They look like ordinary dice, but they are not. In competition Red will systematically beat Green, Green will systematically beat Blue, and the amazingly, Blue will systematically beat Red. Is that really possible?

My son showed me three amazing dice the other day – Grime Dice. They are six-sided cubic dice but they don’t have the usual numbers 1 to 6 on each side. Each dice has a different combination of numbers which retain the same average value (3.5) as a normal dice.

Table showing the numbers on the face of each of the dice. The colour code is A = Red, B = Green and C = Blue

The amazing property of these dice is discernible when you use them competitively – i.e. you roll one dice against another. If you roll each of them against a normal dice then as you might expect, each dice will win as often as it will lose. But if you roll them against each other something amazing happens.

Dice A will systematically beat Dice B

Dice B will systematically beat Dice C

and amazingly

Dice C will systematically beat Dice A

So the fact that Dice A beats Dice B, and Dice Bbeats Dice Cdoes not ensure that Dice Awill beat Dice C. Wow!

And how about this: If you ‘double up’ and roll 2 Dice A‘s against 2 Dice B‘s – the odds change around and now the B‘s will beat the A‘s ! Is that really possible? Well yes, and just to convince myself I wrote a Spreadsheet (.xlsx file) and generated the tables at the bottom of the article. If you download it you can change the numbers to try out other combinations.

There are lots of sets of non-transitive dice and they have many other surprising properties. This web page has more detail and the video below includes a chat with Mr. Grime himself.

I really don’t know what to make of these dice – but they did surprise me, so I thought I would share that surprise with you.

Table of results when Dice A is rolled against Dice B

Table of results when Dice B is rolled against Dice C

Table of Results when Dice A is rolled against Dice C

This entry was posted on January 8, 2013 at 8:53 pm and is filed under Personal, Simple Science. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

36 Responses to “Amazing Dice: Rediscovering surprise”

@shawn, that term is no longer used in general conversation and has been phased out really (as with a lot of terms and words in all languages over time). That’s like correcting someone that gay actually means happy (if you were around long enough to remember that)

When you are looking at averages you are looking at numbers. When you are looking at wins you are looking at a boolean conditional. So if you calculated a score based on how much you won by the dice would all be even. You are essentially creating circumstances where you win barely and lose by a lot.

The die in the top image are either incorrect or not these actual dice. One has fours (must be Dice C), one has twos (must be Dice B), and one has fives (must be dice B). Am I missing something here?

byron, I doubt that anything misses you. The photograph is a photograph of a different set of grime dice, the ones the people on the web site sell. I didn’t discuss these dice because their averages aren’t equal in magnitude, which I feel is ‘cheating’. The set of grime dice I featured all have the same average value which is what makes their non-transitive result interesting.

It’s confusing that the picture at the top of the post shows the red and blue dice with values they DON’T have in your table. Or maybe I’m misunderstanding the data presented?

I had trouble reconciling the picture of the 3 die to the coloured table of possible numbers. The picture shows a green 5, red 4 and blue 2 – which doesn’t look possible according to the table.

The table says that you cannot get a 5 and 2 except from the same coloured dice (green)…
There is no 4 for red (which belongs to blue)
and there is no 2 for blue (which belongs to green).

I think I must have misunderstood something – or I’m badly in need of my morning coffee….

As a teacher, there’s a nice test / investigative challenge question there. It’ll go nicely with the Sicherman dice.

On the issue of plurals, dice is most definitely a plural as the singular is used in other areas. A die is used to mint coins, for example, and we still say ‘the die is cast’.

Rather than encourage this mistake, fixing the article would be doubly entertaining and educational.

Thanks: I think I used a random number generator to make Monte Carlo estimates of the odds. Thanks for the exact numbers – I couldn’t quite see to calculate them.

It’s the three dimensionality of the third variable. By the way, if you want to win overall with a single Grime Die that averages to 3.5, you’re best served by one with three sixes and three ones.

Statistically speaking, it will win more rounds against die A, Die B, and it will be a draw on Die C.

Interesting post. These dice are illustrating a very basic statistics/logic function. The are really quite simple- we spend too much time making math mysterious, if these dice are useful for anything its to illustrate that math and statistics can be very “fun” and very very simple.

It seems much less mysterious if you just think of it this way… if Die A has a number on it on any face that is higher than any number on any face of die B it wins 1/X of the time period (if X is the number of faces with non equal numeric values between the two dice)… just add up the number of faces that are a guaranteed win and put that number over the number of faces that are a guaranteed win on the other die.. the fraction of times Die A beats Die B (A guaranteed winners/B guaranteed winners) is revealed (we ignored ties right?!).

