The Most Powerful Dice - Numberphile

Поділитися
Вставка
  • Опубліковано 18 вер 2024
  • Tadashi explores a special set of dice... And has a powerful lesson.
    More links & stuff in full description below ↓↓↓
    Subscribe to Numberphile: bit.ly/Numberph...
    More dice videos: bit.ly/Dice_Videos
    Playlist of Tadashi Tokieda videos: bit.ly/tadashi_...
    Want to buy some dice like this - our friends at Maths Gear have some: bit.ly/MathsGea...
    Discuss this video on Brady's subreddit: redd.it/53mhyi
    Support us on Patreon: / numberphile
    NUMBERPHILE
    Website: www.numberphile...
    Numberphile on Facebook: / numberphile
    Numberphile tweets: / numberphile
    Subscribe: bit.ly/Numberph...
    Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumb...
    Videos by Brady Haran
    Brady's videos subreddit: / bradyharan
    Brady's latest videos across all channels: www.bradyharanb...
    Sign up for (occasional) emails: eepurl.com/YdjL9
    Numberphile T-Shirts: teespring.com/...
    Other merchandise: store.dftba.co...

КОМЕНТАРІ • 919

  • @YostPeter
    @YostPeter 8 років тому +489

    "We want to avoid arithmetic."
    -Tadashi Tokieda

    • @dielaughing73
      @dielaughing73 3 роки тому +10

      Wee I hope so, I can't even count to ten gazillion!

    • @MarcTelang
      @MarcTelang 3 роки тому +2

      @@dielaughing73 but it's NUMBER phile

    • @DrKaii
      @DrKaii Рік тому +1

      ​@@MarcTelang he's from arithmephile

  • @jkazos
    @jkazos 8 років тому +117

    "If the other one were careless enough to show its zero." I love this guy's choice of phrasing!

    • @chriswebster24
      @chriswebster24 2 роки тому +2

      Yeah, that would be pretty careless.

  • @ExhaustedElox
    @ExhaustedElox 8 років тому +201

    Once again, Tadashi-sensei explains a complex concept well before ever referring to it so the audience isn't disengaged by complicated sounding terms. He is a great educator.

    • @aurelia8028
      @aurelia8028 Рік тому

      It really isn't that complicated

  • @Cadrid
    @Cadrid 8 років тому +852

    I'm a simple man.
    When I see a Tadashi video I watch it, my mind is blown, and then I hit "Like" in a math-induced stupor.

    • @fossilfighters101
      @fossilfighters101 8 років тому

      +

    • @Klayhamn
      @Klayhamn 5 років тому +5

      I'm a simple man.
      When I see descriptions of what other simple men do, I like, and comment - telling my story as a simple man.

    • @morgiewthelord8648
      @morgiewthelord8648 3 роки тому

      @@Klayhamn its just that simple

    • @Triantalex
      @Triantalex 11 місяців тому

      false.

  • @Bluedragon2513
    @Bluedragon2513 5 років тому +41

    5:04 There it is. The legendary anime, glass-setting middle finger. It appears once every few months to even years.
    I applaud. He is great.

  • @PauLtus_B
    @PauLtus_B 8 років тому +177

    5:04 did you just give us the finger?

    • @SecretofTerra
      @SecretofTerra 8 років тому +95

      No, it was meant specifically for you ;)

    • @PauLtus_B
      @PauLtus_B 8 років тому +2

      XPossseidon
      I know XD

    • @PauLtus_B
      @PauLtus_B 8 років тому +24

      Secreto Terra
      I'd feel honored to get the finger from Tadashi.

    • @SapphireCrook
      @SapphireCrook 8 років тому +3

      As a glasses wearer myself...
      That sometimes happens and I instantly panic if anyone took it personal.

    • @PauLtus_B
      @PauLtus_B 8 років тому +10

      Sapphire Crook
      I'd be surprised if anyone took that as an insult.

  • @robertb3672
    @robertb3672 8 років тому +40

    One of the more profound quotes by an undergraduate professor that taught me a few years ago:
    "The problem with real life is that friendship is not transitive."

    • @sommers6235
      @sommers6235 3 роки тому +5

      I once made an argument in one of my sketchbooks that the closer human relationships were to following equivalence relations, the stronger those relationships are. It's been many years and many ponderings, but I still think there may be some truth to it.

  • @iytbazzie7723
    @iytbazzie7723 5 років тому +53

    Surprisingly, I first learned about this phenomenon when compairing stats for guns in Borderlands 2.

  • @brianpso
    @brianpso 8 років тому +130

    I just can't get enough of his lessons, I need more Brady! Please!

    • @numberphile
      @numberphile  8 років тому +34

      we're working on it

    • @brianpso
      @brianpso 8 років тому

      Derek Leung Thanks a lot! I'm gonna watch it later today =D

    • @tauceti8341
      @tauceti8341 8 років тому +4

      Professor Tadashi has a great general maths lecture with toys that's definitely worth checking out. I had great fun with Tadashis toy lecture.

  • @GameNationRDF
    @GameNationRDF 8 років тому +26

    Yet again Tadashi blows my mind

  • @vicr123
    @vicr123 8 років тому +584

    I don't know why I found that "ten gazillion" thing so funny... :)

    • @minimanimo7239
      @minimanimo7239 8 років тому +2

      Why not m8? :)

    • @spider853
      @spider853 8 років тому +9

      Because of Godzilla? Tadashi.... Godzilla.... Tadashi... Godzilla ;) you see the connection?

    • @litigioussociety4249
      @litigioussociety4249 8 років тому +13

      I want one of those 10 gazillion dice.

    • @vicr123
      @vicr123 8 років тому +7

      Litigious Society Imagine playing Snakes and Ladders with that :P

    • @bytefu
      @bytefu 8 років тому +9

      Probably because 10 is not much compared to gazillion, which is supposed to be some very large number, so it doesn't really make any difference, thus putting it before the gazillion is meaningless and therefore funny, just like shooting a single bacteria from a giant cannon.

