Polyomino Puns

I realized I dont own any any Polyomino games (different shaped "tetris" pieces that you place on a board) and was wondering what everyone's favourites are? So far doing some research here are the ones I've been looking at:

• Patchwork
• Bärenpark
• New York Zoo
• Paris: La Cité de la Lumière
• Kingdomino/Queendomino
• Silver & Gold
• Cartographers
• Copenhagen
• Isle of Cats

Though Im sure theres more, but curious on what everyone's favourites are!

Is there a robust way to determine the smallest area and layout that can hold any polyomino of a given size? Or have there been publicized lists of such areas and layouts? (Such problems come from too much time in rooms with tiled walls and floors.)

I know the areas(vertically oriented) up to n=4 look like:
n=1: (Area 1)
X
n=2: (Area 2)
X
X
n=3: (Area 4)
X
XX
X
n=4: (Area 6)
X
XX
XX
X

I think n=5 can be the following or its mirror, but I have less confidence in its completeness, as well as less of an idea how to expand to n=6:
X
XX
XXX
XXX
XX

I would appreciate guidance towards prior work. Thank you!

I'm working on a game where you build a grid out of polyominos of arbitrary sizes shapes. It should support drag and drop within a grid. One example of a similar game is Isle of Cats.

What I'm after is a component that allows me to create styled polyonimos that have pixel-perfect shapes so they can be dragged from any part of them. I'd also like to be able to style them as a whole component, rather than building it from an asset per square.

Is there any library or code approach that'd allow this use case? I have DnD resolved by react-dnd or interact.js, so as long as the components can have those concave shapes I look for it should all work together.

Last night finishing a solo version of Era: Medieval Age, I looked around my apartment and realized that there are a lot of polyomino games that have come out lately. Or there are games that include polyominos. I have acquired:

• Cartographers
• Era: Medieval Age
• My City
• Tapestry*
• Tiny Towns

^(*Not a polyomino game but you have a city mat that you place buildings onto so it incorporates a rather cool polyomino mini-game.)

I've played and enjoyed Patchwork at a friends. I've seen but not played A Feast for Odin, Cathedral and Copenhagen. There are others out there that I would like to get eventually like Isle of Cats. This however is a lot of games about polyominos, or at least involving them. And this is a non-exhaustive list. The experience so far is different with each of the ones that I have. (I haven't played My City or Tiny Towns yet. Dang pandemic.) Boardgamegeek.com lists over 180 polyomino games, although the numbers are skewed because every expansion is listed separately, so I suspect it's more like 100.

There is however only one that I can find coming in 2021. (This may be due to poor Google-Fu.) Game designers are pretty intuitive, and I have yet to see a science fiction polyomino game, although I think I spotted one on the BGG list. So my questions are:

Have we reached peak polyomino games?

Regardless of whether we have or haven't, what's a good "next polyomino" game aside from Isle of Cats in your experience?

Hello! I'm not sure if this is the right sub for this, but here goes.

I'm working on the expansion for a board game that I Kickstarted this summer (link here if you're interested). Basically it's a filmmaking themed game that uses dice with custom faces as crew, and you set up your scenes for shooting by arranging the dice in the setup indicated on the scene card, which looks like a polyomino. Example:

Scene setup diagram on the right side

I'd like each scene card to have a unique setup diagram so that there's always a challenge in figuring out how to get the dice in that particular arrangement. This was fine for the initial game, which only has 25 scene cards and I could just do it by hand, but for the expansion I'm planning to include up to 100 additional scenes. So I basically want to have an image that shows all possible shapes that the polyomino could take on a card, so that as I'm designing the expansion cards I can start crossing off the ones I've used and always have new ones to work with.

Note that it's NOT important for me to have every combination of every die face; just the different possible shapes that the dice can make as a group. For this purpose they might as well all be blank squares. I can assign the faces later.

The rules for the setup diagrams are:

• Must fit onto a 3x4 grid (that's the space that the scene card allows, graphically, for the diagram, as you can see above)
• Only combinations of 4 and 5 dice (or polyomino squares)
• ALL combinations of 4 and 5 dice - linked together orthogonally, diagonally, spaced apart, 2 spaced and 3 diagonal, etc etc etc - as long as they fit on the grid space
• Mirrored shapes would be considered unique from each other, but not rotated shapes (the rules of the game state that you CAN arrange dice in a pattern that is rotated (ie, 90 or 180 degrees) from what's displayed on the card, but you CANNOT set it up flipped / mirror image)

I'm hoping for a simple diagram that shows all the possible combinations.