These are a great teaching tool to introduce (among other things) the concept of how overused and uninformative an “average” value can be (e.g. using it to presume an expectation of the results- which quite obviously fails) and to introduce statistics! Which is quite amazing. My only complaint is the title and text tends to presume that this is in some way a surprising result, or that it requires something beyond the ability to compare numbers (you don’t even need to know what the difference is between the numbers, just recognize that a 6 is more than a 5, for instance). The same is true even of the fraction you compute to determine which die wins more often, you do not need math, all you need is to recognize which number is bigger- if the top number is bigger that die wins more (the percentage of times it wins requires you complete the division but the determination of which die will win most often requires only that you can compare numeric value for equality.).

A great tool to introduce these concepts but hardly amazing… thats my take for the utility of these guys.

I note that on these dice we have no ties… so the X just becomes the number of sides (6) and its even less complicated.

Yes, I wondered whether there were more natural sequences in which the differences between averages didn’t express some key physical or numerical property of the system. But I haven’t thought of an example yet

If you find this interesting, you might enjoy taking a shot at solving this Project Euler problem: http://projecteuler.net/problem=376 . I beat my head against that for a while but couldn’t find an efficient solution.

[…] Red beats green 4 out of 6 times. Green beats blue 4 out of 6 times. Blue beats red 4 out of 6 times. All the colored dice tie the “normal” die. More here. […]

FYI- theres a new dice game soon to be released on kickstarter that is a new and different shape. the producers have a indorcment from the commisser of gambling,of Nevada,to put it on the Vegas floor if a couple of requirments are met. How exciting( move over craps)

January 9, 2013 at 1:55 pm |

Paper scissors stone for the 21st C!

February 25, 2013 at 8:38 am |

Interesting!

BTW, “dice” is plural, the singular is “die”. Die A systematically beats die B.

March 14, 2013 at 7:44 pm |

[…] These three dice don't follow the rules…of reality. […]

March 14, 2013 at 8:04 pm |

@shawn, that term is no longer used in general conversation and has been phased out really (as with a lot of terms and words in all languages over time). That’s like correcting someone that gay actually means happy (if you were around long enough to remember that)

March 15, 2013 at 3:08 pm |

Surely this is troll. No one is so stupid.

March 15, 2013 at 4:44 pm |

Really? no uses the word die any more? who decided that? The gamers I play with use the word “die” all the time.

April 8, 2013 at 9:34 pm |

Sorry, you could’t be more wrong. People use the singular “die” all the time.

March 14, 2013 at 8:14 pm |

When you are looking at averages you are looking at numbers. When you are looking at wins you are looking at a boolean conditional. So if you calculated a score based on how much you won by the dice would all be even. You are essentially creating circumstances where you win barely and lose by a lot.

March 14, 2013 at 10:29 pm |

Absolutely correct and that is the non-linearity makes the dice work, Nonetheless I still find them surprising.

March 14, 2013 at 8:54 pm |

The die in the top image are either incorrect or not these actual dice. One has fours (must be Dice C), one has twos (must be Dice B), and one has fives (must be dice B). Am I missing something here?

March 14, 2013 at 10:31 pm |

byron, I doubt that anything misses you. The photograph is a photograph of a different set of grime dice, the ones the people on the web site sell. I didn’t discuss these dice because their averages aren’t equal in magnitude, which I feel is ‘cheating’. The set of grime dice I featured all have the same average value which is what makes their non-transitive result interesting.

Well observed :-)

March 14, 2013 at 8:57 pm |

It’s confusing that the picture at the top of the post shows the red and blue dice with values they DON’T have in your table. Or maybe I’m misunderstanding the data presented?

March 14, 2013 at 10:32 pm |

No like ‘byron’ you have spotted that the grime dice I photographed correspond to a different set of grime dice.

March 14, 2013 at 10:11 pm |

I had trouble reconciling the picture of the 3 die to the coloured table of possible numbers. The picture shows a green 5, red 4 and blue 2 – which doesn’t look possible according to the table.

The table says that you cannot get a 5 and 2 except from the same coloured dice (green)…

There is no 4 for red (which belongs to blue)

and there is no 2 for blue (which belongs to green).

I think I must have misunderstood something – or I’m badly in need of my morning coffee….

March 15, 2013 at 5:43 pm |

No – you are just very observant. The photograph is of a different set of Grime Dice. Sorry for the confusion.

March 14, 2013 at 11:20 pm |

The singular of “dice” is in fact “die.” Please fix, despite what clueless up there said.

March 15, 2013 at 2:37 pm |

The singular of dice is most commonly “dice” nowadays. At least in UK english.

http://oxforddictionaries.com/definition/english/die–2

March 15, 2013 at 8:58 am |

@Markhov – die is still in common use.

March 15, 2013 at 1:31 pm |

Fascinating! Brilliant!

As a teacher, there’s a nice test / investigative challenge question there. It’ll go nicely with the Sicherman dice.

