It's very important to tile a plane with pentominoes and not tetrominoes. Tiling a plane with tetrominoes causes the plane to disappear, but you get the highest score possible in tetris.
Fun fact, Tetris was originally supposed to be Pentris and it was based on pentomino, but Soviet computers scientists realized their hardware specification they chose for the project doesn't have enough of memory to handle pentomino pieces, so they downscaled to tetromino pieces and created Tetris.
I love the rectangle tessellations you show at around 10:59. There's something just so satisfying about the whole gang showing up, in a neat-and-tidy space!
Tetris was actually inspired by a puzzle its creator owned, which involved fitting pentaminos into a box. He figured that pentaminos were too complicated, and switched them out for tetraminos. guy saw pentaminos and was like “its got too much… 1 thing its exactly 1 thing too much
tetronimoes actually get used all the time in geometry teachings. wonder what could have possibly caused such a specific interest in so many mathematicians... it's a mystery.
@@maeve-wav I had the GBA game for that, and I really wish they did more with the idea. As a kid I really liked fitting the shapes together in different ways. Because of the die unfolding theme you specifically get the 11 cube net hexominos, rather than the set of 35 total hexominos, but it still ends up being a wide variety. I don't remember much about the game other than the dice, so I couldn't say whether it was actually fun as a game, haha.
Guys, I have an ideia for a video game. So we make a 2D box and let random pentominos fall slowly till they reach the bottom, the player can move the piece left and right, and spin the pentomino. Once a line is filled the game give points and clear the line leaving the space where the pentominos above fall. The game is over when there's no more free space. The game will be called Petris
One very important pentomino fact that you forgot is the parity of the pieces. If you imagine putting each piece on top of a chessboard, where the cells cover up the black and white squares, you’ll find that all but 1 pentomino has a 3-2 parity, where it covers 3 black, 2 white, or 3 white 2 black. The only piece with a different parity is the X, which has a 4-1 parity. This means that if you place an X somewhere, you’ll need to place 3 non-X pentominos to get back to covering an equal amount of white and black squares, or just by placing one additional X pentomino.
I remember learning about this in the context of Tetris, where the parity of the board is an invariant except when you place a T piece. This is part of what makes it so difficult to clear checkerboard-pattern garbage, since it's as far away from 0 parity (aka an empty board) as possible.
Blokus is a fun board game, where you are given 1 of each pentomino, tetromino, triomino, domino, and monomino, and try to place as many as you can while only expanding via corners, and you can't touch edges. It's impossible to fit them all, so you have to compete with other players for space.
P-pentomino is the only one that has a perimeter of 10, while all the others have 12. My fourth grade teacher claimed that they all had a perimeter of 12 and asked students to try and prove her wrong. None of them tried the P shape and concluded she was right, but I discovered it later. I was too shy at that age to argue against a whole class so I’ve spent my whole life without vindication.
My favorite type of youtube video is one that is just listing off facts partaining to a particular math subject. My favorite part of math is simply just how much you can do just by playing around and having fun with interesting constructions. And then... what's that? A Patricia Taxxon background song?? My favorite musical artist?? This is a perfect video.
I hope I'm not too late for this, I've found a way to get an upper bound for a given nxn grid and a given number of pieces. Basically, each square on the best path is flanked by 2 walls, these walls might not be directly adjacent, but surrounding each square there will be 1 exit square, 1 entry square, and 2 other squares that either have a wall, or are an an empty path towards a wall. If the paths were not empty (ie they contained a cell that was part of the maximal path) then the path would simply choose to go towards the current cell via this empty path, meaning the maximal path is not maximal. If that made sense, the next step is to see that each piece provides exactly 12 blocks of adjacency, when you account for multiplicities (ie the L block gives 1 square in the elbow with a multiplicity 2 adjacency). Also, there is free adjacency provided by the exterior wall, equal to n * 4. So the basic adjacency score is 12 * b + n * 4, where b is the number of blocks. So the max path length is less than (12 * b + n*4)/2 However we can adjust the adjacency score to make it more accurate. First, notice that when pieces touch each other to form a connected wall, they lose one adjacency (unless they touch on their corners). This means that each non-corner touching gives a -2 to adjacency score. Also, we should assume that there is a square on the outside rim, so that is another -1. Also Also, the last sections of the path require 3 adjacent walls, instead of 2. So that takes 1 more adjacency score. We are left with an equation that looks like: (12 * b + n*4 - 2 * (b-c) - 1 - 1 )/2. where c is how many corner connections are between the blocks. Note that c
The P piece uniquely does not have the same adjacency score as the rest of the pieces, at 10 instead of 12, making this slightly incorrect, and the upper bound could be reduced further.