  • @12tone
    @12tone 8 років тому +102

    Neat! Would love to see a follow-up with James Grime, who has his own set of these with additional interesting properties.

    • @mzadro7
      @mzadro7 3 роки тому +1

      Well it’s wierd to see you here

    • @_wetwillyinc
      @_wetwillyinc 3 роки тому +11

      @@mzadro7 nah man
      Music theory is just group theory mod12

    • @WilliamWizer
      @WilliamWizer 2 роки тому +2

      to be precise he has a set of five that have 2 different cycles. one of the two gets reversed if you play with two dice instead of one.
      this makes it fun because you can play with two other players and let them choose die so long as they let you decide how many dice will be used on the roll. you are guaranteed to have a stronger chance to win no matter what they choose.
      he made me wonder if it's possible to do it for more players (you plus 3 others). haven't found a solution but it's fun to try.

  • @TheDarkFalcon
    @TheDarkFalcon 8 років тому +64

    Tadashi needs to host a podcast, I could listen to him all day.

    • @lmiddleman
      @lmiddleman 8 років тому +13

      He has a great voice and an excellent command of English... a natural.

    • @fossilfighters101
      @fossilfighters101 8 років тому

      +

    • @bambiwhite6385
      @bambiwhite6385 5 років тому +2

      I prefer that he gets some rest and lives a long life to be around and do his work

  • @hedonisticzen
    @hedonisticzen 5 років тому +11

    I love the way he uses language at times. "Careless enough to show it's 0" also gave me a new perspective on probability! I feel like I gain so much from his videos.

  • @jimmykeffer7401
    @jimmykeffer7401 8 років тому +56

    This needs to be a game where you design die and try to beat your opponents' die.
    More tadashi!

    • @WilliamPeytz
      @WilliamPeytz 8 років тому +24

      666666

    • @Dolkarr
      @Dolkarr 8 років тому +32

      9,9,9,9,0,0 - same total value, but beats yours (6,6,6,6,6,6)
      This really could be an interesting game mechanic, if you distribute a fixed amount of points between a set of values, since it has nice rock-paper-scissor behavior.
      10,10,4,4,4,4 - same value again, beats the above, but loses against yours.
      If we continue...
      10,10,7,7,1,1 - beats all of the above, but...
      8,8,8,8,2,2 - beats that, but not the others except for the 6's.
      Now.. given a fixed value, which distribution of points beats the most other configurations?

    • @Maharani1991
      @Maharani1991 8 років тому +1

      +

    • @jimmykeffer7401
      @jimmykeffer7401 8 років тому +3

      Dolkarr How does 10,10,4,4,4,4 beat
      9,9,9,9,0,0? wouldnt it lose.... nvm it wins 55%
      So let's go with 36 as the number of given points to distribute. What combinations would be the best?
      9,9,10,8,0,0 - beats 6,6,6,6,6,6, and ties 9,9,9,9,0,0, I'm too lazy to do the rest
      11,11,11,2,1,0 - another combination that can beat the 10,10,4,4,4,4
      *metagame intensifies*

    • @meneldal
      @meneldal 8 років тому

      If you allow negative numbers, I propose 1 gazillion 5 times and -5 gazillion once. Wins 5 out of 6 against everyone here.

  • @mspeir
    @mspeir 8 років тому +261

    I knew it! I knew gazillion was a number! A mathematician just said so!

    • @MrAllenCrown
      @MrAllenCrown 8 років тому +14

      Who want's to be a Gazillionaire? I know I do.

    • @MrRedtail26
      @MrRedtail26 8 років тому +3

      finally

    • @fossilfighters101
      @fossilfighters101 8 років тому

      +

    • @lythd
      @lythd 8 років тому +1

      what is that number? how many zeros? ya i thought so

    • @HaloInverse
      @HaloInverse 8 років тому +35

      It doesn't have a fixed number of zeros. A gazillion is defined as "the smallest positive finite power of ten that is effectively infinite for practical purposes", i.e. slightly bigger than anything else you'll be dealing with, but not actually infinite. It's part of the family of numbers known as "Parker constants".

  • @RPMRosie
    @RPMRosie 8 років тому +24

    i honestly love the tadashi videos

  • @NiLowther
    @NiLowther 6 років тому +103

    Spent most of the video thinking "how could these dice be incorporated into D&D..."

    • @minecraftermad
      @minecraftermad 5 років тому

      Easy sand off everything between 5 and 18 from the d20 and let people choose wether to use d20 or d6 for rolls

    • @pedroscoponi4905
      @pedroscoponi4905 3 роки тому +7

      You could build strange magical weapons that use these weird dice instead of the ordinary roll. Maybe a shortsword from Mechanus always rolls the average of 3, for example.
      You could also have the dice be magical objects themselves in-fiction, and rolling certain faces on certain dies would have specific effects. Then you just give the actual dice set out as a prop ;)

  • @venkatchait007
    @venkatchait007 8 років тому +12

    Amazing, you know the teacher is good when the solution seems obvious once explained :O

  • @ZimoNitrome
    @ZimoNitrome 8 років тому +55

    Oh yeah Tadashi be my waifu

    • @Nessa-939
      @Nessa-939 8 років тому

      Hej Simon

    • @ZimoNitrome
      @ZimoNitrome 8 років тому

      Anka Hej Ankan

    • @ichbinein123
      @ichbinein123 8 років тому +2

      ZimoNitrome, du er en af de fugtigste svenskere jeg kender på den anden side at Øresund! Keep it up!

  • @LeviJohansen
    @LeviJohansen 7 років тому +39

    5:1 and 3:3 is equal.

    • @safrprojects
      @safrprojects 3 роки тому +13

      Yet if you compare the other opposites, 662222 is stronger than 444400

    • @natew.7951
      @natew.7951 3 роки тому +5

      @@safrprojects yeah that's what I wish had been mentioned. It's obvious that one of the two comparisons that weren't made in the video (333333 vs 555111) is equal, but the other one isn't as obvious. I had to do the math myself to see if it was equal.

