Where to next?

At this point I've written pretty much everything there is to write regarding stuff like my solving process. And a lot of times when I solve something with hexominoes or heptominoes now, it isn't anything particularly interesting or challenging or worth telling the internet about - just more of the same.
So the plan for 2020 might be to sort of vary thing a little bit, still keeping it firmly to do with polyforms but not just a monthly 'Here's a bunch of pictures of heptomino constructions with not much in the way of descriptions to go with them' type posts. Which is what this place has a very real risk of becoming.

Fig. 1: Three 16x16 squares with the heptominoes.

See?

That one was actually solved with laser cutting in mind, if I ever wanted a nice new set of heptominoes, cutting this design from 3 small (180x180mm) pieces would be way more cost-effective than trying to cut them any other way. Probably.

Then a few weeks ago I solved (mostly) the one below. Instead of doing what I usually do and spreading out the set of heptominoes out all over the desk before solving manually, I did this by drawing it directly in MS Paint and crossing off the pieces from a list as I went. This has a few advantages - since I'm drawing the complete outer shape first I can guarantee I've put the centre hole in the right place. And it saves me having to redraw the solution once I've found it. And that's a fair enough trade off for not being able to backtrack nicely. And there being a risk of drawing pieces in wrong as well. For all the shortcomings of using a big ol' set of physical pieces, there's never the possibility that you'll place an n-omino with the wrong n while you're solving.

Fig. 2: Three 11x23 parallelograms. First two by hand, last one partially completed with computer search.

It's putting the last 15 or so pieces in that are the worst when you're doing this without a physical set. Backtracking and keeping track of which pieces have and haven't been used is just too much for my little mind. Especially when I'm tired and it's an evening and I've been at work all day. Which is most of the times I do things like this.

So yeah. Expect a more varied polyominoes blog in the new year hopefully.

Oh. And another thing. The future may involve octominoes a bit more...

Octominoes: The Hall of Shame

Hoo boy, where do I start with this one?

Fig. 1:  ;~;

This is what happens when you solve without a physical set of pieces. Pieces that don't belong in the set (in this case, the two blue nonominoes) creep in undetected, and only when you're down to the last little corner do you realise that the remaining space is the wrong size. I do a quick manual check to see if the number of free squares are divisible by 8 at about this point, and in this case it wasn't so I engaged panic mode (i.e. looked for the offending pieces, found that they were nowhere near the edge of the solution, and gave up.)

This was my second crack at the same solution, a month or so later:

Fig. 2: Another 4 hours I'll never get back.

This time, the area of the final space was divisible by 8, but it wasn't the multiple of 8 I was expecting. I had 15 octominoes left to place, and an area of 14*8 = 112 unit squares. So I reckon I've used a piece twice somehow. I've looked over the solution but can't see it, so if anyone actually reads this and has a better eye than me, see if you can spot what I did wrong here.

A while after all this, I read somewhere that octominoes have parity constraints of sorts. Or at least the 363 unholey ones do. (See here, click through to 'Other octomino constructions' and it's about 2/3 the way down.) I'm still trying to get my head around this; I'm still not sure if and how it will impact constructions like this. Maybe this shape wasn't even solvable to begin with, since I put down the holey octominoes first, reducing the construction to a solution with the set of 363 unholey ones?

Hardware Upgrade

Exciting times! Well, depending on your definition of excitement anyway. This blog sets the bar for exciting pretty damn low.

To cut a short story even shorter, I found a laser cutting place a little while ago and got a set of pentominoes and hexominoes cut from acrylic. It's one of those times it really hits me we're living in the future, the fact that I can just draw up a .svg file of whatever polyominoes I want, click a few buttons then a week or so later those exact polyominoes rock up at the house in physical form.
(Actually I was out when they attempted delivery so I had to trail right out the the sorting office, but it's still pretty impressive. That or I'm just easily impressed.) Anyways, here they are:

Just look at all that sticky protective stuff on the perspex - that's on both sides of the pieces, which took an absolute age to manually peel off each individual piece. Worth it though, they're all lovely and pretty and shiny.

Not that you can tell, mind you, thanks to the amazing fuzzy blurriness of my phone camera. I've been meaning to get a proper camera for ages now. But then again I'd been meaning to get polyominoes laser cut since about June so that might be a way off yet. I just have a habit of putting off doing things for no real reason, which isn't good.
What is good however is the way these hexominoes are when you use them. My original set were cut on a CNC routing machine, and as a result have these weird beveled edges thanks to the width of the drill. Which means that when you turn pieces over they look weird, and sometimes pieces just don't comfortably fit together, mainly interlocking pieces with C-pentomino-like indents in them. But these are all nice and precise and fit together flawlessly, it's just so satisfying to sit there building stuff with them. Oh yeah, and they're scaled to 1cm squares too, so I can use that cutting mat to assist with construction. (Not that it helps much clearly, given the amount of patterns I've cocked up due to misaligning things in the past...)

