Merry Christmas! (feat. some heptominoes)

This is the problem since getting the new website set up: Every time I have something I want to post I'm unsure of whether to do it here or put it there. And in the indecision I end up doing neither. So the blog looks abandoned and the site doesn't get updated. You'd think after like what, eight months, I'd have figured out a strategy for doing both, like updating here first then working the new stuff into the web page at a later date, but nope. I'll have to add that to the ever-growing list of New Year's Resolutions.

I'm still solving things with the heptominoes and such like. (And still sobbing into large sets of polyiamonds 'cause the pieces just won't go...) But there's just not so much happening that I feel the need to tell the world about. Some day there will be - some day there'll be a little momentarily lapse in judgement and I'll find myself ordering a set of enneominoes so big they'd practically tile the entire floorspace of my flat - but until then it's just making similar shapes with the same sets of pieces and there's nothing to say about that which the past fifty blog posts haven't already said.

So for now, have this square ring with the internal holes not quite evenly spaced. They're spaced enough so that it looks pretty at first glance, but then you realise the spacings are 5-6-5 and once you notice that it'll just irk you for ever more.

And now it's telling me 'spacings' isn't a word. Lovely.

Hexahexes

I always sort of overlooked polyhexes in the past. Polyominoes were the main event, so to speak, and the first polyforms I really got properly into, and on the rare days I wanted a really infuriating challenge I would usually turn to the polyiamonds. But polyhexes for whatever reason just weren't really on my radar. Sure, I had a physical set of the 1- through 5-hexes from Kadon, and I solved a couple of things with them. And I even made a half-arsed blog post a while back. But that was about it really. Until now.

A few weeks ago on a whim I got a set of hexahexes cut out of the cheapest MDF money could buy. I didn't even shell out the extra two quid for the laser cutting people to cover the wood with protective masking tape, instead opting to let the bits get gently toasted around the edges by the laser. And then I took them on holiday, to a chilly weekend in a caravan in Northumberland where I knew I'd be a captive audience in the evenings. And while there I slowly began to realise that I'd missed out... Polyhexes were fun. In fact they weren't just fun, but were in fact... very fun.

This photo doesn't really give any sense of scale, but each hexagon is 8mm to an edge, and the full solution has a diameter of about 40cm or so on average. I think. Nice and chunky. I actually checked the scale this time before cutting unlike my positively tiny enneiamonds.

The slight browning of the edges turned out to be something of a blessing in disguise - it makes the borders between adjacent 'hexes stand out a bit in photos which is handy. Sometimes I'm too lazy to draw up a pen-and-paper record of a solution, so just being able to take an aerial photo that I can work from to create a digital image is a nice time-saver. And talking of digital images:

Here are solutions to two different hexagons, the more compact one is the shape of the solution Kadon uses for Hexnut II; the larger thinner hexagon I haven't seen anywhere before. I haven't ran the numbers for hexagons larger than this; it could be that there is an even bigger thinner (and therefore harder to solve) hexagon ring out there waiting to be found.

Speaking of, solve difficulty is the best thing about the hexahexes. It's somewhere between that of hexominoes and heptominoes, I'd say. A good, meaty challenge but one that I don't need to set aside a whole evening for. There are a couple of kinks to be ironed out with my solving technique, though, mostly the fact I'm not used to hexagons so it's often not immediately obvious whether a piece will fit in a certain place without actually trying it a few different ways.

Unholey Hexahexes

If you discard the holey hexahex (as we sort of unintentionally did for the rings above) you get 81 pieces and a total area of 486 hexagons, which divides up very nicely indeed. So far all I've done with this set is the really easy stuff - a couple of approximations of parallelograms, of which one is shown below for your perusal.

The 81 unholey hexahexes squeezed into an 18x21 parallelogram. Solve time approx. 45 minutes manually.