@@Aimlesswaves. since you are not forced to use the p piece, opting not to will give the current upper bound. you could have two upper bounds, one for if you use the p piece and one for not using it, although this doesn't seem too useful since you aim to get the longest path possible.
As someone who (casually) plays Pentris, this is a fun analysis! Pentominos are definitely a lot more complex and interesting than their 4 tiled counterparts, and it really spices up the amount of thought you have to put in (especially during the faster phases when you have to make split second moves with awkward shaped pieces). Highest score I've ever gotten has only been a bit over a couple thousand points, so this could definitely be improved with a good bit of theory.
not surprising, considering each X is a water block with cane on each side! the fact that this tiles the plane exactly with no gaps corresponds with the fact that this is the maximum amount of cane that can exist in that space for that amount of water, which is of course true since each water block cannot possibly connect to more than 4 sand/dirt blocks! neat!
And now you know that a trivial game design choice created a scenario that due to the underlying and often ignored structure of mathematics has exactly one solution. Isn't that crazy? The plane can be tiled so many ways by so many monotiles, but due to the inherent geometry of a grid and this gameplay restriction there's exactly one way to build such a farm, and everyone must eventually stumble upon it. If the shape was any different there could be infinite farm designs, but this one provides no room for personal choice without sacrificing efficiency.
wait wtf this only has 500 views?? this video was great it feels like the type of video that should have like a million or something!! Keep up the great work !!
So lucky to find this video. Throughly intersting for its entire runtime with nice editing and good sound quality. Reminds me of Kuvina Saydaki, but higher energy. Subscribed in a heartbeat.
thank you very much!! kuvina was one of the inspirations for this video- I'm a massive fan of their sorting algorithm explaining video so I'm happy to be compared to them lmao
17:47 For board sizes n=42 and above, as soon as you’ve found the perfect solution for one board you’ve found the perfect solution for all of them because the game becomes a matter of “hiding” the two ends of the path. The optimal solution will just keep growing a bigger empty middle section the higher n gets. Why 42? Because 41 is the sum of the lengths of the “long sides” of every pentomino, the longest length a shape built from one of each can reach. 42 makes it impossible to block off the middle area.
@@jakobr_ basically, once the grid has doubled, the longest distancethe line can travel is no longer the adjacent corner, but instead the opposite diagonal. Because of this, its meaningless to make the stacked line anymore, and instead it's best to make dense mazes in either one corner or opposite corners
@@YeaCloth I don’t understand why the longest distance wouldn’t be to the opposite corner (or inside structures near opposite corners) from sizes 42-84
That pentomino game sounds like something that would be on an old school website and have a leaderboard on it where the people with the highest scores are shown.
I quite enjoyed this. The letter names you gave are in the Wikipedia article. I like the alternative naming they give by John Conway which uses the consecutive letters O through Z, though assigning O to the long straight pentomino is a bit dodgy. Other than that, my main math objection is the use of "regular symmetry" which is better known as "reflectional symmetry". Not that I know much, I'm an engineer.
i feel like this was one of the most pointless videos ive ever watched and simultaneously one of the most insightful and interesting and one of my personal favorites
Got excited to see a pentomino video on my feed out of nowhere! Gave me a bit of a throwback to a book series from Blue Balliet that got me interested in them. A character in that series used them for all sorts of things, a couple of which were mentioned here (I think rectangles of pentominoes were brought up a few times) and a few more unconventional things. Actually, OP, as an artist you might enjoy those books, as they're art-themed mysteries and treat the pentominoes as halfway a math object and halfway an art object, kind of like you did here.
When I was a kid, we did a Halloween school project where we had to make mixed media presentations on a mystery book. I found a book called Finding Vermeer about kids solving an art theft, and one of the kids was obsessed with pentominos, using them as a metaphor for conceots throughout the book. I made a fake newspaper clipping decorated with cardboard pentomimos, and ever since then I've been thinking about pentominos (and also Johannes Vermeer lol) at every possibly relevant time. I feel seen
I'm pretty sure the reason that grouping kept appearing was because of the underlying graph structure, ILNVUWZ are all lines, and FTY all have a central square with two branches of length 1 and one of length 2
@N-U-T-Ton the left side of the screen where the like middle shade of purple meets the lighter purple. You can see a bit of green. Also on the left side of the furthest left white tile is another.
Not only are there several of those games already, but technically it would be called Pentis since the R in Tetris is from the "Tetra" part of the name.
@@mateuszszulecki5206 Yeah but without the R that T is holding back all the dyslexic mistakes a reader could make all by itself. I don't think T has it in him, do you?