Here's the full set. Hexominoes, pentominoes, and a bunch of little monominoes and dominoes which are useful for marking out pattern boundaries and hole locations and other such things.

Of course the real goal here wasn't just to have a nice spanking new set of hexominoes. Lord no! These were just a test run really, to see what kind of quality the pieces would be and how much everything would cost, stuff like that. But now that I know this works, the plan is to get myself some octominoes made. Never mind that there's not a big enough flat surface in the house to use them on, that's besides the point. Octominoes! Picture it, all done in fluorescent clear plastic so you can see the boundaries between pieces nicely - that's the one flaw with these hexominoes, but I chose a solid colour on purpose so when I eventually make the octominoes they're visually distinct.

And try not to think about how long it'll take to manually remove the scratch-protection sticky business from all 369 octominoes. On both sides.

Oh yeah, almost forgot, here's a couple of little hexomino things, just since I've been playing with the new set quite a bit recently. Here's a better illustration of the 11-hole rectangle from the photo above, because due to a combination of lighting and piece colour you couldn't really see what's going on:

And here's a 5-cell high parallelogram that was a ball-ache to complete. In fact I used a program to place the last 8 pieces in a fit of laziness. It was getting late and I had other stuff to do.

And then I found a bunch of different pattern variations based on a 15x15 square with 15 holes. There's some quite nice challenges here, analogous to the pentominoes in an 8x8 square with 4 holes that you can place wherever. It's hexominoes though so parity constraints mean you can't just stick the holes anywhere, but it still leaves enough room for creativity. Hell, now that I can laser cut stuff I'm thinking about the possibility of making a little tray to hold a 15x15 solution, and 15 monominoes in a very different colour that can be used as a little self-contained puzzle.
Here's three example solutions for you to feast your eyes on, arranged from left to right in increasing order of fiendishness.

More heptomino things

Recently I've been way busier than usual with non-polyform-related things. There haven't been as many evenings where I could just whip out the ol' heptominoes set, so constructions have been a little thin on the ground. Additionally, I can't help but feel I'm beginning to exhaust all the possibilities with just heptominoes (well, all the interesting possibilities anyway, I'm sure there's plenty more rectangles with holes in them possible but after the first five or so they don't make for particularly gripping blog posts.)
Anyway, to kick things off here's a squarish shape with minimal holes and maximal possible symmetry, that I'm surprised I hadn't thought of sooner:

Here's two parallelograms, using the full set of 108 split between them. Getting the holes centered here was hard, it's impossible to eyeball it and it's a bit of a pain to work out where they should be by counting. And it never looks quite right either, just due to the sloping nature of the diagonal sides. Still, far as I can tell this is correct:

For reference, the first one took about 10-15 minutes, and the second one took a bloody age because I'd foolishly forgot to use the [ shaped piece up earlier. And it doesn't play nicely with other pieces.
Things like three 11x23 or 4 10x19 parallelograms are probably possible too. Next time I've got a free evening I'll have a crack at one of them maybe.

A while ago I found a nice hexomino solution (about half-way down that page) that had a 7x9 rectangular hole in it, which could fit a set of pentominoes inside it with 3 cells to spare. Well, I went one better and found a heptomino frame that could accommodate that inside it, creating the following three-layer pattern. With the holes spaced nicely around the edge too. Forget 45° zig-zag edges - five-cell-high tubes with heptominoes are my new least favourite thing to solve.

To me right now, this just screams 'find a fourth layer with the octominoes!' I haven't checked yet if it's mathematically possible to, or at least if it's possible without introducing more holes than the 6 required for the six holey octominoes. I'll have to have a check.

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.

"There's No Way I'd've Found That By Hand!" Vol. 1

At work I have a notebook. And in it there's surprisingly little pertaining to actual work, but lots of ideas, calculations and sketches for possible shapes that could be filled up with various sets of polyominoes or other polyforms. Most of these end up solved at home, then drawn up all pretty and posted on here, but there are some that I either try and can't do, or that just look so intimidating I don't even try. It's those jagged diagonal edges... I just have a right job doing them.

So, here's a little collection of solutions found with various programs, that I was too weak to suss out by myself.