But there's a lot more out there than just parallelograms. It's fairly easy to work out formulae tying the edge lengths to the area for various hexagons, triangles and other such shapes that hexagons lend themselves well to. And from there just a little bit of searching for edge lengths that give the magic number, 486.

Which will all be a nice excuse to post a bunch more blog posts. I need to pick up the pace - this year my posting rate on here has gone right down. That's partly because I've been putting some things directly to polyominoes.co.uk (and discovering the joys of trying to display characters like '°' in html), but it's also partly because my interest in polyforms seems to come and go in phases. And summer this year I've just been preoccupied with other things (recording an album, teaching myself to read Japanese, and dusting off the Rubik's cubes and getting back into speedsolving). But now with winter drawing in, and with its long cold rainy evenings with nowhere else to go and not much else to do, there's a non-zero chance I'll dedicate a bit more time to the sacred art of polyform-ing. And to the subsequent rambling about it on here.

Tetracubes, revisited

A while ago I wrote a post exploring polycubes up to and including the tetracubes. And I didn't really go too far in depth with it. I found a few things with the tetracubes then just kind of stuffed them in a cupboard and forgot about them. But recently I dug them out and all it took to rekindle my interest with them was a few minutes scribbling on them in felt tip pen so that the individual pieces could be told apart in a construction. Behold:

It's messy and it looks like a six year old did it, but it's a slight improvement on how they looked before. From a solving point of view anyway.

After doing this I decided I needed to solve a few things to test out how they looked. And this quickly showed me that there was a lot more overlooked potential in this set of shapes than I ever realised.

Scaled-up tetracubes

The total volume of the tetracubes is 32 units, which is just enough to construct a big tetracube scaled up by a factor of two. For four of the five planar tetracubes and two of the non-planar ones this is trivial once we've got the two 2x2x4 blocks above solved, just put them together in whichever configuration. For the two remaining tetracubes - the T-tetromino and the cyan one bottom left in this image - it's a little more tricky.

For the diagram on the right, a dot indicates that that piece extends into the layer below there, and a square indicates that it extends into the layer above.

The solution for the T-tetromino is shown above, but the final tetracube is a challenge for the reader. It's possible, but I can't be bothered to draw out another diagram for it so you'll have to find it yourself.

Almost-cuboids

There are also the two cuboids-with-holes-in-them that can be done. There's the 2x3x6 with a 1x1x4 hole, and the 3x3x4 with a 1x2x2 hole. I imagine there are several ways to solve each, but I've included once possibility for each, in that not immediately readable notation everyone uses for polycube constructions.

Octomino towers

Imagine an octomino. Any octomino you like. Then imagine it made of cubes as opposed to squares, the planar octacube equivalent of the octomino. Now imagine stacking four of these perfectly on top of each other, creating a prism with volume 32 units squared and the octomino as its cross section. There is a chance that this resulting shape can be filled with the tetracubes. I mean, sometimes there's not, the I-octomino in this instance corresponds to a 1x4x8 rectangle which clearly can't fit the non-planar tetracubes. But The tetracubes show a surprising versatility when it comes to most of the other octominoes.

Some highlights shown in the diagrams below.

The stairstep octomino came as a surprise to me, I really didn't think it would be possible.

A useful solution is the one above to the doughnut-shaped one. This can be broken into three small 4-unit high towers as shown below, and these can be pushed together in various ways to make a whole host of octomino stacks.

Which begs the question: which of the octominoes can we solve this way, and which ones can't be? The above 'kit' of three pieces probably covers the majority, but it'd be interesting to see the set that just can't be done, either through the kind of impossibility the I-tetromino demonstrates immediately, or just cases where the pieces won't go despite there not being a clear reason for it.

This felt like the kind of problem where if I posted it to the Puzzle Fun Facebook group someone would get back with the results of an exhaustive search by the end of the day. So I tried that.

Pretty quickly Edo Timmermans had found all of the octominoes that could be created by combining the monomino, triomino and L-tetromino that make up the solution to the ring octomino way up there in the previous bit. These are marked in green in the above image. He also showed that any 'L' shaped octomino with a single bend in it was impossible, as the three nonplanar pieces would all have to occupy that bend.