    • @Aggressiphyst
      @Aggressiphyst 3 роки тому +6

      This all needs to be implemented in a board game of some kind.

    • @armcie5080
      @armcie5080 3 роки тому +11

      @@Aggressiphyst Throw away the board.... how about if you somehow made hand symbols to represent the dice. And maybe just use three of them.

    • @a.velderrain8849
      @a.velderrain8849 3 роки тому +1

      @@safrprojects Indeed. 662222 beats out 444400 20/36 times, or 5/9 times, for anyone who was interested and didn't want to do the math.

  • @yungml
    @yungml 8 років тому +39

    When Tadashi says "nifty"

  • @Zince
    @Zince 8 років тому +8

    probably my favorite guy for this channel

    • @subh1
      @subh1 8 років тому +7

      what's the probability?

  • @Hogibaer
    @Hogibaer 8 років тому +149

    I always knew "10 gazillion" was a legit number! :-)

    • @MrDasbinich123
      @MrDasbinich123 8 років тому +11

      yeah that part made me smile ... which hurts because I just got my wisdom teeth removed ._.

    • @johancouder8013
      @johancouder8013 8 років тому +10

      Of course it is ! Economists use it all the time, as in: "The US debt is 10 gazillion dollars" :)

  • @gui1521
    @gui1521 8 років тому +229

    Got the answer, most powerful dice is the one with 10 gazillion on all faces. Ez

    • @fishecod2552
      @fishecod2552 8 років тому +16

      sorry, i think you missed the one with 10 gazillion 1 on it.

    • @gui1521
      @gui1521 8 років тому +9

      fishecod
      Must be quite a huge dice ._.

    • @cam32189
      @cam32189 8 років тому +7

      Still would rather have a 10 gazillion-sided die with 10 gazillion on all faces

    • @lxathu
      @lxathu 8 років тому +4

      Should you try to throw it, you would be about to die.

    • @lythd
      @lythd 8 років тому

      or maybe oo on all faces!

  • @firstnamelastname4752
    @firstnamelastname4752 8 років тому +488

    Gonna like before watching because Tadashi is always the dankest.

    • @pietervanderplas9592
      @pietervanderplas9592 8 років тому +27

      Yeh, his videos are awesome. Love this guy.

    • @nerdcedric
      @nerdcedric 8 років тому +7

      Now I can tell my math teacher that one of the most liked Numberphile mathematicians said "A Gazillion"

    • @zachell1991
      @zachell1991 8 років тому

      I found that ironic when he used 10 gazillion.

    • @jesusthroughmary
      @jesusthroughmary 8 років тому +1

      After watching the video, I can say that your confidence is well placed.

    • @eustatianwings
      @eustatianwings 8 років тому +3

      Does anyone else have non-transitive preferences between mathematicians?

  • @bogdanyer
    @bogdanyer 8 років тому +6

    wow! that is so counterintuitive!

  • @xSaucecode
    @xSaucecode 8 років тому +2

    I love these videos with Tadashi! So simple and elegant.

  • @rich1051414
    @rich1051414 8 років тому +4

    Love this video. I was just talking to someone the other day how logic and common sense are not the same thing, and sometimes logic completely contradicts common sense. This is a great example of that.
    If A is better than B, and B is better than C, then common sense would dictate A is better than C, but logically you cannot be certain about such a claim.

    • @losveratos
      @losveratos Рік тому

      @@eltodesukane No it isn't.

  • @TheRealPunkachu
    @TheRealPunkachu 5 років тому +1

    This is something I study a lot in game design. It's important to look at breakpoints like this when you look at balancing statistics.

  • @JamesSpeiser
    @JamesSpeiser 8 років тому +7

    Very counterintuitive and interesting!

  • @gsurfer04
    @gsurfer04 8 років тому +3

    It's always nice to start the day with Tadashi-san.

    • @cortster12
      @cortster12 8 років тому +2

      No, he's more of a teacher role. So sensei would fit better.

  • @ber2996
    @ber2996 5 років тому +35

    "A is stronger than B and B is stronger than C but it's not necessarily means that A is stronger than C"
    I might use this example in a certain situations.

    • @losveratos
      @losveratos Рік тому

      @@eltodesukane No it isn't.

  • @YC80087
    @YC80087 8 років тому +1

    Tadashi's lessons never ceased to amaze me! I learn every single time I watch his video!

  • @PauLtus_B
    @PauLtus_B 8 років тому +22

    I'd love a d20 with 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,10 gazillion

    • @nahuelmat
      @nahuelmat 4 роки тому +4

      The ultimate critical hit

  • @FlyingTurtleLP
    @FlyingTurtleLP 8 років тому +4

    Numberphile, I really love your vids. Mr. Tokieda explained that extremely well. Thank you!

  • @JazzRadioFfm
    @JazzRadioFfm 8 років тому +3

    i always knew that soccer (or any other sport) tournaments don't mean anything. depends on which team you play; if your strengths match the weaknesses of your opponents.

  • @TheShadowOfMars
    @TheShadowOfMars 8 років тому +1

    6:02 Two comparisons were missed out: The left die (662222) beats the right die (444400); The upper die (555111) and the lower die (333333) are equal.

  • @zhangpe1
    @zhangpe1 8 років тому +6

    The 5,1 dice and the all 3's dice relationship could be a clue as well. One dice can't not be declared stronger than the other.

  • @adizmal
    @adizmal 8 років тому +21

    Tadashi is a boss.

  • @alexare_
    @alexare_ 8 років тому +148

    A video about non-transitive dice on Numberphile, without everyone's favorite star, Dr. James Grime?? Not even a mention, or a link? Cmon Brady... Tadashi is great, but Dr. Grime has a great set of non-transitive dice named after him.