In hindsight, this one doesn't look that bad, and I've solved similar in the past. I guess at the time I was just not feeling up for the challenge. I think (judging by the colour and scale of the image) this one was found using the solver on Peter Esser's site here.

I'll be honest, I didn't even attempt this one. I did the calculations to make sure that it was permitted by the parity constraints, then just despaired at the thought of having to actually solve it, central holes and all. But FlatPoly2 made short work of it, finding this in about five seconds flat.

Then there's this family of solutions I have no recollection of looking for but are in the folder called 'Polyominoes' so I may as well just post them for the hell of it.

 I think at this point, hexomino solutions just aren't the most impressive thing in the world any more, computer-found or not. But more interesting things like heptominoes take a while to do, and I've got driving lessons and band-related business to contend with on most evenings, so that stuff tends to take a back seat.
Actually, I've got a nice new shiny set of octiamonds that I've yet to do anything of note with... I'll have to have a little play around with those, see if I can create anything worth posting on here with them.

Hexominoes - Scraping the Bottom of the Barrel

In a previous post I made some pretty shapes with the hexominoes and had a good whinge about parity while I was at it. And at the time I thought I'd pretty much exhausted all the possibilities for nice things that could be done with hexominoes. But then a few weeks after that, I found myself on holiday in a little caravan, with the weather being utterly atrocious outside. Luckily, I had decided to bring my little homemade hexomino set along with me, so in the absence of anything else to do I uh... constructed some more constructions, I suppose.

First off, a good (but probably not 100% sure-fire) technique for finding things that are actually constructible (and aren't ruled out by parity constraints) is to start with a rectangle that has odd x odd dimensions (and an area > 210 units) then carve out a central hole until the total remaining area is 210 units. As long as you're fairly careful here, the resulting shape should be solvable.
(Some day I'll do a write up of parity the way I understand it on here... honestly more for my own benefit than anyone else's.)

Fig. 1: Three rectangles with nice symmetrical cross-shaped holes hacked out of them. That third one, I've seen a different solution to before (I think it might be in Solomon Golomb's book).

The third solution, the 15x17 one, took an outrageous amount of time and attempts to solve. It's a shame that processing-power-wise we're a long way off being able to enumerate solutions for a given shape with hexominoes (like we can for pentominoes) - it'd be interesting to see if the ones that take me ages have fewer solutions and are genuinely harder.

I also found a solution to a 13x21 rectangle with a 7x9 removed from the centre. With the central hole being 63 unit squares, that means there's enough space to get all the pentominoes stuffed in there with a little bit of wiggle room to spare.

Fig. 2: This took a while too, as far as I remember. Hexominoes and thin sections (the 3xn bits above and below the pentominoes) don't seem to mix too well.

Then I realised that I didn't need to restrict myself to taking the excess squares out of the middle of the rectangle - I could just as easily round off the edges, and make some stretched-out octagons.
Which I did.

Fig. 3: A way stretched-out octagon.

And here's a slightly more regular, more octagonal-looking one I found a little while later:

Fig. 4: I bet there's a way of doing this with the six internal holes all in a diagonal line, but I'll leave that as an exercise for the reader 'cause I can't be arsed right now.

And just for an extra Brucie bonus, while we're on the subject of octagons, here's one made out of heptominoes that I found a few weeks ago:

Fig. 5: For whatever reason I have no recollection whatsoever of the actual solving process that led to this.

Congruent Rectangles with Heptominoes

(This post might be Part 1 of many, depending on how much free time I have to look for other sets of rectangles.)

A few weeks ago I found this:

Fig. 1 - Four 10x19 rectangles using one complete set of heptominoes.

There's a number of ways of making sets of same-size identical holey rectangles with the heptominoes. First off, there's three 11x23s which I've done before in an earlier post (and since 11 and 23 are both odd you can get the hole slap bang in the centre of each rectangle.)

Then, there's four 10x19s as above. Then it looks like the next set of rectangles that might be possible is nine 5x17s (with potential for centered holes too!) but whether that's even solvable is another question entirely. I mean, it goes without saying it'll be a frustrating ordeal knocking it out by hand. Actually, Patrick Hamlyn managed twelve 8x8 squares (three quarters of the way down this page) so anything's possible.

***

Also, a thing I've found that works really well for heptominoes if you're just after solutions and not the glory of finding them entirely by hand. Solving the first 90% of a shape by hand, saving the nice clumpy blocky bits of course (that goes without saying), then crack open some software like FlatPoly2 to brute-force the last 10 or so pieces. This works with heptominoes especially, since the really wide search tree that is placing the first ninety-odd pieces doesn't seem to play particularly nice with my computer. (More expensive computers may get better results!)