I then found that my solution for the zigzag octomino could be similarly partitioned into a monomino, triomino and Z-tetromino which allowed solutions for a further five octominoes, those in dark blue.

George Sicherman then found the set of the 48 octominoes for which the prism has no solution, these were marked in red in the diagram. Which left nine octominoes, all of which had solutions, but which hadn't yet been found. I managed to pick off some of these by hand, as shown in the diagrams below, and the remaining five I verified in Aad van de Wetering's 'Poly3D' software. But I'll not put the solutions here, just in case anyone reading has a set of tetracubes themselves and fancies a nice challenge.

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I've been updating this blog and the website sort of in tandem; I prefer the site for its more flexible formatting and the fact I can interlink and break things into pages a bit better, but this blog probably gets more traffic (it's not much, but it's traffic nonetheless) so I'll keep adding stuff here too. I'll figure something out.

One-Sided Hexominoes

It's been a long time since I last posted anything on here. Too long. But sadly there just hasn't been that much in the way of polyomino-related goodness to post. Partly because I've just been too busy doing other things, and partly because there's only so much you can do with polyominoes - the easy stuff isn't interesting enough to write blog posts about, the hard stuff is too hard to do without the assistance of a beefy computer, and the stuff in the sweet spot is hard to come by.

I've never written up anything about the one-sided hexominoes before on here. Basically, earlier this year I got a second set of heptominoes laser cut, which opened me up to the possibility of solving shapes using the set of one-sided heptominoes by combining the sets and then trying not to flip the pieces over. Then I realised the same thing could be done using my two sets of hexominoes too. And the set of one sided hexominoes is a much more versatile set than the plain ol' regular hexominoes.

There are 60 one-sided hexominoes in comparison to the standard set's 35. But this new set doesn't have those pesky parity constraints which means that a lot more shapes are possible to tile - rectangles without unsightly internal holes, for example.

The total area is 60x6 = 360, which means that rectangles of size 4x90, 5x72, 6x60, 8x45, 9x40, 10x36, 12x30, 15x24 and 18x20 should all be possible. Sadly, due to the length of the perimeter compared to the amount of perimeter squares the pieces can provide, 4x90 isn't possible.

Solving manually is a little trickier than the normal hexominoes - each piece having only one accepted 'right side up' means that any given piece is slightly less practical than its two-sided equivalent, and you do get those cases where you're down to one piece left and the hole is the mirror-image of the piece you're holding.

Here's two 9x20's which can be combined to make either a 9x40 or an 18x20, in a two-rectangles-for-one type deal. As 360 divides up really nicely, this gives a lot of possibilities for tiling groups of congruent shapes, but that'll be a blog post for the future. Others have already solved congruent sets of ten or twelve shapes, so go look at those. Scroll about three-quarters of the way down the page for them. In fact just read the entire page, it's all good.

Something else nice you can do with the one-sided hexominoes is square rings, i.e. squares with a centred square hole. Here are three possibilities (these might be the only three actually), with the rings getting progressively thinner, and as a result a little harder to solve manually:

And here's one more shape, a diamond with a central hole and those tricky diagonal edges.

And I'm purposely leaving this post a bit less thorough than usual so I've got an excuse to post 'One sided Hexominoes - Part Deux' in a few weeks. Or a few months, if my recent posting schedule is anything to go by.

Nine 9x37 Rectangles with Octominoes

The second in the 'Nine congruent rectangles with the octominoes' series. Last time the rectangles were closer to square and had less holes. but this time I decided to tackle 9x37 rectangles with 5-hole configurations in the centre, which would make it even harder than the last batch. And that last batch was hardly a walk in the cake.