This is a great video, with one exception: it's so annoying that there's a lot of text that only pops up for a split second (you have to pause or go frame-by-frame to read it)
i am watching because i have refreshed my recommendation list over 10 times today but this is always at the top so i will watch in the hopes youtube will finally recommend something else.
important detail on the tetrominoes is that you often see them referred to as being 7, JZ TOILS, but J and Z are mirrors of L and S which are only distinct if mirroring isn't allowed but rotation is (as in the video game Tetris, which is where most people first encounter this subject)
interesting stuff, I'm pretty familliar with pentominoes since I play a lot of tetris variants with them, but it really surprised me to see some of these connections that I've sort of understood intuitvely layed out like this. You're proposed puzzle is also really damn interesting, played around a bit myself but it's a lot more initmidating than it first seems, do wish to see someone take a brute force computer though, if only for the satisfaction of knowing.
For a while in high school i was obsessed with fitting pentominoes (specifically including their mirrors too) into large rectangles. After succeeding a couple times, I eventually tried the same with hexominoes and heptonimoes but to no avail, they’re so much harder to work with the further you increase cell count
Huh, that pentomino path packing puzzle is pretty interesting, wouldn't be surprised if you could get it into a recreational maths journal with a bit of write up.
The mathematical properties of these objects are actually surprisingly interesting, thank you for sharing all these fun facts! They remind me of symmetry groups, especially at the start of this video of course. What i mean by groups is that field of math with algebraic symetries or whatever it's called, like the ever-mysterious monster-group (which is like mathematical cosmic horror imho, the smaller groups tend to be more comprehensible to mere 3d mortals 😅).
Your channel is very underrated! This was a great, well-edited video that I found very interesting and well-written. You deserve at least a couple thousand subs. I'm gonna help get you closer :)
Fascinating video! I really liked this format of just sharing what you have discovered about a specific topic. Pentominoes are incredibly interesting, and this scratched that itch I've had for a few years now.
where? this arrangement doesn't work, if that's what you're thinking of vv o=path ==wall _=empty o o o o o o _ o = = = = o _ o = o o = o _ o = o = = o _ o = o o o o _ o o = = = = = _ o o o o o o
the visual explanations/infographics are perfect and you did a great job with them. i also cannot imagine being able to say all of the things you said without needing like 46 takes for each so thats also impressive.
I can’t believe that you called the R-pentomino the “F-pentomino”… (wait… Conway calls an R-pentomino) Anyway, starting from the R-pentomino, one can do an RF28B to a B-heptomino, then BFx59H to a Herschel, then HL95P into a pi-heptomino (releasing a glider), then PF35W into a wing, then WFx46H into a Herschel again, and finally HRx65R back into an R-pentomino (releasing another glider)
@@dmcdouga07 I guess I’m a member of the ConwayLife community, and everyone there calls it the “R-pentomino” since that’s what Conway calls it In fact, Conway called the I, L, F, and N pentominos “O”, “Q”, “R”, and “S”, so that the pentominos would be OPQRSTUVWXYZ
Personally it looks more like a lowercase f to me, doesn't really look like an R at least in the font of YT comments, so the label of " F-pentomino" makes more sense to me. Also I haven't ever played conway's game of life but I have heard of it and know a game inspired by it called "Cell Machine" (yes that Cell Machine that was made by Sam Hogan)
Dude, this is genuinely so good, you 100% deserve way more attention. Your passion for the topic really comes through and I love how easy it was to follow your logic through the different avenues. The graphics you used really helped make the video comprehensible. If I could make one suggestion, your music choices felt kinda inconsistent in vibe and were a bit loud at times. That being said, if you keep creating videos of this quality you will blow up, I'm sure. Amazing work!
I’m sure you have heard this a lot by now, but the tiles that when flipped are different are known as chiral. Saying chiral gets across your point about parity in an easier way. This means there can be chiral and or achiral tile patterns for the shapes.
Here's one more potentially interesting thing: Take any one pentomino; can you use nine of the other pentominoes to make an extra large version of that pentomino, where one square of the pentomino is comprised of nine squares in the large pentomino?
12 year old me would have gone CRAZY over this lol. This video's lots of fun. That puzzle you showed off at the end is very interesting; I'd like to see someone tackle that in more technical depth.
It's very important to tile a plane with pentominoes and not tetrominoes.
Tiling a plane with tetrominoes causes the plane to disappear, but you get the highest score possible in tetris.
take your like and get out
.
Do not fill a box with pentominoes
Fun fact, Tetris was originally supposed to be Pentris and it was based on pentomino, but Soviet computers scientists realized their hardware specification they chose for the project doesn't have enough of memory to handle pentomino pieces, so they downscaled to tetromino pieces and created Tetris.