    • @numberphile
      @numberphile  8 років тому +226

      1. There is a link to BUY Grime dice in the video description
      2. I've been asking Dr Grime to a do a Grime Dice video with me for YEARS without success.

    • @taba1950
      @taba1950 8 років тому +2

      I think he made a video in his own channel a while ago

    • @SlyMaelstrom
      @SlyMaelstrom 8 років тому +6

      James Grime definitely made a video about non-transitive dice at some point. I was going to comment that he did it on Numberphile, but seeing this thread it makes more sense that he did it on his own channel. It also appears he did another video specifically about Grime dice which is actually shown on that Mathsgear website on the product page. I could understand why he wouldn't care to be redundant on the topic.

    • @Wegnerrobert2
      @Wegnerrobert2 7 років тому

      vaego haha before I researched those dice I had assumed it's some set of 10^25 dice with a dimension of 2^363728949-1 or something like that because that's often the scale of mathematical discoveries lol.

  • @TRUCKLOADOFPAIN
    @TRUCKLOADOFPAIN 8 років тому +2

    Tadashi could literally make a video of him just telling the entire world how terrible of a person I am and I would still love it. Dude's awesome.

  • @SheezyBites
    @SheezyBites 8 років тому +36

    The 6,2 dice is the best though because it's the only one statistically stronger than 2 other dice... at least until people start using the all 3 dice to try and win the meta game.

    • @Moorb0y52
      @Moorb0y52 8 років тому +7

      And then there would be a small number of people running the niche 4,2 strat to counter the 3s

    • @randomdogdog
      @randomdogdog 8 років тому +7

      just doing the maths for those who want it,
      the dice are:
      3 3 3 3 3 3 (3), 4 4 4 4 0 0 (4), 5 5 5 1 1 1 (5), 6 6 2 2 2 2 (6)
      3 beats 6, ties 5, loses 4
      4 beats 3, loses 5, 6(5/9th of the time)
      5 beats 4, ties 3, loses 6
      6 beats 5, 4, loses 3

    • @awyoung706
      @awyoung706 8 років тому +4

      To answer that last part: If all 4 dice were pitted together in one final battle to the death, The 6 and 5 die are the best with a 33% chance of winning. The 4 die has a 22% chance of winning, and the 3 die has an 11% chance of winning (don't pick that one!)

    • @djbrowny
      @djbrowny 7 років тому

      Thank you for saving my brain from a cognitive dissonance meltdown. I couldn't comprehend his analogy with drugs and implying that C>A>B>C. That's not possible with discrete data.

    • @ComradeTiki
      @ComradeTiki 7 років тому +4

      Here's a game for you:
      Two players discreetly select a die in order. The first player receives what they selected. If the second player requested the same, they pick again (knowing which die the first player chose).
      Since 6,2 is the best in a battle royale, and player-A knows this, but if player-B chooses the same, then for their second pick they're guaranteed to pick all-3's... so player-A could set an ambush by choosing all-3's. Unless the second player tries this as well and then learns to choose 4,0. So player-A might as well set a trap by starting on 4,0. But it will lose if player-B starts on 6,2.
      From a random Monty-Hall problem standpoint, it looks like player-B will win more often, but something tells me that in applied setting it would balance toward Rock-Paper-Scissors (50%).

  • @rosejuliette9180
    @rosejuliette9180 3 роки тому +2

    This would make a really fun game of rock paper scissors because even if you pick the stronger dice you still have to make the roll.

    • @sommers6235
      @sommers6235 3 роки тому

      Rock paper scissors is exactly what I was thinking too

  • @Unparadoxability
    @Unparadoxability 7 років тому +4

    This has been known for a long time for pokémon enthusiasts.
    Charizard beats Venusaur.
    Venusaur beats Blastoise.
    But no-one would conclude that this means Charizard beats Blastoise!

  • @WCairforce
    @WCairforce 8 років тому +1

    I'm loving all the dice related videos!
    These set up some fun experiments for students to explore.
    Cheers

  • @MrAntieMatter
    @MrAntieMatter 8 років тому +18

    5:03 Tadashi, how dare you sir!

  • @qaz010wsx
    @qaz010wsx 8 років тому +30

    "...This is something that should be know by all CITIZEN..." lol

    • @Theraot
      @Theraot 8 років тому +1

      League of Legends?

    • @E1craZ4life
      @E1craZ4life 8 років тому +2

      Laugh Out Loud

    • @rstriker21
      @rstriker21 8 років тому +3

      Lots of love

    • @marshmelows
      @marshmelows 8 років тому

      and then he flips the bird ahah

    • @ruben307
      @ruben307 8 років тому +2

      logarithmic or linear

  • @Robin_Nixon
    @Robin_Nixon 8 років тому +10

    Professor Tadashi got me thinking again - this time about how to improve the design of a game I'm working on :)

    • @chyousub
      @chyousub 8 років тому +2

      You might like to map it out. Just did it for interest because i like to see well balanced games too. The dices with the 5 and 6 win over 2 other dices. But the dices with the 3 and 4 only win over one other dice. That is still useful for games. The 3 and 4 can be healer and tank like in mmos. The 6 and 5 range and melee attack.

  • @SamBegich
    @SamBegich 5 років тому

    Tadeshi Blows My Mind Every Time. He does it so graciously too.

  • @spacegothgirl
    @spacegothgirl 5 років тому +5

    I would love a follow-up to this comparing the dice in Super Mario Party. Each character has a unique dice to give them all a different feel. It would be interesting if you could determine who's the strongest!

  • @JMcMillen
    @JMcMillen 8 років тому +4

    I noticed something when I compared the dice to their non-adjacent die. 333333 vs 555111 is even odds, half the time you win and half the time you lose. But with 662222 vs 444400, 662222 has a slight edge over 444400.

