Which DICE beat the others? Nobody knows.
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- Опубликовано: 26 сен 2024
- i'm flabbergasted. They are non-transitive dice!
Tadashi Tokieda's Numberphile video on non-transitive dice: • The Most Powerful Dice...
singingbanana's video on Simpson's Paradox: • Maths: Simpson's Paradox
GitHub repo with this visualization's source code: github.com/car...
10:30
"There's this region between 3 and 5 known as 4"
The way you said it is just hilarious to me.
also 11:09
theres this thing know as youtube
I thought the region between 3 and 5 was a restraining order
“There’s this nice and convenient region between 1 and 3 known as 2”
the problem i have with this set of intransitive dice is that purple beats green 5/9 times, so against a random die, purple is the best and green is the worst
Oh yeah, that is true that the purple die beats the green one 5/9 times! I'm not sure if there's a clean way to make all "opposite dice" come out exactly equal unless you give them a lot more sides.
Wait! I just got a *weird* solution. If all the dice are D12s (dodecahedrons), you could have:
Red: 12 threes
Blue: 6 fives, 6 ones
Green: 8 fours, 4 zeros
(these three are essentially the same)
Purple: 9 twos, 3 sixes.
(purple now has proportionally more twos.)
In this instance, purple and green are exactly tied against each other. (In order for green to win, green must roll a four (8/12) and purple must roll a two (9/12), which multiply to 1/2). Red and blue are also tied. However, the odds of blue-purple and purple-red are changed - so you win some, you lose some?
@@carykh Wait @8:25 it shows purple vs green being 2/3. But mathematically it isn't that? Is that a issue with arranging the order of the numbers in the chart?
@@carykh I think I've just proved that a perfect set of 4 non-transitive dice with the numbers we have (0-4, 1-5, 2-6, 3-3) isn't possible with any number of faces, but I'm going to double check.
@@carykh Yup, here's the proof:
The probability of green beating red is just the probability of green rolling a 4, which we'll call p.
The probability of red beating purple is just the probability of purple rolling a 2, which must also be p if the set is perfect.
I'll abbreviate these as P(g4) = p and P(p2) = p. Then, P(g0) and P(p6) must be 1-p.
The probability of purple beating green is P(p6) + P(p2)*P(g0), and we want this to equal 1/2.
Plugging in, we get the equation: 1-p + p(1-p) = 1/2, which becomes 1 - p^2 = 1/2.
This means that p must be 1/sqrt2. But this is a problem because p is now irrational, so no die with finite sides can have probability p of rolling something.
Therefore, no perfect set of 4 dice with these numbers is possible, regardless of the number of faces.
@@mjohnson2807 you need to rotate one of them 90 degrees to show the range of match-ups.
Its actually interesting how much math you can put into statistically getting a win with die that aren't 1-6
Finally, a mathematical way to choose the best character in Mario party!
oh my gosh yes
Is not about who moves further, but where you move.
Shy Guy will nearly always roll a 4, that's very handy.
It's either shy guy for consistency or Bowser for average roll
I'm glad someone else thought about this
@@segadoeswhatnintendontand wario for schmoovement
To make this non-transitive property more widely known, we should come up with a simple game (maybe less dice, and things instead of dice), like a rock cuts through paper, scissors chisel a rock, and paper envelops scissors?
Yeah it’d be very interesting to decide things
Maybe "Rock, Paper, Scissors!" could be the name
Or maybe "Grey ball of stone known as a boulder which can get defeated by a sheet of paper, that same sheet of paper that was once a tree and can be halfed by the item known as scissors as well as the scissors, which can be destroyed by the grey ball of stone."
@@Egglet-st3ox ex-tree vs. big rock the size of small rock vs. Ctrl+C is a strong contender for a decision-making method. It's up against the infamous "coin toss".
Which one will be the winner?
Should we decide usiing the first one or the second one?
6:20 The second you explained this, I thought about gerrymandering. Creating matchups where one side is having its "potential" deliberately wasted to skew the results of what would have otherwise been an overall more proportional result.
Non-transitive dice and voting is indeed the math behind gerrymandering. It also explains why swing states are emphasized so much in presidential elections, because a marginal win is still the same as a landslide victory in 48 states + DC (because electoral college). So if a politician is limited in the people they can sway with their campaign, they want to target people who are
1) On the fence (could vote either way, but you get one vote rather than 60% vote for one candidate and 40% for another).
2) In a swing state that can change the electoral college vote (because >50% = 100%).
It's like an avalanche where you could be the changed vote that changes your state, therefore the president, BUT it's planned out so that you're less likely to be in this situation unless prior politicians wanted people in your area to have this power. Otherwise it goes like you said, you're moved to an area where your vote cannot make a difference. Or you criminally charge people for bogus crimes like drug possession to disenfranchise groups like the poor and minorities.