@@minskghoul what? I'm scared now
in my head it went like: 'this is a square; also known as a monomino..." which I'm laughing at just thinking about it.
Do doo, da do do.
A monomino, more commonly known as a mahjong tile or a Scrabble piece.
The O monomino
The "." mononino@@newcantinacrispychickentac7754
@@ArcheoLumiereThank you, that was my immediate thought upon seeing this comment
I love the rectangle tessellations you show at around 10:59. There's something just so satisfying about the whole gang showing up, in a neat-and-tidy space!
I love how absolutely no one liked or replied (expect me)
Hi cary
3rd reply?
I knew i'd find you down here!
I know you will be here, also bfdi is the best
guy saw tetris and was like "its missing something... 1 thing its missing exactly 1 thing
Specifically speaking, exactly 1 thing per piece
Tetris was actually inspired by a puzzle its creator owned, which involved fitting pentaminos into a box. He figured that pentaminos were too complicated, and switched them out for tetraminos.
guy saw pentaminos and was like “its got too much… 1 thing its exactly 1 thing too much
@@FizzyChalice So it went from Pentaminoes, to Tetraminoes in Tetris, then back to Pentaminoes in Pentris? Interesting
katamino has been around longer than tetris . look it up
@@FizzyChaliceTETROMINOES AND PENTROMINOS. NOT TETRA AND PENTA, TETRO AND PENTRO.
i like how you alternate between "zed" and "zee"
oh the woes of being a Canadian
@@v.deckard I can’t believe that you called the R-pentomino the “F-pentomino”…
@@ValkyRiverunacceptable
@@VivianAttler You’re clearly not someone from the ConwayLife community…
@@VivianAttler you are lemongrab’s 3rd cousin (twice removed)
the not-allowed pentomino at 0:16 is a conway's game of life glider! it would move diagonally up and to the left if it advanced
Now that you said it I cannot unsee it
God I love nerds like you
THE GLIDER HAS BEEN OUTLAWED
a wild glider has been spotted!
thats such a niche thing to know and I hate that I instantly knew what you were talking about
"it's not all that important" *introduces a mathematical research question*
Ikr
this feels like an entire untapped branch of mathematics and i’m all for it
tetronimoes actually get used all the time in geometry teachings. wonder what could have possibly caused such a specific interest in so many mathematicians... it's a mystery.
heavily tapped, but still very tappable, plenty more to research
Yu-Gi-Oh Dungeon Dice does some stuff with hexominos-you unfold a die into different shapes depending on where you want to go
@@maeve-wav I had the GBA game for that, and I really wish they did more with the idea. As a kid I really liked fitting the shapes together in different ways. Because of the die unfolding theme you specifically get the 11 cube net hexominos, rather than the set of 35 total hexominos, but it still ends up being a wide variety. I don't remember much about the game other than the dice, so I couldn't say whether it was actually fun as a game, haha.
well it's definitely not that
I’ve just always called the P pentomino “Utah”
As a Canadian who doesn't know he shape of each individual US state, I assume Utah looks like a P
@@djangel3108it’s more like a lowercase b
@@djangel3108
¤¤
¤¤¤
¤¤¤
@@djangel3108It looks like the P pentominobut not a P
As a Utahn I approve
Guys, I have an ideia for a video game. So we make a 2D box and let random pentominos fall slowly till they reach the bottom, the player can move the piece left and right, and spin the pentomino. Once a line is filled the game give points and clear the line leaving the space where the pentominos above fall. The game is over when there's no more free space. The game will be called Petris
Tetris with Pentominoes has been implemented and it's really really really hard. Tiling pentominoes is so much harder than tiling tetrominoes.
its called tetris because theyre tetrominos. as in tetr-is. so naturally, the pentomino game should be called penis.
It's Petris, not tetris. They're completely unrelated and unique
@@megapussi PENTRIS, NOT PEE
@@danielthecake8617 "they're completely unrelated"
You could say that the P pentomino contains LOTS of tetrominoes
Omg your a genius XD
Lmao I thought of that too
Oh crap you are right
P can fit the T, J, Z and O tetriminos
NO
@@AerianTelevisionThat isn’t what he meant…
One very important pentomino fact that you forgot is the parity of the pieces. If you imagine putting each piece on top of a chessboard, where the cells cover up the black and white squares, you’ll find that all but 1 pentomino has a 3-2 parity, where it covers 3 black, 2 white, or 3 white 2 black. The only piece with a different parity is the X, which has a 4-1 parity.
This means that if you place an X somewhere, you’ll need to place 3 non-X pentominos to get back to covering an equal amount of white and black squares, or just by placing one additional X pentomino.