Initially I thought I could do this by hand, and I've already created 7 pages of a document, each page looking like a variation of this:

[This is roughly what I'm looking to get at the end](https://preview.redd.it/b8fbq931uc161.jpg?width=2732&format=pjpg&auto=webp

For some reason Polyomino games have fallen flat with everyone (although they seemed well loved by many on the internet)... I think can summarise my main gripe with this:

1. Its nearly impossible to read what your opennents are doing at the start. eg Patchwork, Isle of Skye. Theres just too many possibitilies to consider at the start, and analysis will usually drag the tame. In the end, most of us just focus on our own boards.

2. That being said, towards the end, it does become easier to read your opponents. But perhaps thats also a "flaw" I find about polyomino games. Whereby the decision space is extremely large and complex at the start, and towards the end you have no more decisions to make. If you lose early, you're essentially just left twiddling your thumbs (eg Tiny Towns). I find that really bums out my more casual gamers, because the initial planning required at the start is way too intense. And usually makes for awful end games for newer players.

3. Polyomino games tend to have small tiles which makes it even harder to read what your opponents are doing. In 5p games like Isle of Skye, noone bothers to look at what other players beyond their immediate left or right are doing.

4. It sometimes tends to feel more like a puzzle than a game. Not sure how I can explain this but perhaps this is just really more of a taste of my various groups. We prefer games that dont feel like "chess" or "sudoku" in a sense.

Again, Im just sharing my pain points about polyomino games and wondering if someone can help me see this in a new light. It seems like polyominoes are all the rage these days and itll be great if I can find a way to get myself and my group to like such games. I like the themes of Tiny Towns and I want to get New York Zoo for its theme. But somehow the actual plays of the game truly falls flat most of the time despite it seeming to work for most other players...

I'd like to get a game with polyomino but I'm not sure which one to get. If I knew people who liked to play board games, I'd get Copenhagen, but I'm a solo player, and all I've heard about are the Uwe Rosenberg games, and Isle of Cats. So I'm just looking for suggestions on what solo polyomino game to get.

I was trying to understand gerrymandering and my approach is usually to simplify the situation as much as possible. And that led me to the question posted above.

If you have a 1x1 grid, then there is only 1 way to tile the grid using a single monimo piece.

If you have a 2x2 grid, then there are 2 ways to tile the grid using 2 dominos.

If you have a 3x3 grid, then there are 10 ways to tile the grid using 3 tronimos. ...I think. I could be wrong about this because I’m just drawing everything out by exhaustion.

If you have a 4x4 grid, then there seems to be a lot more ways to tile the grid using 4, uh, quatrominos (look, I’m just making up the words by this point). The main jump in 4x4 seems to be that there are 16 blocks to fill. When you divide them up, the fact that 12 (or 3x4) isn’t relatively prime with 16 adds a lot of permutations.

My suspicion—without checking—is that a 5x5 grid won’t have as many ways to tile OR it won’t have as large of a jump as the case change from n=3 to n=4. This is because 5 is a prime number.

In any event, does anyone have any ideas or resources regarding this problem? I guess the fastest way to say what I’m looking for is the sequence of “number of permutations.” So we’d have

1, 2, 10, ...

or, if we know that function, then

(1, 1), (2, 2), (3, 10), (4, ?), ...

Any thoughts?

I'm almost certainly not the first person to discover this, but I figured it out this morning and thought it was pretty cool.

Choose any two nonzero real numbers a and b, and make a tile by adjoining squares of side length a and b. Picture

Now, let's tile the plane with this! Picture

Connect corresponding points on the tiles to form a bunch of squares (shown in red here).

The side length of this square, c, is the hypotenuse of a triangle with side lengths a (horizontal distance) and b (vertical distance). But since the two tilings of the plane have a one-to-one correspondence, each tile has the same area. So a^(2)+b^(2)=c^(2).

So, I'm a comp sci major, (senior, but going to need a super senior year) and I'm looking at trying to get attached to a research project, mainly because I think it would be a good experience and also it might look good on a resume for grad school if I decide to go that route. I'm looking at research projects in the math department and there is one that looks interesting. It had to do with studying the shape of random polyforms. I'm thinking about the essay I'm going to write about in my application will link back to one time where I was playing around in the canvas package and how I would create fractal patterns not because it was an assigned project, but because I personally found it interesting, and how that will transfer over to studying for this project. (I am told this is a common intro to comp sci project, and I don't know if my current university teaches it, but I know my previous community college didn't touch fractals, or even canvas in general.)