On the issue of plurals, dice is most definitely a plural as the singular is used in other areas. A die is used to mint coins, for example, and we still say ‘the die is cast’.

Rather than encourage this mistake, fixing the article would be doubly entertaining and educational.

March 15, 2013 at 2:21 pm |

I thought it was d6. :\

March 15, 2013 at 3:11 pm |

Your doubledice results are a little beside the odds:

For the 1296 possible outcomes the results are

A vs. B : 531:765

B vs.C: 531:765

C vs. A: 625:671

Best regards

March 15, 2013 at 5:41 pm |

Thanks: I think I used a random number generator to make Monte Carlo estimates of the odds. Thanks for the exact numbers – I couldn’t quite see to calculate them.

March 15, 2013 at 3:31 pm |

It’s the three dimensionality of the third variable. By the way, if you want to win overall with a single Grime Die that averages to 3.5, you’re best served by one with three sixes and three ones.

Statistically speaking, it will win more rounds against die A, Die B, and it will be a draw on Die C.

March 15, 2013 at 3:39 pm |

Interesting post. These dice are illustrating a very basic statistics/logic function. The are really quite simple- we spend too much time making math mysterious, if these dice are useful for anything its to illustrate that math and statistics can be very “fun” and very very simple.

It seems much less mysterious if you just think of it this way… if Die A has a number on it on any face that is higher than any number on any face of die B it wins 1/X of the time period (if X is the number of faces with non equal numeric values between the two dice)… just add up the number of faces that are a guaranteed win and put that number over the number of faces that are a guaranteed win on the other die.. the fraction of times Die A beats Die B (A guaranteed winners/B guaranteed winners) is revealed (we ignored ties right?!).

These are a great teaching tool to introduce (among other things) the concept of how overused and uninformative an “average” value can be (e.g. using it to presume an expectation of the results- which quite obviously fails) and to introduce statistics! Which is quite amazing. My only complaint is the title and text tends to presume that this is in some way a surprising result, or that it requires something beyond the ability to compare numbers (you don’t even need to know what the difference is between the numbers, just recognize that a 6 is more than a 5, for instance). The same is true even of the fraction you compute to determine which die wins more often, you do not need math, all you need is to recognize which number is bigger- if the top number is bigger that die wins more (the percentage of times it wins requires you complete the division but the determination of which die will win most often requires only that you can compare numeric value for equality.).

A great tool to introduce these concepts but hardly amazing… thats my take for the utility of these guys.

I note that on these dice we have no ties… so the X just becomes the number of sides (6) and its even less complicated.

March 15, 2013 at 5:39 pm |

Yes, I wondered whether there were more natural sequences in which the differences between averages didn’t express some key physical or numerical property of the system. But I haven’t thought of an example yet

M

March 15, 2013 at 5:59 pm

three sided “dice” with the numbers one through nine.

die A :1,5,9 die B:2,6,7 die C:3,4,8

also with the pleasant feature that the sum of all the sides is 15 on each die

March 15, 2013 at 5:53 pm |

more of a brain teaser i think is trying to accomplish the same thing with the numbers 1-18 used only once per side.

at least the puzzle is harder I think.

March 15, 2013 at 11:55 pm |

[…] really good and easy-to-understand lesson on probability and how results can be somewhat […]

March 16, 2013 at 1:09 am |

If you find this interesting, you might enjoy taking a shot at solving this Project Euler problem: http://projecteuler.net/problem=376 . I beat my head against that for a while but couldn’t find an efficient solution.

March 16, 2013 at 5:13 pm |

Where the toast can I buy these?!

March 16, 2013 at 5:19 pm |

Aha! Found them… but they’re UK-based. Nuts.

http://www.mathsgear.co.uk/non-transitive-dice/

Ah, there we go. The secret is to just search for ‘non-transitive dice’ but NOT in the Google Shopping thingy.

March 16, 2013 at 10:02 pm

But be warned: they sell a more complicated set than the simple set I described – that’s why the photograph is not correct :-)

March 16, 2013 at 10:56 pm |

[…] Red beats green 4 out of 6 times. Green beats blue 4 out of 6 times. Blue beats red 4 out of 6 times. All the colored dice tie the “normal” die. More here. […]

May 3, 2013 at 10:01 am |

[…] Amazing Dice: Rediscovering surprise […]

July 10, 2013 at 5:17 pm |

FYI- theres a new dice game soon to be released on kickstarter that is a new and different shape. the producers have a indorcment from the commisser of gambling,of Nevada,to put it on the Vegas floor if a couple of requirments are met. How exciting( move over craps)

September 8, 2015 at 9:38 pm |

south-wind.chicappa.JpAmazing Dice: Rediscovering surprise | Protons for Breakfast Blog