Which DICE beat the others? Nobody knows.

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...

[@Diriector_Doc]

10:30 "There's this region between 3 and 5 known as 4" The way you said it is just hilarious to me.

[@ibrahimali3192]

also 11:09

[@SixtyStone]

theres this thing know as youtube

[@tylerrichter6760]

I thought the region between 3 and 5 was a restraining order

[@Arnikaaa]

“There’s this nice and convenient region between 1 and 3 known as 2”

[@matheweon]

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

[@carykh]

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?

[@mjohnson2807]

@@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?

[@dsgowo]

@@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.

[@dsgowo]

@@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.

[@tk8364]

@@mjohnson2807 you need to rotate one of them 90 degrees to show the range of match-ups.

[@Greeny109]

Its actually interesting how much math you can put into statistically getting a win with die that aren't 1-6

[@pieguy5692]

Finally, a mathematical way to choose the best character in Mario party!

[@razkrispies]

oh my gosh yes

[@TunaBear64]

Is not about who moves further, but where you move. Shy Guy will nearly always roll a 4, that's very handy.

[@segadoeswhatnintendont]

It's either shy guy for consistency or Bowser for average roll

[@adamantii]

I'm glad someone else thought about this

[@CMan185]

@@segadoeswhatnintendontand wario for schmoovement

[@matthiaskontsevitch4553]

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?

[@mathguy37]

Yeah it’d be very interesting to decide things

[@Egglet-st3ox]

Maybe "Rock, Paper, Scissors!" could be the name

[@Egglet-st3ox]

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."

[@matthiaskontsevitch4553]

@@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?

[@galacticfish6380]

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

[@carykh]

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)

[@technospyform1578]

​@@carykh time to bring out four then, cause that means it's 4D time... shit pun

[@wrongwayscout]

@@carykh it does portray it, just that they cross over multiple times instead of just once.

[@Johnny-tw5pr]

@@carykhnothing stops you from rotating the green or purple plane

[@Medlek]

​@@technospyform1578Was super funny to me actually, dw, have your like

[@stephfassel6967]

0:31 ”Obviously yellow Cary because that motherf###er rigged this game again for me” That got me dying of laughter

[@ChiEsp619]

MATHEMATICALLY SPEAKING!!! Mathematically speaking okay I fcking died of laugh

[@m3morizes]

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.

[@ncedwards1234]

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.

[@cogwheel42]

Yeah 1 minute in and I went searching the comments to make sure gerrymandering came up :)

[@MeesterTweester]

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.

[@carykh]

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

[@TunaBear64]

Exactly, Pokémon is also a clear example of non-transitive matchups, the starters being the most obvious example.

[@rokkraljkolesa9317]

@@TunaBear64 doesn't Porygon have an extremely niche use in gen 1 competitive due to being a counter to a specific pokemon?

[@Ciurk]

a lot of games are very rock paper scissors i notice

[@Roverdrive_X]

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).

[@agreenkirby201]

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

[@greencreeper9144]

Second Strongest Mole!!! I find the actual blood on their fingers scary

[@pengusplayz5756]

3rd strongest mole moment

[@skld-xm]

was not expecting EarthBound in this comment section.

[@adamboyd1132]

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.

[@EvelynFTTE]

Same tbh

[@wrob08]

Swords beat axes, axes beat lances, lances beat swords.

[@burneyarts]

Genji beats bastion, winston beats genji, bastion beats winston. At least that’s the way it was like 4 years ago lol

[@NonJohns]

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

[@Fourtle]

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

[@bgdy-pw8xj]

True I just watched it

[@hy7864]

carykh short for cary krashes hard

[@Squares2]

@@hy7864big brain

[@Gamer23car]

idk why but i just like numbers with percentages and analytics like this so these videos are always fun :D

[@razkrispies]

yeah, cary is part of the reason i can be in math class without dying so thanks cary 👍

[@zache001]

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

[@MO1M_Official]

​@SonicDavid3 Same

[@The-Mikester]

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

[@DS-tv2fi]

Bold prediction. Let’s see if you’re correct.

[@The-Mikester]

@@DS-tv2fi everything in the world is made up of at least some of em 🤓

[@blue-cuboid]

12 edges, 8 corners

[@The-Mikester]

@@blue-cuboid circle or triangle

[@MichaelDarrow-tr1mn]

@nelsonojserkis219 dice is also singular

[@elijahbedinger1222]

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!

[@darrenwu3507]

0:01 "CARY Krashes Hard here!"

[@alphabetlorefan436]

Kikoriki fans: KRASH!!!???

[@darrenwu3507]

@@alphabetlorefan436 Krash with a K, not a C.

[@darrenwu3507]

@@alphabetlorefan436 Another KH Joke!

[@alphabetlorefan436]

Hey @@darrenwu3507, i already know it's a kh joke, i just thought about it and posted it.