New floor tiling just dropped!
Holy hell
how do you even know this
I remember learning about this in the context of Tetris, where the parity of the board is an invariant except when you place a T piece. This is part of what makes it so difficult to clear checkerboard-pattern garbage, since it's as far away from 0 parity (aka an empty board) as possible.
@@GarryDumblowski i feel like understanding what you mean will make me a better tetris player so please how does that work?
16:27 some pentominos? (Changes position) PERRY THE PENTOMINOS!?!?
Hilarious
Glad I'm not the only one who noticed that haha. Pattern recognition go brr
“Ah, Perry the Pentominos, what an unneeded cameo. And by unneeded, I mean COMPLETELY NEEDED!”
OH WAIT NOW I SEE IT TOO AHAHAH-
Blokus is a fun board game, where you are given 1 of each pentomino, tetromino, triomino, domino, and monomino, and try to place as many as you can while only expanding via corners, and you can't touch edges. It's impossible to fit them all, so you have to compete with other players for space.
Such a fun, simple game!
Can confirm, used to play it fairly often with family when I lived at my father's place.
It is a very fun game! I play it with my family on weekends sometimes
Ah yes, the domino.
OH WAIT THATS WHY ITS CALLED A DOMINO??
P-pentomino is the only one that has a perimeter of 10, while all the others have 12. My fourth grade teacher claimed that they all had a perimeter of 12 and asked students to try and prove her wrong. None of them tried the P shape and concluded she was right, but I discovered it later. I was too shy at that age to argue against a whole class so I’ve spent my whole life without vindication.
Very interesting
I will tell her 🫡
@@VivianAttler 🫡
My favorite type of youtube video is one that is just listing off facts partaining to a particular math subject. My favorite part of math is simply just how much you can do just by playing around and having fun with interesting constructions. And then... what's that? A Patricia Taxxon background song?? My favorite musical artist?? This is a perfect video.
I hope I'm not too late for this, I've found a way to get an upper bound for a given nxn grid and a given number of pieces. Basically, each square on the best path is flanked by 2 walls, these walls might not be directly adjacent, but surrounding each square there will be 1 exit square, 1 entry square, and 2 other squares that either have a wall, or are an an empty path towards a wall. If the paths were not empty (ie they contained a cell that was part of the maximal path) then the path would simply choose to go towards the current cell via this empty path, meaning the maximal path is not maximal.
If that made sense, the next step is to see that each piece provides exactly 12 blocks of adjacency, when you account for multiplicities (ie the L block gives 1 square in the elbow with a multiplicity 2 adjacency). Also, there is free adjacency provided by the exterior wall, equal to n * 4. So the basic adjacency score is 12 * b + n * 4, where b is the number of blocks. So the max path length is less than (12 * b + n*4)/2
However we can adjust the adjacency score to make it more accurate. First, notice that when pieces touch each other to form a connected wall, they lose one adjacency (unless they touch on their corners). This means that each non-corner touching gives a -2 to adjacency score. Also, we should assume that there is a square on the outside rim, so that is another -1. Also Also, the last sections of the path require 3 adjacent walls, instead of 2. So that takes 1 more adjacency score. We are left with an equation that looks like: (12 * b + n*4 - 2 * (b-c) - 1 - 1 )/2. where c is how many corner connections are between the blocks. Note that c
😮
🤓👆
@@nicholasstanton9575 wtf man
The P piece uniquely does not have the same adjacency score as the rest of the pieces, at 10 instead of 12, making this slightly incorrect, and the upper bound could be reduced further.
@@Aimlesswaves. since you are not forced to use the p piece, opting not to will give the current upper bound. you could have two upper bounds, one for if you use the p piece and one for not using it, although this doesn't seem too useful since you aim to get the longest path possible.
As someone who (casually) plays Pentris, this is a fun analysis! Pentominos are definitely a lot more complex and interesting than their 4 tiled counterparts, and it really spices up the amount of thought you have to put in (especially during the faster phases when you have to make split second moves with awkward shaped pieces). Highest score I've ever gotten has only been a bit over a couple thousand points, so this could definitely be improved with a good bit of theory.
Yo aren't you the channel(besides me) that comments on all those Matt Rose videos?
@@BinglesP I comment on a lot of videos yeah but for some reason people know me from Matt Rose (probably because I watched him way before he blew up)
WAIT ITS REAL?
The X pentomino tiling is exactly how I build my sugarcane farms in Minecraft!
Same!
I'd imagine Pentominos working well for Minecraft building in general
not surprising, considering each X is a water block with cane on each side! the fact that this tiles the plane exactly with no gaps corresponds with the fact that this is the maximum amount of cane that can exist in that space for that amount of water, which is of course true since each water block cannot possibly connect to more than 4 sand/dirt blocks! neat!