Then there's that other thing. The one that feels partially like cheating, just a teensy little bit. Picture the scene, you're just spend two hours slogging away at a particularly gnarly construction and just can't get the last few pieces in. There's a point at which you start to wonder if what you've got left here is even possible. So occasionally I've succumbed to the temptation to input in the shape of the remaining hole and the last five or six pieces, not to find a solution, but just to check if one exists. Then if it transpires you're looking for a solution in a space where none exist, I tear out one or two pieces, input the new, bigger hole and keep checking until we've got a gap where there is definitely at least one possible solution. Finding that will still be an ordeal though.

So it's computer-assisted solving. I don't know whether it counts as truly properly solving by hand, but when I've just sunk a whole evening into a construction and it's pushing eleven and I've got work tomorrow, I'm usually not too bothered.

Baby steps with Polyhexes

I know, I know, the blog name is 'polyominoes' and this isn't strictly polyominoes but hear me out. A few weeks ago I got my grubby mitts on a set of these:

Polyhexes! The 1- through 5- hexes to be more specific, fresh from Kadon Enterprises. And after playing with nothing but polyominoes for years, switching to these is weeeird. The whole '120-degree angles' thing.
With months of solving polyomino constructions I had developed a kind of sixth sense for instinctively knowing whether a piece would fit in a given place, and I had a pretty good idea of which pieces I needed to hold onto for late in the solution. With this bunch, no such luck however. It didn't help that I had no familiarity with the pieces as a set either - with hexominoes (and even heptominoes to a degree) you start to individually know each piece in the set, and can generally rely on memory to get a vague idea of which pieces have been used so far. And the pieces end up with little nicknames based on their shape, so that when I'm frantically scrabbling around looking for a piece I can better remember exactly which one I'm after. With polyhexes it was like starting from scratch again. So I started with just the easy pieces and worked my way up...

Tetrahexes, then.

There are seven of these, and they can do a surprising amount. Their total area is 28 units, meaning that a 4x7 parallelogram should be possible... and it is. While there is a sort of restriction similar to the parity issue with polyominoes that can occur in polyhexes, it doesn't impact constructions like this the way it does tetrominoes (I think it might be responsible for the triangle with side length 7 not being possible though.)

(Also surprisingly challenging: drawing hexagonal things in MS Paint.)

There's also this 3-cell-high pattern too. There are two possible solutions for this; finding the second one is an exercise for the reader.

 And here's two patterns based on the 5x6 parallelogram with symmetrical holes.

Difficulty-wise, I'd put these somewhere between tetrominoes and pentominoes. Which sort of makes sense, as there are 7 of these, right between the 5 tetrominoes and 12 pentominoes. And that propeller-looking piece is a royal pain in the arse.

There's bound to be more fun stuff to be done with these pieces, but this was all I managed to find before the allure of the pentahexes became much too strong to resist.


Pentahexes, for those not in the know, are the shapes made by sticking five hexagons together edge-to-edge. And there are 22 of them, giving a total coverage of 110 units. Which is promising, since 110 can be divided up in various nice ways - we ought to be able to get a nice selection of parallelograms out of them.

Sadly, I've been a tad lazy and only attempted the 10x11 so far; my solution is shown below.

If you look at the top-left you'll see that I've tried to carry over my usual technique for polyominoes, which is holding onto the clumpy, blocky bits. But this technique... needs work. This was still a right hassle to solve, I think it took about an hour by hand (and just to rub it in, search software finds solutions to this in like 3 seconds.)

Oh yeah, and there's one other fun thing I noticed with the pentahexes. None of them extend for more than 3 cells in more than one direction. They all could fit in a three-cell-high construction, if someone was masochistic enough to go look for it...
I remembered how deceptively tricky getting the tetrahexes into that 3-cell hexagon thing was. And at this point I could have done the right thing and put down the pentahexes and, I don't know, gone outside and talked to girls or something. But~! Once a challenge like this presents itself, you can't just back down, so I began knocking together little segments of three-cell-high, to be hopefully worked into one big long construction. Remember the infuriating propeller-shaped piece in the tetrahexes? (Maybe you own a set, and know the frustration first-hand!) Well, the pentahexes have a good selection of pieces related to the propeller but with an extra hexagon tacked on, and these have all the infuriating properties of their 'parent' tetrahex, and then some!

So after quite a while (I lost track of time, as tends to happen once you get right into a good polyform construction) I eventually stumbled upon the following solution. And vowed never to tackle something like this again - not for next few hours anyway.

Fig. 6 - The 22 pentahexes squeezed into a little narrow construction that I'm stunned actually works.