    • @UnCavi
      @UnCavi 8 років тому

      Are you sure? I think it's even odds as well

    • @anothermoth
      @anothermoth 8 років тому +1

      It can't be even. Both dice could be mapped onto 3 sided dice without changing the probabilities. When you roll 2 d3, there are 9 possible outcomes which is an odd number. If there are no draws then one must win more than the other. 662222 beats 444400 5/9ths of the time.

    • @alkoholhaltigdryck
      @alkoholhaltigdryck 8 років тому +1

      662222 has a 1/3 probability to roll a 6, which always wins. It has a 2/3 probability to roll a 2, in which case it wins 1/3 times. 1/3 + (2/3)*(1/3) = 0.555...
      662222 indeed has a slight edge.

    • @SPACKlick
      @SPACKlick 8 років тому

      555111 cannot be mapped onto a 3 sided die. 333333 and 555111 can be mapped to a two sided die/coin as 33 and 51.

  • @zelda12346
    @zelda12346 5 років тому +1

    6 is the strongest because in the entire, uniterated meta-game it beats 4 and 5 where 3 ties with 5. So, while the loop is a non-transitive tournament, there are still two more edges to the full graph. It is in these two edges we can make a ranking. Additionally, if we zoom out and consider pick rates for the prior information, even when considering this non-transitive loop, we can rank the four dice as equally strong, according to the definition given.

    • @reesverleur9546
      @reesverleur9546 5 років тому

      Would you mind giving your calculation? I had a feeling this was true from some cursory calculations but I wasn't able to totally convince myself.

    • @zelda12346
      @zelda12346 5 років тому

      @@reesverleur9546 Sure. 6 beats 5 as show in the video. 6 beats 4 because:
      The probability of A rolling 0 and B rolling 6 = 1/3*1/3 = 1/9
      P of A=4 and B=6 = 2/3*1/3 = 2/9
      P of A=0 and B=2 = 1/3*2/3 = 2/9
      P of A=4 and B=2 = 2/3*2/3 = 4/9
      As you can see, only one of these events ends up with A winning, event #4 which has a probability of 4/9, less than 0.5. Thus, A is weaker than B.
      3 is equally strong as 5 because 3 is a constant die and the 1's and 5's are equally probable on the 5 die. Hence, a both dice are equally likely to win when head-to-head.

  • @Birkirrey
    @Birkirrey 8 років тому +3

    Wow... The necessity for cross validation of statistical models just clicked for me in a new way. Mind blown!

  • @pawebielinski4903
    @pawebielinski4903 7 років тому

    I object to the analogy with medicine. The quantity, percentage, etc. of people cured can be expressed by a number, so of course a>b>c implies a>c. The situation is quite different than that of these interesting dice.

  • @Ocklepod
    @Ocklepod 8 років тому +44

    James grimes did the same thing on his own channel, and now it's the third video of exactly the same dice.

    • @Bringidon
      @Bringidon 8 років тому +3

      Yes he did

    • @ricepatch
      @ricepatch 6 років тому

      This one's more powerful though.

    • @Varksterable
      @Varksterable 6 років тому +3

      James Yang Each one beats the other, non-transitively?

  • @kevincross4302
    @kevincross4302 3 роки тому +1

    love that '10 Gazillion' example to explain averages

  • @RainnySkies
    @RainnySkies 8 років тому +36

    Tadashi has taught me more than all of my math teachers ever have.

    • @ThatGuy-nv2wo
      @ThatGuy-nv2wo 8 років тому

      And you're English teachers...

    • @rupertlosl122
      @rupertlosl122 8 років тому +13

      The irony...

    • @ichbinein123
      @ichbinein123 8 років тому +2

      My math teacher in highschool was educated in philosophy, but he said that the two subjects are so closely related, that one compliments the other in a surprising amount of ways.

    • @ichbinein123
      @ichbinein123 8 років тому

      That Guy Nah, friendo, he had a point. A lot of real logical problems can be quantized with mathematics. It doesn't even have to be particularly complicated.
      Besides, it's not as if he wasn't educated in math as well. He had a masters in philosophy, and a bachelor in mathematics, so he had his credentials in order.

    • @ThatGuy-nv2wo
      @ThatGuy-nv2wo 8 років тому

      IchBinEin Fair enough, but I still hold my point that philosophy is nothing like maths.

  • @mattp.158
    @mattp.158 8 років тому +2

    Tadashi-san just flipped us off at 5:04.

  • @imveryangryitsnotbutter
    @imveryangryitsnotbutter 8 років тому +8

    When I saw this video, my first thought was to design a rock-paper-scissors type game around it. I immediately came up with a game in which two people randomly draw a die from the set of four presented in this video, and then roll their selected dice to determine the winner.
    Unfortunately, I found that in a series of random matchups, the [6/2] die is overall the strongest.
    [4/0] beats [3], odds 2:1
    [5/1] beats [4/0], odds 2:1
    [6/2] beats [5/1], odds 2:1
    [3] beats [6/2], odds 2:1
    [3] and [5/1] are matched 1:1.
    [6/2] beats [4/0], odds 5:4.
    Based on the above, we can see that [3] and [5/1] have average performance; they are weak to one die, strong against another, and evenly matched by another.
    But [6/2] performs slightly better than all the others; it is weak to one die, but strong against the other two dice.
    Conversely, [4/0] performs slightly worse than all the others; it is strong to one die, but weak against the other two dice.
    But luckily, there's a way to fix this! You can even out this power balance if you make a small alteration: change the [4,4,4,4,0,0] die to a [4,4,4,4,0,X] die.
    X, in most cases, is equivalent to a 0, and loses to any other roll... *except* a 6, which it trumps. In a matchup between this die and [6/2], the chance of rolling 6 vs. X is 1-in-18; this alters the odds exactly enough that each die is equally likely to win against the other.
    So, by changing one of the 0's to an X, and giving the X the properties stated above, you have a game in which every die is strong against one, weak against another, and equalled by another. A perfect balance.
    ...Okay, so it would be a pretty boring game on its own. But you could use it as the basis for a tabletop RPG's combat system!