Through this the process of politicians weighting votes unevenly, they are indirectly biasing votes for future politicians (such as themselves and their cronies). I sure hope they don't attempt to remain in power against the desire of the average voter's desire. 🤔
Then we can get into propaganda because some of us are swayed to another side with less effort, so we will be targeted more than the stubborn ones who need only the occasional reminder to keep voting the way their parents taught them to (because their grandparents taught their parents how to think).
New customers cost more than maintaining old ones, so allocate resources carefully in a zero-sum game. And remember, lies are cheaper than truths; and winner takes all because the past winners said that's how it should be.
Yeah 1 minute in and I went searching the comments to make sure gerrymandering came up :)
This honestly reminds me of a group of five Earthbound bosses where each believes they're the third strongest because they all won against two of their brothers and lost against the other two. Not sure how exactly the explainations match up, but I've got a feeling
Second Strongest Mole!!! I find the actual blood on their fingers scary
3rd strongest mole moment
was not expecting EarthBound in this comment section.
Talking about "non-transative" reminds me of ranking players or characters in games like Super Smash Bros. Just because A beats B and B beats C, does not necessarily mean A will beat C or is better than C.
That's a really good point, Smash characters could easily be non-transitive! Like, if three characters have three different types of attacks, and they have shields against other types of attacks, it could be like a Rock-Paper-Scissors sort of thing
Exactly, Pokémon is also a clear example of non-transitive matchups, the starters being the most obvious example.
@@TunaBear64 doesn't Porygon have an extremely niche use in gen 1 competitive due to being a counter to a specific pokemon?
a lot of games are very rock paper scissors i notice
Smash Ultimate even has a system like this for Spirits, where they come in three different types: Attack, Shield, and Grab. Shield beats Attack (because you can block attacks), Grab beats Shield (because grabs pass through shields), and Attack beats Grab (because attacks are longer ranged than grabs I suppose).
The beginning of the video is so random and funny
I love that math can sometimes be so complicated and confusing yet always so fun to research around it
True I just watched it
carykh short for cary krashes hard
@@hy7864big brain
Somewhat surprised that this video didn't mention Rock-Paper-Scissors, or the many Strategy games where there are units that can be good counters to another, but weak to a second unit. My favorite version is Archers beat Infantry beat Cavalry beat Archers.
Same tbh
Swords beat axes, axes beat lances, lances beat swords.
Genji beats bastion, winston beats genji, bastion beats winston.
At least that’s the way it was like 4 years ago lol
funny how he brought up boxing too because in smash melee, (as mentioned in the melee documentary) players would beat other players and then it was assumed that it was linear when it was actually a triangle like rps due to different styles of combat
0:01 "CARY Krashes Hard here!"
Kikoriki fans: KRASH!!!???
@@alphabetlorefan436 Krash with a K, not a C.
@@alphabetlorefan436 Another KH Joke!
Hey @@darrenwu3507, i already know it's a kh joke, i just thought about it and posted it.
@@darrenwu3507It was a k all along
0:31 ”Obviously yellow Cary because that motherf###er rigged this game again for me”
That got me dying of laughter
MATHEMATICALLY SPEAKING!!! Mathematically speaking okay
I fcking died of laugh
the dice that wins always has at least 6 sides, 12 corners, and will be flat
edit: i have found out cubes has 8 corners, not 12
Bold prediction. Let’s see if you’re correct.
@@DS-tv2fi everything in the world is made up of at least some of em 🤓
12 edges, 8 corners
@@blue-cuboid circle or triangle
@nelsonojserkis219 dice is also singular
idk why but i just like numbers with percentages and analytics like this so these videos are always fun :D
yeah, cary is part of the reason i can be in math class without dying so thanks cary 👍
It was really cool to see a 3d visual of this non transitive problem! It really helps give an intuitive understanding of why this paradox actually makes sense!
I hadn't thought about dice in this way. It's cool how math can be used like that.
Somehow, the fact that each die’s score adds up to 100% across the two graphs it’s present in reminded me of how Yellow Cary and Orange Cary are each other’s fathers.
yo cary kumon helper this video is fascinating! i didn’t know that colorful dices and plots could be so interesting! i learned quite a bit from this video
@SonicDavid3 Same
Since it wasn't mentioned in the video, I looked at it myself: The purple die has an edge over the green die winning 5/9 (20/36) of the time
That's true! Purple does win against green 5/9 of the time. (My visualization doesn't portray that though because green and purple are both on the X-axis, oh no)
@@carykh
time to bring out four then, cause that means it's 4D time...
shit pun
@@carykh it does portray it, just that they cross over multiple times instead of just once.