And now you know that a trivial game design choice created a scenario that due to the underlying and often ignored structure of mathematics has exactly one solution. Isn't that crazy? The plane can be tiled so many ways by so many monotiles, but due to the inherent geometry of a grid and this gameplay restriction there's exactly one way to build such a farm, and everyone must eventually stumble upon it. If the shape was any different there could be infinite farm designs, but this one provides no room for personal choice without sacrificing efficiency.
No!
THE X PENTOMINO TILING CAN’T
I'm going to try and create a program for your pentomino game! I'll report back to explain how it goes.
good luck
Good luck
Good_er_ luck
Waiting response
Good luck
The P Pentomino has "LOTS" Of Tetronimos in it.
wait wtf this only has 500 views?? this video was great it feels like the type of video that should have like a million or something!! Keep up the great work !!
my thoughts as well
Not anymore
fr
Yay, the video's up to 10k views - but I bet it'll get way more than that lol
-Paintspot Infez
Wasabi!
its 83k now
So lucky to find this video. Throughly intersting for its entire runtime with nice editing and good sound quality. Reminds me of Kuvina Saydaki, but higher energy. Subscribed in a heartbeat.
thank you very much!! kuvina was one of the inspirations for this video- I'm a massive fan of their sorting algorithm explaining video so I'm happy to be compared to them lmao
I love watching people gush about special interests.
Same!!!
me too
very swag pfp :3
@@sweetmesaJS Thanks! I'm not really sure what yours is but it's cool!
hey you're that one celeste person
the work done to reach the conclusion you did at 3:06 did not go unnoticed by me!
The color theory for the tetronimos combinging into the pentominos goes crazy, peak.
By the end of this video, I really felt like this group of shapes were my old friends. I knew so much about them.
17:47 For board sizes n=42 and above, as soon as you’ve found the perfect solution for one board you’ve found the perfect solution for all of them because the game becomes a matter of “hiding” the two ends of the path. The optimal solution will just keep growing a bigger empty middle section the higher n gets. Why 42? Because 41 is the sum of the lengths of the “long sides” of every pentomino, the longest length a shape built from one of each can reach. 42 makes it impossible to block off the middle area.
this was looked in to, and that play only works from 42x42 up to around 84x84, when it's no longer the longest distance
@@YeaCloth I’m curious to know what happens when the size is doubled that makes the strategy change
@@jakobr_ basically, once the grid has doubled, the longest distancethe line can travel is no longer the adjacent corner, but instead the opposite diagonal. Because of this, its meaningless to make the stacked line anymore, and instead it's best to make dense mazes in either one corner or opposite corners
@@YeaCloth I don’t understand why the longest distance wouldn’t be to the opposite corner (or inside structures near opposite corners) from sizes 42-84
Your strategy only works for 16 board sizes, at 58x58, it is more optimal to create a line that leads directly to a corner, unlike 42x42.
The very video that got me into a rabbit hole about pentominoes and eventually let me make my own game:
link?
@@YeaCloth still making it so none yet
link?
I've gone down a rabbit hole of interesting underrated math related youtubers and I'm all for it.
this channel gives off carykh vibes
agreed
Yeah
even the voice is similar!
Definitely!
CARY MENTIONED 🎉🎉🎉🎉
7:00 this makes me want to make a factory game where you merge ominos
Yes omg that’s so good what
That pentomino game sounds like something that would be on an old school website and have a leaderboard on it where the people with the highest scores are shown.
For some reason my friend group has colloquially named the F pentomino the 'Seahorse'
I quite enjoyed this. The letter names you gave are in the Wikipedia article. I like the alternative naming they give by John Conway which uses the consecutive letters O through Z, though assigning O to the long straight pentomino is a bit dodgy. Other than that, my main math objection is the use of "regular symmetry" which is better known as "reflectional symmetry". Not that I know much, I'm an engineer.
i feel like this was one of the most pointless videos ive ever watched and simultaneously one of the most insightful and interesting and one of my personal favorites
0:39 Canadian spotted
What I was thinking
😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂😅😂
This feels like a Wikipedia page if it was audio, but it’s very relaxing to listen to
this is exactly what i need as source material for my Sokoban x Game Of Life mashup
Got excited to see a pentomino video on my feed out of nowhere! Gave me a bit of a throwback to a book series from Blue Balliet that got me interested in them. A character in that series used them for all sorts of things, a couple of which were mentioned here (I think rectangles of pentominoes were brought up a few times) and a few more unconventional things. Actually, OP, as an artist you might enjoy those books, as they're art-themed mysteries and treat the pentominoes as halfway a math object and halfway an art object, kind of like you did here.
