Wide Tetrominoes, and a Different Kind of Parity

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

Introduction

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

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

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

Wide Tetrominoes

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

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

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

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

A Good Reason for It

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

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

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

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

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

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

What next?

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

---

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

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

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.