The First Eight

The more skinny a rectangle is, the harder it gets to tile with polyominoes. Because thin rectangles favour pieces with a long flat side, and these pieces run out fast. Then, every time you get to the end of a rectangle, the two corners there usually demand a piece with a clear right angle. And there aren't too many of those either. Well, there are a few, but they're generally the simpler more cooperative pieces, the kind you want to save for the dreaded final rectangle. It was painful: when I'd put all but two pieces into a rectangle I'd then mentally dissect the remaining 16-omino shaped hole, trying to partition it in such a way that both halves were awkward wiggly pieces. But it pretty much never happens that way - it's either one nice, practically rectangular, piece and a wiggly piece, or two wigglies, one of which has been already used elsewhere in the construction. So inevitably, each completion of a rectangle left the pool of nice endgame pieces a little emptier than before. And that means when you get to the final rectangle, you get problems.

Fig. 1: Getting the easy bit out of the way with.

The Last Rectangle

I'm sure I've said before on here that for solutions like these 20% of the pieces take 80% of the time. In this case it was more that 10% of the pieces take 90% of the time, or possibly more.

The pieces I was left with after tackling the first eight were... less than ideal, let's just say. Some were the kind of pieces it's easy enough to use up in a large, wide rectangle, but that don't play so nicely in a construction like this where it's all edges. Some were just hideous. Check out numbers 2, 6, 7 and 41 in the image below.

Fig. 2: The remaining pieces, and the shape to fit them into.

I gave the last rectangle a valiant shot bare-handed but really struggled. I only ever managed to get about two thirds of the rectangle done, just past the central dots, and even then I'd find myself at either a dead end or with some really unappealing pieces left. Accepting that I wasn't going to get anywhere continuing in this vein, I threw the pieces and board into mops.exe (from Peter Esser's website) and let it run for a while, to see if it fared any better than I did.

It didn't. The best it managed was positioning 39 of the 41 pieces - which it had managed several times - but no better. Which concerned me. The lack of all-but-one-piece solutions was a concern, especially because given the amount of 39s I was seeing it seemed statistically likely that we'd see at least one 40. It was at this point I thought maybe there could be some additional restriction, something like parity, that I'd overlooked.

I posted the above image of the 41 pieces and the board to the Puzzle Fun Facebook group, asking if there was any restrictions that made it impossible but that I had missed. And not long after, it turned out that not only was there no such issue preventing it from being possible, but Patrick Hamlyn (I don't think he has a website or I'd link it here) had managed to find a solution using his search program! It's the rectangle in orange at the bottom of the following image:

Fig. 3: The full construction, with the rectangle solved by Patrick Hamlyn shown in orange.

Apparently his software hit upon 866,000 almost-solutions with 40 of the 41 pieces placed before finding this, so I would have had to have been very patient with mops.exe to even have a hope of turning this up.

One Sided Heptominoes (and a new website!)

If you happen to own two compatible sets of heptominoes, then by discarding the duplicate mirror-symmetric pieces and turning a bunch of the remaining bits over so you've got distinct mirror image pairs, you have yourself a set of the one-sided heptominoes. Difficulty-wise they seem to slot in somewhere between heptominoes and octominoes - the main challenge now being resisting the urge to flip over pieces. The total area they cover is 7x196 = 1372 but as one of the pieces has a hole an extra unit cell must be accounted for when deciding on shapes to fill. (I suppose you could just discard the holey heptomino and see what you can do with an area of 1365. Maybe a 37x37 with 4 corner squares removed?)

Anyway, I found this 32x43 rectangle, just since it was my first time doing anything with the one-sided heptominoes and I didn't want to take on anything too elaborate straight out of the gate. The first two-thirds of the solution process feel pretty much the same as solving with the standard turn-over-able heptominoes, but when you get right down to the last few bits there are a lot more near misses - times where the remaining hole could be filled with the mirror images of the remaining pieces but not the pieces I was holding.

Fig. 1: A rectangle with the one-sided heptominoes. As far as I know there are no pieces which have been accidentally turned over, but I can't say for definite and I can't be bothered to check either.