@@carykhnothing stops you from rotating the green or purple plane
@@technospyform1578Was super funny to me actually, dw, have your like
When I saw these dice, I immediately thought about Super Mario Party on the switch, because every single character in that game has their own unique dice with different values along side the standard di, for example bowser as 0,2,4,7,8,9 and wario has 0,0,0,6,6,7. its really cool to see this and think about the game with this in mind
"Luckily for us, there's this region between 3 and 5 known as 4."
Thank you Cary, we wouldn't have known otherwise.
Cary is the only person somehow able to get me anywhere near interested in math or statistics. Interesting how all this can stem from a 6-sided die!
Man, cary's math videos are so fun!
I agree with this comment.
we need to see pink cary more
They're the third twin
Great visualization. This week has been great for dice-related content. Numberphile also just released a new dice video.
Nice video. Glad to see such a classic internet math paradox animated like this for the first time
his videos are more educational than what i lear at school
i love this video!!
in highschool i had to write a math essay on a topic of my choice and i originally chose this topic. i used colored matrices as well, but did not think to use 3 dimensions which was genius! instead i created a new 36-sided "die" by subtracting one result from the other, but this is about where i changed topics to just "addition of discrete random variables" because so little is known about these.
thanks so much for making an awesome video about this! if i had one suggestion it would be to render the dice results as thick rectangular boxes that do not connect so that you could see the stacking without rotating. or add some transparency
I find it interesting that in the final one if you compare all 4, the blue and purple die win 12 times each, and the other two win 6 times each. Which makes me curious if you can create a set of dice that has this property but if you roll them all simultaneously, they have an equal chance of winning. 36 is divisible by 4, tho something about each die winning 9 times out of 36 feels off. So I imagine it might be easier to pull off with 3 dice each winning 12/36 times or 6 dice winning 6 out of 36 times.
not the answer youre looking for but the easiest way to accomplish this is to use 4 (or however many you want) of the same dice
purple beats blue, meaning purple is the best.
@@er4795 4 of the same dice doesn't have this property
Something I noticed is that the purple d[y]e beats the green d[y]e, but blue and red are evenly matched. Just thought that was interesting.
this is a great proof of why different tools/methods are better for different problems- things are often not one dimensional in complexity
I love how when you represented red you did but you’re the bloon instead of btd6
I really liked how visual and well thought your video was. Fantastic work.
0:30 Who’s more likely to get the higher score?
Pink Cary: *OBVIOUSLY YELLOW CARY, BECAUSE THAT MOTHER [bleep] ER RIGGED THIS GAME AGAINST ME*
'Wait, they're all threes?'
'Always has been.'
3:42
The ultimate Mario Party strategy video
I am watching this at 12 am and my brain cannot understand but I love it
“obviously yellow Cary because that mother-🤬 rigged this game agaisnt me 😢”
“MATHEMATICALLY SPEAKING MATHEMATICALLY SPEAKING OK-“
8:17 Right here you see blue and purple are tied covering 12 squares each, the other 2 covering 6. So purple and blue beats the rest. But purple beats blue so purple beats the rest.
When you’re a kid you watch BFDI when you’re older you watch Cary explain mathematics
This is like rock paper scissors:
Person A chooses rock
Person B chooses paper
Person C chooses scissors
“Scissors beat paper” (pin in bfdi 1a), paper beats rock and rock beats scissors
I think that, because you can mirror it and it looks the same, it's _not_ chiral.
I saw this at 26s…
Although, if you keep track of the colours, the shape does have Chirality because the shape isnt superimposable with the order of each die intact. If you treat each die as the order of priority then the order. Green, blue, purple, red is different to purple blue green red and thus has chirality.
Now I wonder just how balanced the Super Mario Party character dice blocks are. Which ones beat which other ones, and is there one that we could consider the objective "best" because it beats the most of the other dice?
notably that's a lot more complicated because 0s can be good due to reactivating a space it would normally take an entire trip around the board to return to, and then also coins have to be factored in
"There's this region between three and five, known as FOUR"
yes.
Should I tell my math teacher about this?
Only Cary can make math fun
why is this so interesting
even though i don't even have a big interest in stuff like this
I'm not using this for like reports or anything
but i love it
"I don't get it!"
> Hey wanna play rock paper scissors?
"... Okay I get it."
holy crap carykh video !!!! so excite
The "wasted potential" thing to me feels like gerrymandering lol
could maybe be a funny basis to create a game with this? idk i dont want to do it...
So many different ways to beat another color
that processing sketch blew me away. nice job
Lovely video! I think intransitive die got brought up on a 109 pset. Have been interested by them ever since. Thank you for this!!!
This honestly reminds me of the dice from super mario party, where the dice are all different. I'll bet that the mario party dice are gonna be way more complicated in dice rolls than these 4 dice.
Edit: NEW RECORD OF LIKES: 38.
Edit 2: 41 likes now. I guess this will be hard to beat. (I guess people like super mario party)
I thought the exact same thing. When I played that game and got multiple dice I always calculated the expected value of all the dice in order to determine the "best" one.
