the key to cracking this one.

I'm toying with the idea of laser-cutting a fresh new set of hexominoes (again...) this time using transparent acrylic so I can see the boundaries between pieces. It's tempting to make a tray for them shaped like the above pattern, especially since it's approximately square so the entire tray would be relatively compact. I could maybe have the central heptomino 'hole' in a contrasting colour too. The only drawback of all this would be that every time I wanted to tidy the pieces away into the tray I'd have to go through the ordeal of solving it.

 

 29th? Yesss! didn't miss a month!

More Octominoes

Here's a few more (okay, two) octomino constructions from over the past few weeks. Mainly boring shapes like rectangles, just because I'm limited by the size of the desk in my room. A big diamond with diagonal edges would be awesome but it's just not going to fit on the available working space, in fact this one with height 57 really pushes the limit of what I can do, it was practically overhanging the edge of the desk:

Fig. 1: A 57x63 frame that fits the 23x33 heptomino rectangle perfectly inside it.

An odd thing with n-ominoes is this - as n increases, not only does the number of n-ominoes with a particular feature increase, but the proportion of all n-ominoes with that feature increases too. Take polyominoes with holes for instance. With heptominoes there's one, giving a proportion of 1/108 = 0.93%. With octominoes it jumps to 6/369 = 1.63%. And this trend seems to keep going, as shown below.

Counts for the really big polyominoes taken from here. Sorry it's an image and not a copy and paste-able format, blogger hates tables for some reason.

My gut feeling is that this keeps going, approaching 100% but never quite getting there, instead tapering off in a big s-shaped curve.

Here's a graph that has everything wrong with it. Default OpenOffice design, tidied up in MSPaint, uses percentages which I assume isn't good form for actual scientific purposes... I may as well have gone the whole hog and done the captions using WordArt.

The same trend seems to hold true for other features too, such as polyominoes containing a 2x2 block inside them somewhere as a subset of their squares, or pieces with a two unit deep well (like the pi heptomino.) I guess it could be in part down to the fact that as n increases there's more scope for a given piece to exhibit several of these features at the same time, e.g. have an internal hole as well as a 2x2 sub-section. Maybe? This feels like the kind of territory that would need approaching from a rigorous mathematical angle, but I have no idea how to do that so the best you're going to get is me guessing wildly based on a hunch I got from spending too long playing with glorified jigsaw puzzles.

Here's a 29x102 rectangle with the octominoes. David Bird beat me to the punch by several decades but I'm still going to share it here anyway, dammit.

Ooh, and we get a final bonus guest solution this time too... I had shared an older polyomino solution (the one with concentric layers of 5- through 8-ominoes) to the Puzzle Fun facebook group, mentioning that there was a possibility for a 121x129 layer of enneominoes to surround the entire construction. Sadly, there was no arrangement of holes that would preserve the symmetry of the hole distributions in the inner layers, but I settled on something close enough and Patrick Hamlyn found the following solution for it:

(Click for full-size.)

Interesting to note are the pieces his search program saved for last - the ones just below the bottom-left corner of the octominoes section. There's a lot of the kind of pieces I'd expect, blocky, rectangular pieces, but also a lot of long thin bits with a completely smooth edge on one side. I'd never considered holding onto these pieces that late in a solution, but clearly it gets results so it's maybe something worth me experimenting with while solving by hand.

And yeah, if anyone fancies wrapping a set of dekominoes around that, that would be grand.

A Fun Little Challenge with Pentominoes and Hexominoes

 I've probably seen this done somewhere else but I can't think where or else I'd drop a link in and let them explain it better than I can. Basically, the idea is to create a rectangle (or another shape, I won't judge) out of the pentominoes and hexominoes together - 270 units total so you've got a few choices here. But, there are a few restrictions you can place on the positioning of the pentominoes.

For the easiest option, forbid the pentominoes from touching each other (either by sharing an edge or touching point-to-point.) Or if you want to ramp up the challenge a bit, prohibit the pentominoes from touching the edge of the rectangle too. This is essentially a hexomino construction with 12 pentomino-shaped holes in it, if you want to think about it that way. Below are pictures for the 15x18 and 10x27 cases, but there are several other aspect ratios possible, the thinner the rectangles become the harder I imagine it'll be to keep the pentominoes away from one another.

Fig. 1: The 15x18 solution in a colour scheme that would have Ikea's lawyers frothing at the mouth

Fig. 2: A 10x27 solution

Enneiamonds

You know that feeling you get the instant it hits you you've bitten off more than you can chew? I may have got just a little bit of that feeling when I first tipped out the set of enneiamonds onto my desk:

The actual pieces themselves are smaller than I'd anticipated, due to me scaling down the SVG files I'd made then forgetting before I sent them off to be cut out. But they still work properly, now with the added bonus that they don't hog my entire tabletop when I'm mid-solution.