    • @avocette
      @avocette 5 років тому

      late comment, but can you explain more in detail how (44440X) beats (666622)?

    • @Seb135-e1i
      @Seb135-e1i 5 років тому

      @@avocette Ah, that's where you've confused yourself and confused me. The 6/2 dice is not 666622, but rather 662222
      Against 44440X, it'll work like this:
      4 dice rolls a 0 with 1/6 chance, guaranteed loss.
      Rolls an X with 1/6 chance, wins with 1/3 chance if 6 dice rolls a 6.
      Rolls a 4 with 2/3 chance, wins with 2/3 chance if 6 dice rolls a 2.
      So let's see if the math checks out this time:
      2/3 * 2/3 is 4/9. That's just under a half. This means that the X roll has to be at least 1/18, which will put it at 1/2 chance to win or lose.
      1/6 * 1/3 is 1/18.
      Oh, that's why. The original comment states that the 4 dice is strong against 1 and weak to 2. This change does, indeed, make it strong against 1 and weak against 1, with it now being equal to the 6 dice.

    • @calembur
      @calembur 5 років тому

      I read your whole explanation.

  • @Zandonus
    @Zandonus 8 років тому +1

    This can be used in say....paradox interactive games, where simulations show that one type of army composition kills more men than the other, but doesn't necessarily win as often...A niche case, but it can happen. Usually more losses means a much lower chance of winning, but when looking at simulation results, one should try to find sweetspots to avoid a too qualitative army and instead go for More army elsewhere. I hope i'm making some sense. There's also dice involved, but finding 2 identical armies with 2 identical dice modifiers is rare, and winning/losing tends to snowball based on too many variables.

  • @jevicci
    @jevicci 8 років тому +5

    This guy's ability to explain things in a language that's foreign to him is extremely impressive.

  • @vipermagi5499
    @vipermagi5499 7 років тому

    I have no idea what those dice are made of, but the sound they make when they hit that table is very pleasing.

  • @MaxKoschuh
    @MaxKoschuh 8 років тому +64

    excellent. very interesting to learn about the cicle.
    5:04 WTF? :-)

    • @ItchyKneeSon
      @ItchyKneeSon 8 років тому +29

      The same flag went up for me even though I've been living in Japan for over 6 years. Tadashi is Japanese and he almost definitely developed that habit before making his way west. In Japan, pop culture from the west has informed Japanese (and other countries, too, I imagine) that the middle finger has a negative connotation, so everybody here knows that it's not good to toss your lone middle finger up at someone, but in using it for anything else, it goes totally unnoticed because it doesn't have the same meaning in Japanese culture. Actually, being the longest finger, it's logically the most convenient and useful to use for a number of tasks, including pointing, referencing, and glasses up-pushing. :D It took a while to get used to, but now I've gotten quite used to it. In this case, it was difficult to miss, though. hahaha

    • @bytefu
      @bytefu 8 років тому +14

      ItchyKneeSon
      Didn't notice the gesture until read comments about it. We don't use it much in Russia either.

    • @NoriMori1992
      @NoriMori1992 8 років тому +3

      +ItchyKneeSon I, a North American, also use my middle finger to push up my glasses, but I have _all_ my fingers extended when I do so. XD
      +Artm Borisovskiy I didn't notice it either. XD

    • @ItchyKneeSon
      @ItchyKneeSon 8 років тому

      NoriMori That's called 'dazzle fingers'.

    • @NoriMori1992
      @NoriMori1992 7 років тому

      +ItchyKneeSon By who? Why?

  • @ber2996
    @ber2996 5 років тому +1

    Dice A, B, C, and D
    B has win rate of 66.7% against A
    C has 66.65% win rate against B
    D has 66.65% against C
    and A has 66.7% against D

  • @victorlamarca5156
    @victorlamarca5156 7 років тому +5

    10 gazillion is my favourite number

  • @FireyDeath4
    @FireyDeath4 5 років тому +2

    Hey! I saw a similar trick in a Murderous Maths book.
    So similar that it had 4 entities with 6 positions as well.

  • @finnberuldsen4798
    @finnberuldsen4798 8 років тому +19

    This man could teach rocket science to preschoolers.

  • @TesserId
    @TesserId 3 роки тому +1

    Now discuss this in various ranked voting algorithms. I'd like to see algorithms that resolve cycles in ranked voting. Is Tarjan's algorithm a candidate?

  • @lemonke8132
    @lemonke8132 8 років тому +27

    He just flipped me off 5:02

  • @NoahSpurrier
    @NoahSpurrier 8 років тому +1

    This reminds me of the False Positive Paradox, for some reason; although, I don't think it's actually related. But it gives me the same sort of pleasure when the counter-intuitive understanding finally clicks in my mind.

  • @ghoulofmetal
    @ghoulofmetal 8 років тому +14

    So this is similar to stone/paper/scissor.

    • @jazzsoul69
      @jazzsoul69 4 роки тому +1

      similar but not intuitive it deserves to be pointed out !

  • @Aleksandar0100
    @Aleksandar0100 5 років тому

    5:04 Thanks for flipping me off Tadashi

  • @vkillion
    @vkillion 8 років тому +13

    Didn't we already see a video on non-transitive dice featuring Dr. James Grime (aka singingbanana)?

    • @vkillion
      @vkillion 8 років тому +9

      I guess it was never on Numberphile. It was on Dr. Grime's personal channel (singingbanana) and the Maths Gear channel (mathsgear).

  • @nix207
    @nix207 5 років тому

    Some of the concepts in this shed some light on the paper i did as an undergraduate. I now appreciate both this video and my own work as well. Now if I only had the time, I would have used some of the ideas here to simplify my calculations massively.

  • @Czeckie
    @Czeckie 8 років тому +31

    I love Tadashi. But this is like the third video on nontransitive dice.