I think boo's die was the best, if you ignore the coin penalty for rolling a 0 lol
@@spektr4625 It was Bowser's, actually. His average roll was 4.67; Boo and Wario were 4.00, the Normal Dice + Mario/Luigi/Waluigi/Dry Bones/Diddy Kong were 3.5, most others (Daisy/Pom Pom/DK/Shy Guy/Peach/Koopa/Bowser Jr/Monty Mole) were 3.33, Yoshi was 3.16, Goomba was 3.00 and Rosalina/Hammer Bro were 2.67.
I do a lot with this game.
This like the pokemon equivalent of BUG, GRASS, WATER, FIRE. Bug beats grass, grass beats water, water beats fire, fire beats bug.
0:33 cary swears again but this time censors it
the fun game begins when you have five dice forming two different cycles. for example A beats BC, B beats CD, C beats DE, D beats EA, E beats AB
Thanks for sharing your intuition!!
instead of thinking of a boxing scenario, i thought of just rock paper scissors
Fun result that pops out: Prove that if we have a non-transitive set of n-sided dice, then n is even.
Whenever you see a Cary showing you dice, you'll know you'll get an extra math class
Nah bro is a better teacher then all my teachers combined
am i overanalysing this or can i smell ewow easter eggs in this video somewhere
Omg same like there has to be something here
This is a dream state for multiplayer game design balance.
Now I wonder if it's possible to create a set of 4 or more dice where for each pairing in the loop, the dice have an exactly 50% chance of winning in a roll...
BONUS CHALLENGE: Each pairing has no possible draws
if you think about it, this is a really overcomplicated mario party guide.
I ran simulations of these die and purple repeatably had the highest sum. That seemed strange until I realized that it was not about having a large sum. It is just about winning a particular roll -- while how MUCH you win by is irrelevant.
The carykh acronyms are still going.
10:30 "Now luckily for us, there's this region between three and five, known as four" Idk why this sounds so hilarious
The intuition is that it doesn't matter how much you win by so long as you win. So you can concentrate high scores over a smaller probability while raising the low scores just barely as to beat the previous die.
In one of the newer Mario party games all of the characters have unique dice with different numbers, I'm curious how those would look in a similar plot.
Now I want to make some sort of 4-player battling game where each player gets an unique dice that's utilized in the gameplay somehow.
I'm not sure how well balanced it is, but some modes of Mario party have character specific dice!
@@nstvntt7410 Oh yeah i was thinking about that, but some characters even have dice that can give you coins so it definitely isn't 100% balanced all the time
Plot twist: Cary actually rolls the dice and inspects who wins!
cary talks about cubes for like 12 minutes
I was hoping you'd get into the part where you can construct rock-paper-scissor-like dice such that the relationship reverses when everybody rolls *two* dice
I first learned about these dice as a teen in a great book by Martin Gardner, The Colossal Book Of Mathematics (highly recommended). Your 3D visualization was great!
this is so awesome i love when you make videos like this. like the palindrome one or the river crossing
You mention "valid" matchups. What exactly about the two that go across the middle of the 'circle' makes them invalid, exactly?
This visualization was incredibly helpful
The " **krashes hard** " is looking like kh when you delete the rashes and ard
So basically nobody “beats” everybody? Interesting…
I saw this at 47s…
"Luckily for us there's this region between 3 and 5 known as '4' " 🤣
"how could we construct these dice"
me, a programmer: just check all (4*6)^6 possibilities until you find one that works!
We dont play rock paper scizzors anymore, the intellectuals play red vs green vs blue vs purple.
This is giving me serious mario party flashbacks.
The other dice were beaten by Colonel Mustard in the Billiard Room with the blue die.
"Well, that was random."
Now that is fascinating!
This is just rock, paper scissors with extra steeps
6:30 me neither cary, me neither
anyways this has taught me so much more than my actual math classes so like
I love when cary swears
That one friend on Mario Party taking the most boring character because it has 0.127% better odds
0:52 I just imagined a yellow dot smacking a purple dot with a belt 😭
real fun to the see the graphs and learn about non-transitive dice, real cool!
cary are you okay, you okay, you okay cary?
Good video! Fighting games are also non-transitive which is part of what makes them so fun
Oh so that’s why Super Mario Party has a tier list
It's interesting that even though the whole point is that no die is best... the 3D model sure makes it look, visually, like purple is the best... considering that it and blue are "on top" more often than the other two, and purple beats blue in the head to head.
yeah this is how I feel looking at it hahah
in my defense, I am a bit too simple-minded
The biggest brainfunk isn’t think, it’s “brainfuck”
cary I'm so glad that you exist on the internet in the way you do because you're so sincere in your love of math
The yellow vs purple war continues