They're a lot like heptominoes, if you were to take everything I hate about heptominoes and turn it up to 11. There's a similar number of pieces (160 vs. 108), including one with a hole, but their complexity seems to be upped just a tiny little bit - I think it could be the mixture of 60° and 120° angles which pushes the difficulty a step beyond. And there's a very small selection of easy looking shapes (see the top left of the photo where I've tried to pick them out). It's less than the amount of nice heptominoes, but enough that it promises an endgame that feels a little less like the random trial-and-error ordeal of heptiamonds.

The total area of the enneiamonds is 1440 triangles, which is 4 less than 38²=1444 which means a triangle with side length 38 and four holes is (in theory) possible. A hole at each corner and a fourth hole dead centre; I decided not to think too hard about possible parity constraints and jumped right in.

There's a 'difficult zone' with polyiamonds. For small sets (hexiamonds and below) there are fewer pieces so usually you can solve puzzles with them fairly quickly. And for much larger sets the amount and variety of pieces is so great that no matter what kind of jagged mess you've got yourself into mid-solution, there's always a piece (maybe several) which will fit perfectly. And there's a good handful of the aforementioned chunky pieces that make the very end easier. Between these two extremes lies the heptiamonds and octiamonds - as mentioned in the previous post these seem unreasonably hard. And I was fearing that the enneiamonds would be the same.

I'm tempted to say that the enneiamonds fall just beyond the far side of the 'difficult zone'. But I can't really since I didn't actually finish solving the triangle manually. I got as far as the last 8 or so pieces then threw in the towel after half an hour or so of trying various possibilities. I resorted to inputting the remainder into a solver (Peter Esser's trisolve) and backtracking one piece at a time until I found a solvable state. From there I could have returned to the puzzle with the knowledge that it was at least doable but I was pushed for time, and I knew the sense of achievement would no longer be there.

The colours look nice, but they still don't allow you to see the borders between pieces very well.

Centering the hole in the middle was tricky too. With polyominoes it's easier; the grid on the green cutting mat is to the same scale as the pieces so I can just stick a monomino wherever the holes are going to be. But with this it was tougher - solve up to the rough place where the holey piece is going to go, then count rows in all three directions and shift the piece around until they're equal.

Had I not been on my lunch break (and conscious of how long I was spending playing around with these) there's a chance - yeah, a low one admittedly - that I could have solved the entire thing by hand. But for now, it's a respectable enough first attempt at solving something with this set. And I learnt some valuable lessons: mainly that some of the pieces I was holding onto weren't as cooperative as I'd previously hoped. There were a few long skinny pieces with fairly smooth edges that I'd kept back and these turned out to be a total ball-ache and part of the reason I gave up when I did.

Here it is, drawn out so you can see the individual pieces:

I tried drawing it in Inkscape instead of MS Paint this time, which was much faster but caused all these little variations in line thickness. Some day I'm going to have to learn how to properly use image editing software.

Polyiamonds: A Spotter's Guide

Introduction

Polyiamonds are the shapes made by connecting equilateral triangles edge-to-edge. What sort of polyiamond you've got depends on how many triangles it contains. If it's just the one triangle, you've got what could be called a moniamond or mono-iamond, but is probably better off just being called an equilateral triangle, for everyone's sake. Two equilateral triangles stuck edge to edge creates a rhombus, or diamond if we're using the 'iamond suffix.[1]

Now stick a third triangle onto one of the sides of this diamond. This creates the legendary triamond, which is just half of a regular hexagon. Now when it comes to adding a fourth triangle, things get interesting. Interesting enough to warrant a diagram.

Fig. 1: Can't beat a bit of MS Paint, eh?

There are three unique ways of attaching that fourth triangle, creating the three tetriamonds. There's not as much variation here as there is with the square-analogues, the tetrominoes. In fact, as you increase the number of triangles the total number of polyiamonds increases, but at a much slower rate than the polyominoes.

Fig. 2: Numbers of free n-ominoes, n-iamonds and n-hexes for n=1, ..., 10. Polyhexes grow stupid fast, but that's a story for another time.

This of course is all assuming we're talking about 'free' polyiamonds anyway. That's free as in they can be rotated and flipped over, not free as in there's a bunch of unlockable shapes as in-game purchases or anything. If we allow the pieces to be rotated but not turned over, we get the sets of 'one-sided' polyiamonds, and if we disallow rotation too we get the 'fixed' polyiamonds. If we disallow rotation but allow pieces to be flipped over it would just be weird.