In other news, I now have a website over at polyominoes.co.uk. Right now it's a complete mess and still very much under construction, but eventually it will have a lot of the information from this blog on it, just tidied up and organised in a slightly more logical way. Posts sorted chronologically make sense for some things but it also makes looking for specific things a complete nightmare, even when I manage to remember to tag them correctly.

More Wiggly-edged Rectangles

There's not a technical term for shapes like this, is there?

Fig. 1: With heptominoes.

Those toothed edges are just as difficult as they look. They start out easy enough - the bottom edge was done first using pieces that it makes sense to put there, the 'H' shape and various other U-pentomino-derived pieces that fit perfectly. But alas there aren't enough of those shapes to go all the way around. And by the time I got to the middle of the top edge desperation had kicked in and I was using any piece I could to try and complete the perimeter. The straight line heptomino for example. And even one or two of the pieces with a 2x2 block which I generally try to hold onto until near the end.

Once the perimeter is done it doesn't get any easier. See, now I'm left with a very irregular cavity to fill, and if we want the central hole to be, well, central that forces the position of the harbour heptomino too. So you're left to squeeze an already awkward batch of pieces into a shape that really wasn't designed to accomodate them, just like last time.

- - -

As I was solving the above I wondered whether this sort of shape would be any easier with octominoes. On paper it felt like it should be - sure, an octomino rectangle would have a longer perimeter but the amount of pieces which would fit nicely into that perimeter would increase too. And the ratio of perimeter to area ought to mean that once the perimeter was completed we'd still have a large variety of pieces for the middle.

Fig. 2: The octominoes. Yeah, the corners are a slightly different style - I'd miscounted the number of teeth needed and offsetting them by one like this was a lucky save.

Turns out it isn't really that much easier after all. Maybe a tiiiny bit easier because of pieces like that 'E' shape in the top edge, but the perimeter got way longer and the number of nice pieces just hadn't increased enough to counterbalance it. Admittedly once I'd cleared that first hurdle, completing the rest of the solution was no worse than any other octomino solve I'd done in the past, so maybe I was partially right. Total time, maybe about 4 hours or so, spread out over one afternoon and evening. I don't know, I don't time these things. I just occasionally glance at the clock and realise I've wasted a whole afternoon and I should have had dinner an hour ago. But hey, there are worse ways to waste time.

Some Assorted Solutions

Polyform-related shenanigans have been on the back burner somewhat recently, mainly because I'm in the process of moving house and don't have table space right now to spread out even a full set of hexominoes, let alone a larger set. So this post is going to be a collection of solutions I found several weeks ago but didn't write about at the time for whatever reason.

First up, continuing with the last post's theme of long skinny rectangles we've got the heptominoes in a 9x85 rectangle with nine holes. I think I found this with the intention of including it in that post, but then I found the 5 high rectangle which was way more interesting and in the interest of not making that post a mile long I took this one out.

In February some time I put together another variant of the well-known side 20 diamond found originally by David Bird - same outer shape, different hole configuration. Nothing particularly groundbreaking to look at, but the solving process was interesting. The jagged outer edges use up pretty much every piece with any stairstep-type edges, leaving a glut of pieces with 2-cell protrusions for the middle. Which is less than ideal.

And finally, I found a few more fun things with the hexominoes, in what I call the 15x15-15 challenge because I'm terrible with names like that. The idea is, make a 15x15 square with 15 holes using the hexominoes. The challenge is to make the 15 holes look nice, given that they can't retain the symmetry of the outer square. There's some examples here (from back in 2019) and a few more just below, from back in March.

A silicon ship looking configuration.

Side note: For whatever reason the word 'quincunx' makes me giggle like an idiot.