I basically want to ask if this is a reasonable thing to write about. Or does it secretly expose me for being out of my depth and not knowing what I'm talking about?

Some time ago I started to wonder how to put polyomino-shaped tiles together as Catalan tilings (i.e. the tiling looks identical from each individual tile). I have an algorithm and here are some interesting results:

https://preview.redd.it/6si1k8h0dux31.png?width=200&format=png&auto=webp&s=ef9dad6d8d8c684c167f67b31dfc7e91705b4233

https://preview.redd.it/dziny272dux31.png?width=200&format=png&auto=webp&s=6ab112b66d786411277f38d155e34b325778d452

https://preview.redd.it/3cdckv86dux31.png?width=200&format=png&auto=webp&s=1ba5ee864dc614a7add1e8884481be92665c5b3f

https://preview.redd.it/49d86j28dux31.png?width=200&format=png&auto=webp&s=4f1b76420db871ebc0129b39d3972d3ed52c41d5

https://preview.redd.it/lf93szabdux31.png?width=200&format=png&auto=webp&s=c262fd3a46ce548e286daf608f36d6dfb7603b32

The last one is extension of the algorithm -- you can specify several shapes and the rule then is just that tiling must look the same from every shape of the same category.

It works for spherical/hyperbolic tilings as well, but it's harder to make good pictures there. Here, for example, is a polyform that tiles one variant of (6,6,6,3) tesselation.

https://preview.redd.it/daa3dnqpdux31.png?width=2000&format=png&auto=webp&s=43e4e77fcca6c83945891256807556dc0fdf4ccd

Polyominos are distinct, unique shapes formed out of fixed number of squares. No rotations or reflections are counted as distinct.

A good example of a polyomino are the tetrominos found in the videogame "Tetris". There are 5 possible tetrominos. Not enough for a writing system, unless there are very few sounds in the language.

Hexominos, however, have 35 possible shapes. More than enough for an alphabet, and even a small syllabary if there are not many syllables.

Here is a picture example of the 35 hexominoes.

https://upload.wikimedia.org/wikipedia/commons/thumb/0/02/All_35_free_hexominoes.svg/360px-All_35_free_hexominoes.svg.png

Move up to Heptominoes, which are all the possible arrangements of 7 adjacent squares, and you have 108 distinct shapes to make a glyph out of. Enough for a large syllabary, say Japanese sized.

https://upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Heptominoes.svg/610px-Heptominoes.svg.png

Octominoes have 369 possible glyphs.

https://upload.wikimedia.org/wikipedia/commons/6/6a/The_369_Free_Octominoes.svg

Nonominoes have 1285 possible glyphs. Perhaps big enough for a small logography.

For an Alphabet used to write American English, I think hexominoes would be "good enough" but not as good as Octominoes. 35 possible glyphs. You'd have to cover 42 phonemes with those 35 glyphs, but it's better than trying to do it with 26. While not perfectly matching spoken English, you could make the alphabet at least more phonetic than the current one. Also, many of the heptominoes look similar to each other, and you don't have a big inventory to filter out the similar ones. That's where heptominoes comes in.

Heptominoes offer a bigger inventory of glyphs to choose from, if you're picky about how you want your letters to look. However, you'd need to discard 66 glyphs- the smart way to go about it would be to eliminate glyphs that are very similar to other glyphs, leaving you with 42 of the most recognizable and distinguishable letters possible.

One possible issue is standardizing the dimensions of each glyph. Not all polyominoes have the same height compared to their width. You could have fat polyominoes, or tall polyominoes to varying degrees. Possibly even square polyominoes with equal width and height.

If you can try making a script based on Hexonimoes, Heptominoes or Octonimoes, please post them here. I would LOVE to see them! Now of course using the polyominoes themselves as the glyphs would yield very pixelly letters, but of

Is there a robust way to determine the smallest area and layout that can hold any polyomino of a given size? Or have there been publicized lists of such areas and layouts? (Such problems come from too much time in rooms with tiled walls and floors.)

I know the areas(vertically oriented) up to n=4 look like:
n=1: (Area 1)
X
n=2: (Area 2)
X
X
n=3: (Area 4)
X
XX X n=4: (Area 6)
X
XX
XX
X

I think n=5 can be the following or its mirror, but I have less confidence in its completeness, as well as less of an idea how to expand to n=6:
X
XX
XXX
XXX
XX

I would appreciate guidance towards prior work. Thank you!