The 3 free, 4 one-sided and 14 fixed tetriamonds.

Pentiamonds

There are four free pentiamonds. And that's just not a big enough set to do a great deal with. But by considering reflections as distinct (i.e. the one-sided pentiamonds) we get a set of six which cover a total area of 30 triangles. This allows the set to be packed into a selection of little shapes, including a 3x5 parallelogram and these wobbly approximations of rectangles. And a dumpy not-quite-regular hexagon.

My recommendation at this point is to hack a set of these out of the back of a cereal box or something, they don't have to be exact as long as you can tell which piece is which, and have a crack at finding alternate solutions for the above shapes, or trying to find your own. It's also the kind of thing that could keep a child (or simple-minded adult, I raise my hand sheepishly) occupied for a while... Whole minutes of action-packed fun.

Hexiamonds

Now we're getting to the good stuff. Grab one of your pentiamonds from before, slap another triangle on there and you've got yourself a hexiamond. Keep doing it and eventually you'll wind up with the full set of 12 different shapes, each more spiky and irregular than the last. Twelve - just like the pentominoes. And if you're familiar with the pentominoes (and let's face it, if you aren't what are you doing on a site called Polyominoes The Blog anyway?) You'll know that you can do an awful lot with those twelve pieces. So this beckons the question, are the hexaimonds as versatile, as co-operative, as fun as their square siblings?

Well, the total area the hexiamonds cover is 6x12 = 72 unit triangles, which is promising. 72 divides up in various ways which suggests a selection of parallelograms at the very least - 2x18, 3x12, 4x9 and 6x6.

Fun fact: centre-aligned parallelograms always look off-centred. Though to be honest I didn't even try in this instance. Just slapped them into Paint and thought 'good enough'.

2x12 doesn't go. Too skinny. 3x12 doesn't either although you can get fairly close - real close in fact, 11 pieces - but the last piece just doesn't fit.

Close, but no cigar. (If the piece left over looked a bit like a cigar then that would have been way cooler.)

4x9 and 6x6 though, these are most definitely doable. There are 74 and 156 ways of solving these respectively (not counting rotations and reflections of the same solution) but they are deceptively tricky. That cereal box you cut your set of one-sided pentiamonds out of? Go find it, dig it out of the bin if necessary, turn it over and make a set of these pieces too. Yeah, you might have to turn the pieces over to find solutions with them and one of them might have 'RN FLAKES' or something written on one side. Too bad.
Alternatively, the board game 'Blokus Trigon' has pieces which are the stes of 1 through 6-iamonds. So if you have that you could use those. You even get a nice little board that allows the pieces to click in place when you're solving. This is what I did until I discovered the wide world of laser cutting (and the wide world of having the money to afford to get stuff custom laser cut.)

Any excuse to show off this set I made, complete with a nice little tray to hold them.

And if you get sick of parallelograms, there's no shortage of other shapes to solve too. There's the above four-row-high stretched hexagon for starters (purposely shown half-solved to entice readers to try and finish it themselves) and loads more - as long as its total area is 72 triangles and it's not too weird you're good to go.

Folks more artistic than me will be able to find all manner of solutions, I just tend to go for the obvious geometric shapes even though the symmetry of the base triangle lends itself well to snowflake-like sixfold symmetry. Think of them like a more frustrating version of Tangrams.

Heptiamonds

A deluxe wooden set of heptiamonds ('Deluxe' even though because wood was cheaper than acrylic...)

There are 24 heptiamonds (from here on in we're just going to concentrate on the 'free' ones) and at this point things get a little bit tougher. The pieces are just that little bit wigglier around the edges and that makes putting them together all the more frustrating. Various parallelograms can be made (see the image above), as well as a triangle with edge length 13 and one little triangular hole in the very centre. There's a bunch of other solution shape ideas over at the Polyform Puzzler site too, if looking at heptiamond solutions is your sort of thing.

One of the few shapes I've actually managed to solve by hand with the heptiamonds.

One of the (many) things that makes this set so tricky to solve with is the fact that there's relatively few pieces that fit nicely in a 60° corner. And when you're making a shape with triangles as a base unit, there's usually a few 60° angles knocking about.
I've noticed that when I solve by hand with these I tend to want to keep the longer thinner smoother pieces for the end. Whether that's any help whatsoever is up for debate, mind you. In fact that might explain why I'm so bad at solving these.