Immediately subscribed, I saw someone else mention carykh and I got those vibes as well. I hope you’re going places, this was a great video
you made me emotionally attached to groups of 5 cells
(the W and X ones are my favourite :) )
P and X are my favourites
Those are my favorites, too! W is also very good.
Good luck playing pentris…
seeing the x pentomino be such an anomaly gives me insurmountable amounts of joy
yooo thank you for hosting your survivor speeds back when you did, glad to see this blew up!
I was never so GLUED to the monitor ever before, totally enchanting, i love pentominoes and the video about them
When I was a kid, we did a Halloween school project where we had to make mixed media presentations on a mystery book. I found a book called Finding Vermeer about kids solving an art theft, and one of the kids was obsessed with pentominos, using them as a metaphor for conceots throughout the book. I made a fake newspaper clipping decorated with cardboard pentomimos, and ever since then I've been thinking about pentominos (and also Johannes Vermeer lol) at every possibly relevant time. I feel seen
I'm pretty sure the reason that grouping kept appearing was because of the underlying graph structure, ILNVUWZ are all lines, and FTY all have a central square with two branches of length 1 and one of length 2
3:20 one of the "rows" is misalligned creating a gap on the left
Where
@N-U-T-Ton the left side of the screen where the like middle shade of purple meets the lighter purple. You can see a bit of green. Also on the left side of the furthest left white tile is another.
Now we need pentris for pentominoes to match tetris for tetrominoes
Not only are there several of those games already, but technically it would be called Pentis since the R in Tetris is from the "Tetra" part of the name.
Someone made Pentatris in an app called SilentWorks Game Creator. You're welcome
@@mateuszszulecki5206 Yeah but without the R that T is holding back all the dyslexic mistakes a reader could make all by itself.
I don't think T has it in him, do you?
a a a I love this video! Its premise, the time you put into it for the game, everything!
this is such a great video for how little recognizment you get, expect to see me on your next video :D
I wish you knew how happy it makes me to find this video after fixating on this at multiple points in my life. Awesome work bro
This is a great video, with one exception: it's so annoying that there's a lot of text that only pops up for a split second (you have to pause or go frame-by-frame to read it)
5:23 Ah, so that’s why it’s called Tetris. I feel stupid now
i am watching because i have refreshed my recommendation list over 10 times today but this is always at the top so i will watch in the hopes youtube will finally recommend something else.
At 1.30 this rather abstract video got insanely interesting for any minecraft builder
Professor Layton-ass video (I mean this in an entirely positive way I love hearing people talk about their interests)
This video reminds me of a puzzle
Hey your comment just got hyperlinked
important detail on the tetrominoes is that you often see them referred to as being 7, JZ TOILS, but J and Z are mirrors of L and S which are only distinct if mirroring isn't allowed but rotation is (as in the video game Tetris, which is where most people first encounter this subject)
interesting stuff, I'm pretty familliar with pentominoes since I play a lot of tetris variants with them, but it really surprised me to see some of these connections that I've sort of understood intuitvely layed out like this. You're proposed puzzle is also really damn interesting, played around a bit myself but it's a lot more initmidating than it first seems, do wish to see someone take a brute force computer though, if only for the satisfaction of knowing.
This is probably one of the most niche videos I have watched on TH-cam, and I am all here for it. Great video!
Don't think I didn't see that "AMONGUS" for exactly 1 frame.
Now I'm curious if a proportional amogus shape can be made using only the ominos from every n-omino set
Among Us didn’t die, it became a part of us
11:34
I absolutely love hyperfixating on certain topics most people would not be interested in. All these facts are so cool!!
7:15 L, O, T and S. That spells Lots!
Thank you for creating this video, it was fun to watch you share your passion, and I learned a few things too!
Lol those two specific frames within 11:34
GOT IT 11:34
AMOUNG US
For a while in high school i was obsessed with fitting pentominoes (specifically including their mirrors too) into large rectangles. After succeeding a couple times, I eventually tried the same with hexominoes and heptonimoes but to no avail, they’re so much harder to work with the further you increase cell count
Huh, that pentomino path packing puzzle is pretty interesting, wouldn't be surprised if you could get it into a recreational maths journal with a bit of write up.
these things were invented to fuck me up in puyo tetris
4:12 For all you stardew valley fans out there
❤
This was so helpful for my Pentomino exam. Thank you so much.
0:16 is that the conway ship?
This was a very well-written and well-presented video! I definitely learned alot, but I have no idea how useful it'll be lol
The mathematical properties of these objects are actually surprisingly interesting, thank you for sharing all these fun facts!