[@CraigTabb-vc8tl]

​@@darrenwu3507It was a k all along

[@justjames4]

I hadn't thought about dice in this way. It's cool how math can be used like that.

[@ghoestrhoest376]

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

[@r3dp9]

Very good morale at the end there. It's REALLY easy to fall into the trap of assuming that everything is transitive. We treat most things in life as transitive, because it's easier to think about things that way. IQ scores, test scores, income, net worth, and subscriber counts are treated as transitive by society. But when you look a little deeper, we find that there are many different kinds of intelligence that excel in different ways, earning money is a different skill than spending money efficiently, and the size of a fanbase is usually inversely proportional to the social experience it offers.

[@rowan404]

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.

[@nicknoteln]

Man, cary's math videos are so fun!

[@skydog0_038]

I agree with this comment.

[@themixedmaster]

Great visualization. This week has been great for dice-related content. Numberphile also just released a new dice video.

[@South-A-Cola62]

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!

[@RealRaizo]

"Luckily for us, there's this region between 3 and 5 known as 4." Thank you Cary, we wouldn't have known otherwise.

[@er4795]

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

[@ColinTimmins]

I really liked how visual and well thought your video was. Fantastic work.

[@tailseatingnails]

his videos are more educational than what i lear at school

[@ChephCakePop]

we need to see pink cary more

[@owinz1]

They're the third twin

[@EPMTUNES]

Nice video. Glad to see such a classic internet math paradox animated like this for the first time

[@aaronconner1440]

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*

[@chaosround]

this is a great proof of why different tools/methods are better for different problems- things are often not one dimensional in complexity

[@Mintthemainecoon]

Should I tell my math teacher about this?

[@javokhir6174]

Lovely video! I think intransitive die got brought up on a 109 pset. Have been interested by them ever since. Thank you for this!!!

[@michaelkindt3288]

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.

[@ballerwithout_theball]

“obviously yellow Cary because that mother-🤬 rigged this game agaisnt me 😢” “MATHEMATICALLY SPEAKING MATHEMATICALLY SPEAKING OK-“

Recommending a 13 year old video, very nice. I actually watched that just a few months ago! KZbin tends to only promote recent stuff, even though older stuff can be just as interesting. In mathematics, it is very common to come up with something, only to find that someone already did it centuries ago. :D

[@Gamingdogmochi234]

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)

[@spektr4625]

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

[@natnew32]

@@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.

[@Karmanthegamer]

I love how when you represented red you did but you’re the bloon instead of btd6

[@saljaf79]

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.

[@lnx0007]

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!

[@yeshejksistn]

I am watching this at 12 am and my brain cannot understand but I love it

[@StaniKaz]

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

[@TheOne_6]

'Wait, they're all threes?' 'Always has been.' 3:42

[@JSG4361]

holy crap carykh video !!!! so excite

[@BlazeMakesGames]

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.

[@er4795]

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

[@theofficialwoohoogamers]

purple beats blue, meaning purple is the best.

[@MichaelDarrow-tr1mn]

@@er4795 4 of the same dice doesn't have this property

[@TheSheep1]

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

[@dracoteu2526]

this is so awesome i love when you make videos like this. like the palindrome one or the river crossing

[@SacsachCCABP]

So, it's rock-paper-scissors, but with chance.

[@olbluelips]

Good video! Fighting games are also non-transitive which is part of what makes them so fun

[@goombaguy4859]

The ultimate Mario Party strategy video

[@DanielRossellSolanes]

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

[@flynnfoil]

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.

[@LiftedStarfish]

I think that, because you can mirror it and it looks the same, it's _not_ chiral.

[@QwertyQwerty-dg5yn]

I saw this at 26s…

[@netzaltali]

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.

[@Nakagolas]

"I don't get it!" > Hey wanna play rock paper scissors? "... Okay I get it."

[@nicreven]

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

[@sillyswrdd]

YAAAY new Cary video!!! Yippie!!! Math is fun and there’s joy in the world

[@godowskygodowsky1155]

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.

[@Snaps12345]

This is kind of the core of strategy game design. The purple die being over kill when it wins reminded me of using high single target damage against many small enemies. You SUPER kill the ones you hit but you hit barely any of them.

[@tanchienhao]

Thanks for sharing your intuition!!

[@ivorynen]

I have no idea what you're even saying but I love your videos in this format

[@Grafyte]

This like the pokemon equivalent of BUG, GRASS, WATER, FIRE. Bug beats grass, grass beats water, water beats fire, fire beats bug.

[@gyroelongatedpentagonalbip728]

10:30 "Now luckily for us, there's this region between three and five, known as four" Idk why this sounds so hilarious

[@derektaylor3726]

"There's this region between three and five, known as FOUR" yes.