Another handy tip of course is just to get some polyiamond solving software going (Peter Esser's trisolve is the one that immediately springs to mind) - these make mincemeat of nearly every outline shape you can throw at it. Mincemeat in the good way.

Octiamonds

Now we're reaching the serious business. There are 66 of them which gives a total area of 528 triangles. And at this point the most popular technique for large polyomino constructions can be applied. That is, the age-old art of saving the more chunky blocky straight-edged pieces for the end. There's not a lot of octiamonds that fit this description but there's a few; enough to make the task of solving by hand a little less daunting than it first seems.

These guys. These absolute legends.

Sets of octiamonds are available from Kadon Enterprises. It's where I got mine, and it can be where you get yours if you're that way inclined. If not, lots of cereal boxes. You know the drill.

Enneiamonds

I don't know why the 'non-' prefix wasn't good enough for this lot but clearly it isn't. Just try googling 'noniamonds' and marvel at the complete lack of results that show up. 'Enneiamonds' on the other hand turns up a few sites (maybe even this blog if it gets popular!) with constructions from polyformists (is that a word?) far more dedicated than I.

An interesting thing happens when you reach the enneiamonds too. It's this:

Nine triangles can be made to completely surround another triangle, allowing for the fabled holey enneiamond to exist. Not quite as infamous as its distant relative, the harbour heptomino, but getting there. This makes it a little harder to fill shapes but it can be worked around, usually by introducing a few more holes and placing them symmetrically throughout the construction.

I've got a set of enneiamonds, laser cut out of four colours of acrylic in a way that makes them look very like primary school teaching aids, but as of yet I haven't finished any constructions with them. Peeling the protective sticky material off both sides of each piece was an ordeal enough.

Deciamonds or Dekiamonds, I'm not sure 

There are 448 of these, 4 of which have holes in them. Nevertheless, some brave and intrepid individuals have made things with the full set. Check out this over the the Poly Pages by Patrick Hamlyn. And this page which has a solution by John Greening, using a combination of several sets, all one inside the other like Russian dolls or something.

The 6- through 10-iamonds, found by John Greening (more information at the above Kadon Enterprises link.) I drew this up based on the photo on that page, but the photo itself is unclear in some places so I may have made mistakes.

Beyond these though, you're pretty much stepping into the unknown, venturing into unexplored territory.
Or if it's not unknown, there's certainly not enough pretty pictures of it up on the internet.

---

[1] FUN FACT: This isn't the etymology of the word diamond. The word 'diamond' came first and the suffix '-iamond' was back-formed from it at a later date.

Little mini post: 4x53 with Hexominoes, Revisited

 Yonks ago I wrote a post about 4xn constructions with the hexominoes, which ended with suggesting a 4x53 with symmetrical placement of holes should be possible. Well, I found one:

Amazingly, the entire solution took all of three minutes by hand. I must have just been insanely lucky and hit on a solution on practically the first try, because I was prepared for this to take quite a while.

I guess they're not technically 'holes' when they're on the very edge of the construction but it looks nice so I don't mind really. And for an additional bonus, it's three-colourable and has no 4-way crossroads where four pieces meet at a point. That wasn't even my intention while solving it, it's just how it turned out. Rrrresult!

Octomino Ring

A 57x57 square with a 17x17 square taken out of the middle leaves a nice square doughnut with a convenient area of 2960. This is the perfect size for all the octominoes as well as eight 1x1 holes sprinkled around for good measure.

A 15x15 hexomino pattern would have fit nicely into that central hole; I wish I'd thought of that before I'd finished drawing up the image because it'd probably look nice, but I can't be bothered to change it now.

I also found a four-colouring for it too, which took a little while and a little bit of trial and error. No two pieces sharing an edge are the same colour, and when four pieces meet at a point none are the same colour there either. (At least, I hope so. There's a chance I've slipped up somewhere and missed it.)

Full solve time must have been close to six hours, with the repeated backtracking phase of putting the last 15 or 20 pieces in taking unreasonably long for whatever reason. About two hours on the 22nd, and another half hour on the 23rd until I got that little 'eureka' moment and the last few bits went in perfectly.

 Edit:

I found a middle bit for it anyway - the 289 square central hole fits the pentominoes and hexominoes with 19 squares to spare, which doesn't allow for any particularly nice configurations. The picture below is as good as I could get; sadly it doesn't preserve the overall symmetry of the octominoes' outer shape.

 

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.

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* 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.

A Few Parallelograms with Heptominoes

Edges with gradient 1:2 are surprisingly forgiving. When I started building this I had no idea just how long and skinny the resulting construction would be; I kind of pictured it in my head looking like the 45-degree parallelogram in this truly ancient blog post until I started building and it sprawled right out across the table.