    • @numberphile
      @numberphile  8 років тому +36

      it's our first!?

    •  8 років тому +12

      I think he just meant dice, but dont listen to him, more dice videos please.

    • @Bronzescorpion
      @Bronzescorpion 8 років тому +31

      I don't think he meant just dice, but rather people following this channel tends to follow James Grime and Matt Parker and they also did similar videos. It can sometimes be hard to remember which channel you saw it on, because of the similarity, hence the confusion and the comment.

    • @Jack-sy6di
      @Jack-sy6di 8 років тому +17

      There are several videos on UA-cam with people who often appear on numberphile showing off nontransitive dice, but none on the actual numberphile channel. I was confused at first too.

  • @cmilla111
    @cmilla111 7 років тому

    I like his final point. So many people think not paying attention to math growing up just means they will need a calculator. They don't realize how stuff like this seeps into so many things in life.

  • @Pika250
    @Pika250 8 років тому +5

    These dice are akin to the Fire, Water, Electric, and Grass types in Pokémon.

    • @Pika250
      @Pika250 8 років тому +1

      333333 Fire
      004444 Water
      111555 Electric
      222266 Grass

    • @tsunamininja
      @tsunamininja 8 років тому +1

      Grass doesn't win or lose to electric tho

    • @Pika250
      @Pika250 8 років тому

      Think defensively. Grass resists electric but not the other way around.

    • @Pika250
      @Pika250 8 років тому

      it's been that way since gen 1

    • @conradbamboozled1653
      @conradbamboozled1653 7 років тому

      Rock, bug, psychic, fighting?

  • @DyingToLive310
    @DyingToLive310 5 років тому

    Avoiding arithmetic? Not using a 10 syllable word and expecting us to know it? Fun easy presentable demonstration? Cartoons? BEST TEACHER EVER

  • @ocallaghandesign
    @ocallaghandesign 8 років тому +45

    this needs to be a game mechanic

    • @oz_jones
      @oz_jones 8 років тому +27

      Rock, paper & scissors?

    • @ocallaghandesign
      @ocallaghandesign 8 років тому +3

      similar but better.

    • @oz_jones
      @oz_jones 8 років тому +11

      A lot of strategy games use a similar mechanic but deeper.

    • @klaxoncow
      @klaxoncow 8 років тому

      Rock, paper, scissors?
      Sort of.

    • @zerid0
      @zerid0 8 років тому +1

      It's called rock-paper-scissors ;)

  • @angusmacdonald4860
    @angusmacdonald4860 Рік тому +1

    This guy is the most interesting mathematician to listen to u
    On numberphile..

  • @strawhatroger1657
    @strawhatroger1657 8 років тому +6

    Isn't the 1st and 3rd die the same (50/50) because it's 555111 and 333333 therefore each die wins half the time and loses the other half :/

    • @timschonning269
      @timschonning269 8 років тому

      It can't be even. Both dice could be mapped onto 3 sided dice without changing the probabilities. When you roll 2 d3, there are 9 possible outcomes which is an odd number. If there are no draws then one must win more than the other. 662222 beats 444400 5/9ths of the time.

    • @snowfloofcathug
      @snowfloofcathug 8 років тому

      I agree

    • @JossLun
      @JossLun 8 років тому

      I agree with Straw, the 1st and the 3rd die are equal. No matter how many you throw, both types will win as many times as the others. And sums also have the same average value

    • @KittyBoom360
      @KittyBoom360 8 років тому +3

      Straw Hat Roger, nice catch! Yes, the dice 555111 vs 333333 have a probability of 1:1 for predicting outcome of each roll.

    • @KittyBoom360
      @KittyBoom360 8 років тому +4

      Tim, no, 555111 cannot be mapped to 3 sided dice, what three numbers, 551, 511, 555, 111??? it could be reduced to a 2 sided/coin tho 51

  • @Seltyk
    @Seltyk 2 роки тому +2

    This is what makes rock-paper-scissors the game that it is

  • @undergroundmonorail
    @undergroundmonorail 8 років тому +4

    I'm familiar with nontransitive dice but can someone explain the medicine example? If A cures more than B and B cures more than C, then why wouldn't A cure more than C? It's not like nontransitive dice where the comparison between two can change over time. Assuming the studies were done properly the results should more or less stay the same, so it's not really like rolling B multiple times. You're doing (effectively) one trial for each and seeing which number is the highest.

    • @kroppyer
      @kroppyer 8 років тому +3

      If you're confident (at some level) that A cures more people than B and that B cures more people than C, I'd say you can conclude you're confident (at the same level) that A cures more people than C, right?
      I don't think the medicine example works.

    • @MasterHigure
      @MasterHigure 8 років тому +1

      It could have to do with confidence intervals, but you also have to take demographics into account. Was A and B compared on women while B and C was compared on men? Or young vs old? Or different social standings? Or any other of probably hundreds of factors that could make the two results incomparable.

    • @undergroundmonorail
      @undergroundmonorail 8 років тому

      That makes sense to me but I assumed the studies in question would be equivalent. If that's not the case it's hard to say the results are really comparable at all.

    • @Jack-sy6di
      @Jack-sy6di 8 років тому +4

      If we're thinking of a drug as basically being a coin flip (it has a certain probability of curing any sick person it's given to), then "A cures more people than B" basically means "A has a higher success probability than B", so you're just ranking numbers, so yes, you can't get non-transitivity. Either he chose a bad example, or he was imagining some more complicated model that would make it work.
      I think one place where you can get nontransitivity is in voting. If when asked to choose between candidates A and B, most people prefer B, and when asked to choose between B and C, they mostly prefer C, that doesn't mean most people will prefer C to A. But that's kind of obvious.

    • @kroppyer
      @kroppyer 8 років тому

      ***** maybe you can point us to a specific paragraph or section in the 16 page document.

  • @patxmcq
    @patxmcq 5 років тому +1

    Hearing Tadashi say "0,0,0,0,0,10 gazillion" made me actually lul

  • @RandomWalker39
    @RandomWalker39 8 років тому +50

    Seeing this peculiarity being demonstrated in the US Presidential election this year.