Oh yeah. One last thing that I almost didn't include since it's a shameful chapter indeed in the history of polyominoes. I had attempted a heptomino construction with the crenelated edges similar to that top hexomino one. And I made the classic mistake of doing the math for it late one night when I was too tired to be trusted with simple tasks like adding a few numbers up. This was the result:

Yep. For whatever reason I'd thought that each edge had one more protrusion than it did which threw my area calculation off by 4. And because it was late and I was just itching to get my solve on, I didn't even think to double-check everything. I just never learn...

Polyominoes: How Thin Can We Go?

What is the narrowest rectangle (or vaguely rectangular shape) that can be constructed with each set of polyominoes?

Lets get the easy cases out of the way first. For pentominoes, there are 4 possible rectangles that can be made, the narrowest of which is 3x20:

Fig. 1: One of two ways of doing this (excluding rotations and reflections).
See if you can find the other way. Go on. I double dare you.

The tetrominoes don't do any rectangles due to parity issues, but if you add the triominoes (and optionally domino) into the mix you get a set that can fit nicely within a 2xn rectangle:

Fig. 2: Apparently there are 84 solutions to this excluding rotations and reflections.

 For hexominoes, each individual piece can fit within a 3xn space but we are able to rule 3xn rectangles out for the entire set together - see this older post for the grisly deets. 4xn however is totally possible as illustrated below:

That's all well and good, but what about heptominoes?

An obvious starting place is the well-known 7x107 rectangle which can be made in a variety of ways using the 107 non-holey heptominoes. There's one a little way down this really old post which I found, but I was hardly the first to do so. A few weeks ago I went one better, building the following 6x127 which ended up being the inspiration for this blog post:

(Click for full size)

Thin rectangles are a fair bit harder to solve (manually at least) than wider, more squarish rectangles of the same area. I'm guessing it's just to do with the higher perimeter-to-area ratio. Every extra square bordered by perimeter is a further restriction on the heptominoes within the shape (as are things like holes). And once you get thin enough it begins to restrict weird things, like the orientation of polyominoes within the shape. A heptomino that spans 6 cells in one direction will touch both the top and the bottom side of the 6x127 rectangle for example, and if placing that shape partitions the rectangle into two areas whose areas aren't a multiple of 7 then you've just rendered the lot unsolvable.

A 5 square high rectangle with a mixture of the 1- through 7-ominoes is possible, as demonstrated in the picture below, solver unknown.

I don't know where I got this image from or who to credit for the solution. If anyone knows anything about its mysterious origins please stick it in a comment below.

Here's a neater version with the polyominoes coloured by size:

(Click to embiggen)

 

I also found the following 5x152 pure heptominoes solution using Aad van de Wetering's FlatPoly2 software.

Click for larger. It was either this or insert it vertically and make this post a mile-long scrolling marathon.

The first three quarters or so were placed by hand, and then the final 26 were input into the solver. For whatever reason (I think it's just my computer) the software takes aaages to find anything if you just give it the full shape and all the heptominoes - in fact I've never actually been patient enough to let it finish - so the only way I can find solutions is to solve manually until there's about 20 (or fewer) pieces then let it solve from there.

And height 5 might be the best we can do.

In a similar way to what we did for hexominoes, we can rule out shapes of width 3 and lower - there's a V-shaped heptomino that reaches out 4 squares in two directions, which isn't going to fit no matter how you place it. And for 4xn we run into issues too. According to the analysis that Peter Esser's 'mops' solver does, the set of heptominoes can cover a total of 373 border squares. And a 4 unit high rectangle has to be at least 190 units long for the area to be >756 (allowing for holes for the harbour heptomino.) This means we have 380 border squares for the two long edges alone (and a few more for the little short edges) which the heptominoes just can't manage. So 5 might just be as thin as we can go.

"Fun" with Polyiamonds

Introduction

In branching out the blog to include not just polyominoes but other polyforms too, I couldn't help but notice one thing: my polyiamond solving game is atrocious. I had rationalised this in all sorts of ways in the past - the paucity of chunky yet asymmetrical pieces to hold onto until the endgame was my go-to excuse - and just treated them as a harder bunch of shapes to work with in general.