Fig. 1: A 23x33 parallelogram

 

It turns out that the gentler slope here permits a much wider variety of pieces so it was easier than I'd anticipated. Solve order was like this: Started at the top-right and worked down the right-hand side, then continued around the perimeter of the shape until I reached the top of the left-hand side. This left a long thin internal section to be filled up last, finishing up at the top edge.

(A random sad fact: If some utter maverick decided to solve a heptomino construction the hard way, by using up all the blocky nice pieces at the start and finishing with the most hideous bits imaginable, everyone would just assume they'd solved it the regular way starting at the opposite end. All that effort for nothing.)

Edges with slope 3 should be easier still, since they're a step closer again to straight edges, and therefore even less restricting as to which pieces can be used to build them. But at this point the parallelograms themselves become loooong, and there's not a surface in my house that can comfortably accommodate them, short of the floor (although there's an idea...)

There's also another consideration here - the flatter the gradient, the fewer pieces can fit into the very thin ends of the construction. This limitation clearly rules out parallelograms with a 1:7 or greater gradient, but could make others impossible too. There aren't too many heptominoes kicking around that can fill a six-cell well, that's for sure.

And of course the sloping edges can go the other way, two squares up for one square along, as in the example below:

Fig. 2: A 19x40 parallelogram that doesn't lean quite as much as the other one.

I know, I know, the central holes aren't perfectly aligned with the long edges, but they're about as good as you can get I think. Holes positioned to match the gradient of the edges tend to like odd-number length vertical spacing between them, which doesn't play nicely with the (even) total height of the construction.

I could find more. I really should do. But I didn't want to go more than a month without a blog post so here. Have this lazy half-baked one.

Nine 15x22 Rectangles with Octominoes

As per the title really.

Fig. 1: I was going to say that this is approaching the limit of what can feasibly be solved by hand, by a human. But then again I probably said that about much simpler hexomino things when I was first starting out so who knows, eh?

The top-centre and middle-left rectangles had to be rebuilt fairly late on in the solve because I'd done them with the holes offset by one square the first time. Total solve time was about 6-7 hours, of which close to two was spent on the last half of the final (bottom-right) rectangle. There were just a few awkward pieces - the three which surround the right-hand hole especially - that I had unintentionally held onto far too late into the solution, and they caused all manner of ball-ache.

Nine 11x30's with similar hole configurations to this should be possible (also 6x55 rectangles, if you want to suffer...), but not right now. After solving something like this there's always that period of a few weeks where I feel like I'd rather be made to eat the octomino set than tackle another huge construction with them. Right now I'm still in that phase. Recovering.

Miscellaneous Solutions That Didn't Deserve Their Own Posts

Sometimes polyomino-related things have a decent story behind them (or, failing that, a really boring story that can be stretched out to blog-post proportions.) But sometimes they don't. Today it's a selection of the latter; digitised solution pictures that were just sat around cluttering up the folder named 'BLOG STUFF' on my desktop, to tide me over while I write up some actually half-decent posts.

Rhombus with Hexominoes

Difficulty level: Mild (approx. two chilies out of five)

I've seen this, or variations of it, done before so it's not really particularly groundbreaking as solutions go (although really, are any of them that groundbreaking?) Including it here because it was a hard-won battle - I kept building the edges wrong, accidentally adding in steps of size 1 or 3 then not realising until right near the end when I was left with an internal hole whose size wasn't a multiple of six.

Heptomino Rectangle with 21 Holes

Difficulty level: Breakin' a Sweat

Solved the middle first, since the closely-packed holes are quite restricting on what pieces can even go there. But then the rest just solved like a normal heptomino construction and I've banged on about those at length in other posts so it wasn't really worth doing another one.

And the harbour heptomino doesn't need to be in the very centre of solutions like these, but it just feels wrong any other way.

5x45 rounded Rectangle with 11 Holes

Difficulty level: Real Tears

This was an utter nightmare, combining two of my worst fears into one shape: 5xn with hexominoes is always an ordeal, and adding that row of holes just pushes it over the edge into the kind of territory where it's actually frustrating and unpleasant to solve. You can see by the way the eight pieces with 2x2 blocks in them are scattered all over the shop that my usual solving technique only got me so far before I was left to fend for myself, desperately applying trial-and-error for several hours of my life I'll never get back.

Recommendation: FlatPoly2 can probably crack this one in under 10 seconds, just do that instead.

Stay tuned, next time I might actually have something a bit more substantial.