    • @claeshenriksson5702
      @claeshenriksson5702 8 років тому

      +

    • @bobdfs5230
      @bobdfs5230 8 років тому +2

      Clinton votes > Bernie votes doesn't mean Trump votes > Bernie votes, as they aren't the same thing; primary votes and general election votes were cast in different contexts. Is that what you mean?

    • @RandomWalker39
      @RandomWalker39 8 років тому +15

      No. Clinton beats Sanders in primary. Trump (potentially) beats Clinton. Sanders would have clobbered Trump.

  • @OriginalPiMan
    @OriginalPiMan 8 років тому +22

    I've seen these dice done by someone else but I can't remember who.

    • @Jesper-Music
      @Jesper-Music 8 років тому +7

      That was Numberphile regular James Grime (Singingbanana) :)
      You can buy his dice online, and they're really cool :)

    • @numberphile
      @numberphile  8 років тому +10

      link in the video description

    • @JoQeZzZ
      @JoQeZzZ 8 років тому +12

      Another numberphile host, James Grime.
      watch?v=u4XNL-uo520

    • @elcuboderusca
      @elcuboderusca 8 років тому +3

      Yup, in maths gear shop. (Grime, Mould and Parker, actually)

    • @ShayBowskill
      @ShayBowskill 8 років тому +1

      Oskarpuzzle has a version of these that he made into a sort of game

  • @kerr.andrew
    @kerr.andrew 8 років тому +5

    this is like rock paper scissors

    • @VapidVulpes
      @VapidVulpes 8 років тому +1

      whoooaaa! you gotta guess what die the other person will throw and then throw yours so it's two layers of chance.
      you'd have to know how every other dice pair's win/lose relationship too though to know how that game would work and I'm too lazy to figure it out right now

  • @the.y.method
    @the.y.method 8 років тому

    I love watching Tadashi videos. He is full of fascinating insights and thoughts. Discovered him before I saw him on Numberphile though, doing a series of lectures in Cape Town at the African Institute for Mathematical Sciences, lecturing on topology.

  • @superj1e2z6
    @superj1e2z6 8 років тому +8

    Diesuki desu.

    • @ItchyKneeSon
      @ItchyKneeSon 8 років тому +2

      Please, do not wish death upon Suki Desu. :D
      dai*suki desu / だいすきです。/ 大好きです。

    • @leonardromano1491
      @leonardromano1491 8 років тому +2

      I think you didn't get the joke...
      Die is the singular form of "Dice" which Tadashi is using.
      No one wishes death upon the phrase "好きです".
      This person seems to like puns and dice.

  • @dankeman5266
    @dankeman5266 3 роки тому

    Brilliant, and brilliantly explained!

  • @eretah
    @eretah 8 років тому +3

    I'd like to buy a die with a 3 on every side

  • @PersonaRandomNumbers
    @PersonaRandomNumbers 8 років тому

    Reminds me of the video about nontransitive dice with James Grime! I'm surprised numberphile hasn't made a video about it until now.

  • @mikikiki
    @mikikiki 8 років тому +5

    His voice and his accent are amazing. He could retire and do voiceover work.

    • @numberphile
      @numberphile  8 років тому +11

      I think he'd miss math and science!

    • @ItsYoji
      @ItsYoji 8 років тому

      Depends. Are Math audiobooks a thing?

  • @scottcomber
    @scottcomber 8 років тому

    I wish I could listen to this man lecture all day. Just perfect explanations.

  • @demonstructie
    @demonstructie 8 років тому +70

    japanese guy > klein bottle guy

    • @twwc960
      @twwc960 8 років тому +120

      It doesn't really matter. You might like Tadashi better than Cliff, you might like Cliff better than James, but you might like James better than Tadashi. It isn't necessarily transitive.

    • @demonstructie
      @demonstructie 8 років тому +3

      +twwc960 YOU don't really matter.

    • @imveryangryitsnotbutter
      @imveryangryitsnotbutter 8 років тому +23

      +demonstructie Boy, you _sure_ showed him. :/

    • @RickySTT
      @RickySTT 8 років тому +1

      I've met the Klein bottle guy in person, but I've never met Tadashi, so I won't make that judgment.

    • @CallMePuddlez
      @CallMePuddlez 8 років тому +10

      debatable

  • @fghsgh
    @fghsgh 6 років тому

    I got wondering what the relation was between two dice on opposite sides of the circle, and figured out that they always have a 1/2 chance, so if A>B>C>D>A, then A=C and B=D.
    It could have been helpful to have said something about this in the video, but it doesn't really matter because like this you can have some exercise yourself.

  • @brokenwave6125
    @brokenwave6125 8 років тому

    At first a I thought "o, another non transitive dice video..."But Tadashi makes everything worth watching

  • @danielleanderson6371
    @danielleanderson6371 8 років тому +2

    Something interesting I noticed: Each winning die wins 2/3 of the time, in all four cases. Similarly, opposite dice (333333 and 555111, 444400 and 662222) are exactly evenly matched, such that each one would win against the other 50% of the time.

    • @evanaubry9040
      @evanaubry9040 8 років тому +5

      That's an interesting observation. However, actually, the 662222 die is slightly stronger than the 444400 die. At first glance, it may seem like 50-50, but the 662222 die wins 20/36 possible outcomes, whereas the 444400 die wins only 16/36 times. So one could argue that 662222 is the strongest of them all, as it is the only one that wins over two different dice.

    • @101jir
      @101jir 8 років тому

      +Evan Aubry that assumes that the dice are blind-picked. In the real world equivalent, such as videogame balancing, things like draft pick make things interesting.

  • @5eurosenelsuelo
    @5eurosenelsuelo 8 років тому

    Great video and conclusion at the end