But realistically it's far more likely that the reason I can't manually solve even a heptiamond construction half the time is down to the fact that, compared to polyominoes, I just haven't put the hours of practice in building up a sort of intuition for the solving process. So I made it a sort of unofficial New Years' Resolution* to try and remedy this.

Bootcamp - Heptiamonds

This is where I discovered my apparent inability to solve polyiamonds. I have a nice little set of wooden heptiamonds I got laser cut a while back, and a little tray that holds them, a 7x12 parallelogram. Except the pieces spend most of their time in a little zip-lock bag because I struggle to get them all into the tray consistently.

Heptiamonds are pretty flexible with the shapes they can do; there's no real parity constraints or anything holding them back so you can go ham - triangles, parallelograms, hexagons, you name it they'll have a go. You can do a triangle with side length 13 and a central hole. I say 'you can' because I can't, and believe me I've tried.

One of the few things I could manage was the two identical parallelograms below, found more by sheer luck and perseverance than by any kind of breakthrough in solving technique:

Fig. 1: Two 6x7 parallelograms.

In general it seems that the trial and error aspect involved in solving heptiamond things overshadows any possible impact from being careful with piece ordering. So I moved on to octiamonds instead.

Octiamonds - a tale of woe

You get a little more choice for pieces with these lot. There's 66 of them so you'd better hope there'd be at least a few which aren't a complete pain in the arse. (See the ones I singled out in the octiamonds section of this older post.)

My first success of the year with these was the 6x44 parallelogram below. I solved this one right to left (well, left to right really, but the final image got flip turned upside down at some point while I was sobbing into my keyboard trying to get InkScape to behave itself.) About two thirds of the way through the solution I realised it might be wise to start solving from both ends and meet in the middle - the thin little 60° points are notoriously tricky, especially when you're down to a very limited set of pieces. And the endgame for this one - the little trial and error period where you just place and backtrack and hope for the best - was agonisingly long, as is always the case with polyiamonds. The last 9 or 10 pieces probably took longer than the 55 or 56 that preceded them. I guess it's the Pareto Principle in action - 20% of the pieces take 80% of the time.

Fig. 2: A 6x44 parallelogram with the 66 octiamonds.

Octiamonds have parity. It works pretty much the same way as hexominoes but this time it's up- and down-pointing triangles you have to keep an eye on. Twenty-two pieces have three of one and five of the other, the rest have 4 up and 4 down, as illustrated in the diagram below. (This seems like it should be too obvious to warrant a diagram but I have doubts about my ability to express things in plain text so I'll draw one anyway.)

From this, we can discover that the total numbers of up and down triangles for the whole construction need to be either equal or differ by 4n where 0 ≤ n ≤ 11. And can you guess when I figured this out?

Camera phones in low light conditions: Not even once.

If you said 'right in the middle of a construction where the numbers of up and down triangles differ by 6' then give yourself a hearty round of applause.

The rough working out that I should have done before I placed 60 pieces...

The next time I could bring myself to attempt an octiamond construction I was a little bit more diligent with my parity checks before I set out, and it paid off. Being a bit sick of parallelograms but not wanting to do anything too adventurous I settled on an eight row high trapezium (or trapezoid as they're known in the US/Canada) which passed all the criteria - it has 8 more 'up' triangles than 'down' ones.

Fig. 6 or something: A height 8 trapezium.

Again, the solution took a while to find, but at least this time there was one.

Enneiamonds - the Final Boss of Polyiamonds**

I'm not sure what possessed me to tackle this one when I was clearly struggling with octiamonds, but I did anyway. Enneiamonds (blogger spell check has a vendetta against that word) have all the frustration of octiamonds and then some, and by far the biggest ball-ache is the introduction of holes. You need a hole for the holey enneiamond (the harbour enneiamond?) and then you need more holes to bring the total area up to something that divides nicely. Adding a further three makes the total coverage 1444 triangles which divides up nicely and allows two rhombi with two holes each to be made (Also 4 triangles with one hole each but let's not try to run before we can walk just yet.)