They remind me of symmetry groups, especially at the start of this video of course. What i mean by groups is that field of math with algebraic symetries or whatever it's called, like the ever-mysterious monster-group (which is like mathematical cosmic horror imho, the smaller groups tend to be more comprehensible to mere 3d mortals 😅).
in chemistry we also use symmetry groups because they are related to the ways in which materials interact with light
Your channel is very underrated! This was a great, well-edited video that I found very interesting and well-written. You deserve at least a couple thousand subs. I'm gonna help get you closer :)
0:25 THATS A C
It’s U
Mainly since C and Z gets confusing audibly
But also, a c is just a U rotated
@@alyx8815 their is no u
@@phungnguyen5249 there is….
im sorry for your challanges in life, i hope it becomes easier
Fun fact: It is impossible to fill a 5×4 or 10×2 rectangle with the five tetrominoes. The proof is left as an exercise for the reader.
5:31 TETRIS
Fascinating video! I really liked this format of just sharing what you have discovered about a specific topic. Pentominoes are incredibly interesting, and this scratched that itch I've had for a few years now.
1:32 that is glowstone
"Here's a deep dive into a weirdly specific topic that I have a possibly unhealthy obsession with" is one of my favourite TH-cam genres.
11:34 among us :D
mogus
ogus
Gus
Us
S
PENTRIS, the game we're never going to get.
15:40 a properly placed U-pentomimo creates a 30-path
where? this arrangement doesn't work, if that's what you're thinking of vv o=path ==wall _=empty
o o o o o o _
o = = = = o _
o = o o = o _
o = o = = o _
o = o o o o _
o o = = = = =
_ o o o o o o
The path could cut around the U, making this solution invalid
@@saintbrownthetrojan It would be 28
yay new math rabbit hole!
Wvtp Nfxzu after reaching 0:32: (⁄ ⁄•⁄ω⁄•⁄ ⁄)
This joke sucks, i loved it, keep up the good work
the visual explanations/infographics are perfect and you did a great job with them. i also cannot imagine being able to say all of the things you said without needing like 46 takes for each so thats also impressive.
my autism brought me here
so did op's i think
T-spins ‘bout to go crazy
I can’t believe that you called the R-pentomino the “F-pentomino”… (wait… Conway calls an R-pentomino)
Anyway, starting from the R-pentomino, one can do an RF28B to a B-heptomino, then BFx59H to a Herschel, then HL95P into a pi-heptomino (releasing a glider), then PF35W into a wing, then WFx46H into a Herschel again, and finally HRx65R back into an R-pentomino (releasing another glider)
I've only heard it been called F before
@@dmcdouga07 I guess I’m a member of the ConwayLife community, and everyone there calls it the “R-pentomino” since that’s what Conway calls it
In fact, Conway called the I, L, F, and N pentominos “O”, “Q”, “R”, and “S”, so that the pentominos would be OPQRSTUVWXYZ
Personally it looks more like a lowercase f to me, doesn't really look like an R at least in the font of YT comments, so the label of " F-pentomino" makes more sense to me.
Also I haven't ever played conway's game of life but I have heard of it and know a game inspired by it called "Cell Machine" (yes that Cell Machine that was made by Sam Hogan)
dude i don't know how you came up with this idea for a video but it's so cool, i never expected 18 minutes of *pentominoes* could be so engaging
The pentomino game you came up with already kinda exists. It's called veggie quest and it's a really good puzzle game, highly recommend
I watched this while recovering from an overdose, thank you
Ngl the U and P pentominoes are bottoms
Dude, this is genuinely so good, you 100% deserve way more attention. Your passion for the topic really comes through and I love how easy it was to follow your logic through the different avenues. The graphics you used really helped make the video comprehensible. If I could make one suggestion, your music choices felt kinda inconsistent in vibe and were a bit loud at times. That being said, if you keep creating videos of this quality you will blow up, I'm sure. Amazing work!
we've found the final boss of autism
nah that would probably be Yenji Jem
I’m sure you have heard this a lot by now, but the tiles that when flipped are different are known as chiral. Saying chiral gets across your point about parity in an easier way. This means there can be chiral and or achiral tile patterns for the shapes.
meow :3
i think N is my favorite pentomino, it just flows like a river. thx for the awesome facts, these are some iconic shapes to me
Here's one more potentially interesting thing:
Take any one pentomino; can you use nine of the other pentominoes to make an extra large version of that pentomino, where one square of the pentomino is comprised of nine squares in the large pentomino?
12 year old me would have gone CRAZY over this lol.
This video's lots of fun. That puzzle you showed off at the end is very interesting; I'd like to see someone tackle that in more technical depth.