[@Bbbbbnc]

When you’re a kid you watch BFDI when you’re older you watch Cary explain mathematics

[@samieb4712]

that processing sketch blew me away. nice job

[@po1nt1776]

This is a dream state for multiplayer game design balance.

[@Luigicat11]

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?

[@egon3705]

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

[@TomatoSaucey95]

real fun to the see the graphs and learn about non-transitive dice, real cool!

[@stoopidapples1596]

The "wasted potential" thing to me feels like gerrymandering lol

[@superultrathanksmom3845]

could maybe be a funny basis to create a game with this? idk i dont want to do it...

[@doctorrodman3872]

Fun result that pops out: Prove that if we have a non-transitive set of n-sided dice, then n is even.

[@booiscaredyou.]

6:30 me neither cary, me neither anyways this has taught me so much more than my actual math classes so like

[@Invalid-user13k]

So many different ways to beat another color

[@SpaceCatAtelier]

0:33 cary swears again but this time censors it

[@pravall985]

if you think about it, this is a really overcomplicated mario party guide.

[@quantuminfinity4260]

Now that is fascinating!

[@mikedoesthings2134]

This would actually be a really fun game, maybe like a board game sort of thing

[@rym36]

how so? also nice henry tophat leader pfp lmao

[@mikedoesthings2134]

@@rym36 like if you could choose your own dice out of like a pool of dices with different combinations of numbers, then go against someone with a different dice maybe an online game would suit it better with the idea I have, also thanks

[@metallsnubben]

There's some sauce there for sure. In its base form you've got "fuzzy rock paper scissors" (just cause you "picked correctly" doesn't mean you win) but you could picture something with drafting dice, or even swappable faces tech like Dice Forge Maybe have different types of "faceoffs" like sometimes it's single dice matched up, sometimes it's the sum of a group etc.

[@TheGraphicalHarmonicist]

Try Mario party where each player has a custom die.

[@rym36]

@mikedoesthings2134 @metallsnubben @naxorissthanksfortheshouto8981 these are all pretty good ideas! im imagining an online game where you see your opponent dice, and have to pick a dice (out of a few options) that is most likely to beat it in little time. or maybe you have a specific number of "points" that you can spend to make your own dice. for example a dice of all 6's will be 36 points. so you would need to budget your points to be able to win. sides with no points put on them will default to 0

[@aglegator5115]

Plot twist: Cary actually rolls the dice and inspects who wins!

[@jerri1918]

cary upload spotted. i can breathe again

[@MrTopLLOfficial]

Nah bro is a better teacher then all my teachers combined

[@maxneevo]

Awesome video, glad your'e back making videos!

[@triPocoPi9576]

This visualization was incredibly helpful

[@binarycat1237]

"how could we construct these dice" me, a programmer: just check all (4*6)^6 possibilities until you find one that works!

[@Cocoawaffle]

Cary: *a few years go* 2016 being 8 years ago in less than a week:

[@jorgerangel2390]

Dude, this video is fire! thanks!

[@crafty453]

The carykh acronyms are still going.

[@name6953]

I never knew I'd understand a Carykh video...

[@gettinggamer5297]

instead of thinking of a boxing scenario, i thought of just rock paper scissors

[@deskseven]

am i overanalysing this or can i smell ewow easter eggs in this video somewhere

[@jacqueseven]

Omg same like there has to be something here

[@jamieelton]

Classic Carykh! Glad to see a new video 🎉

[@Thesnakerox]

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

[@irasac1]

I found it interesting that you choose the worst sport for the example, because boxing, MMA or any combat sport really works just like these dice, when two people face each other it's much like rock paper and scissors, a tall lanky boxer may beat an average sized all rounder, but lose to an short and stocky wrestler, who will then lose to the average size all rounder, because styles make fights. I think an sport such as Athletics would better fit as an example.

[@billiam6398]

a less directly adversarial sport like swimming or track would probably be best.

[@geeteevee7667]

5:30 I love this visualisation!

[@jeffandmark5373]

12:00 no, A is better than C. I'm not a statistician, but the thing with "beats" in the case of the dice is that there is no numerical "skill level", as mentioned earlier in the video. However, the drugs have a very clear numerical skill level: number of people cured.

[@asharpnotbflat]

Math is a beautiful thing. Thank you for introducing me to these awesome dice!

[@mysticartist308]

“I like to think of the red guys as the simple-minded brute of the pack.” *shows mario* Ah yes, Super Mario, the simple-minded brute of the pack.

[@nnnnick]

"Luckily for us there's this region between 3 and 5 known as '4' " 🤣

[@EmberNyxtical]

cant believe i havent decided to watch this until now math ftw i love this

[@zombieturtle17]

Only Cary can make math fun

[@MaxNovOfficial]

I'm at the point where carykh teaches me math more than my school does