But knowing a solution should exist and finding that solution are two very different things.

It looks real pretty. And it looks very human-solved, the pieces getting chunkier and easier towards the top corner of the right hand rhombus. But don't be fooled. the last 15 pieces were done with a little electronic assistance.

I don't like doing this. And in fact when I started keying in the pieces which just wouldn't go, my intention was never to just find a solution - that makes for something of a hollow victory feeling when you put that last piece in and know deep down you didn't properly complete it. I initially just ran a search to check for the existence of a solution, but not the solution itself. If the program came back with '0 solutions found' I'd know to backtrack one more piece and try the search again until I reached a position where one or more solutions existed.

I used to do this back in the early days of heptomino solutions, before I got good enough at them to not need any assistance. And usually I'd need to backtrack 5 or 6 pieces and there'd be a couple of solutions lurking in that piece of search space. Not in this case, though - I had to remove fifteen pieces before it could find even one solution. If that's not irrefutable proof that polyiamonds are just a more obstinate breed than polyominoes I don't know what is. Anyway, I'd found a position where a solution was definitely possible, at least in theory. In reality, I fought on, trying configuration after configuration trying desperately to get the rest of the pieces in. The big triangle belonged in the very corner, that felt like a given; it just fit there so well and didn't seem to sit comfortably anywhere else. But beyond that I was just pushing pieces around pretty much at random, sometimes finding I could get all but one in, other times creating awkward little bays and peninsulas in the edge of the construction that seemed to exclude every other shape.

I spent way too long on this - several days' worth of mornings, lunch breaks and evenings - so I eventually decided I would do the unthinkable. I'd peek at a computer solution for those last 15 pieces. Not the entire thing; just the next few pieces, to point me in the right direction.

Key: Purple pieces were already placed, red bits I looked at the solution to find, yellow bits I did myself after the red bits were placed.

Armed with these handiest of hints, I managed (after another 20 minutes or so) to fit the last of the pieces in and gaze upon the finished construction. It still felt like a bit of a cop-out though. I suppose it's just part of the learning process - I did similar with heptominoes a year and a half ago and it wasn't long after that I was knocking out octomino rectangles in single afternoon sittings. Hopefully the next time I dig out the enneiamonds I'll be able to get a little bit further than this time, and before I know it I'll be able to just solve whatever with them without breaking a sweat.

~

* I don't know why I feel the need to prefix that with 'unofficial', it's not like there's anywhere I can go to make a New Years' Resolution official or legally binding or whatever.

** Until I make myself a set of dekiamonds, that is.

2021! (a.k.a. A Fun Little Challenge with Heptominoes and Hexominoes)

Happy new year! Going to kick off 2021 with a lazy low effort post.

In an older post I found two hexomino rectangles which had twelve holes, each in the shape of a different pentomino. Or it could be thought of as a combined pentomino/hexomino rectangle with the added condition that none of the pentominoes touch each other or the edge of the rectangle.

The topic was brought up on the 'Puzzle Fun' Facebook group, and the gauntlet was thrown down - was the same thing possible for heptominoes with hexomino-shaped holes? I had a go and found the following:

Fig. 1: 22x44. Separating the hexominoes and heptominoes out when dismantling the finished construction was not a fun job.

There's two monomino holes in there, one has to be there to placate the harbour heptomino, the second is there because otherwise the combined area (7*108 + 6*35 + 1 = 967) would be a prime number. And the hexominoes are at a higher concentration in the bottom half of the rectangle. I wasn't sure how much space I had when I started off constructing this so I packed them in really tightly at first, then when I got about three-quarters of the way through I realised I had like 4 hexominoes left so they're a tad sparser up there.

the inevitable next step will be the octomino/heptomino equivalent of this. I'm putting off starting it but I'd give it a month tops before I cave and begin trying to solve it.