Talk:Intransitive dice

Solution to four-player open problem?
My brother and I have solved the four-player version of this puzzle with a set of 19 dice with 171 faces each.

We'd love to edit this page to include something like

"In 2021, amateur mathematicians Matthew and Sam Redmond constructed a non-minimal solution using the techniques of http://old.sztaki.hu/~bozoki/Bozoki-MiskolcMathNotes-2014.pdf: their solution requires 19 dice with 171 faces apiece, and labels from [0, 3248]"

The solution is too large to publish a table here. Should we include the first few columns? — Preceding unsigned comment added by 98.210.79.241 (talk) 00:23, 2 November 2021 (UTC)

Substantial Edit
I've removed some false statements from the introduction, and made substantial cosmetic changes to the "Example" section. If I've added any false statements of my own, or caused other problems, please let me know! Vectornaut (talk) 07:00, 27 February 2013 (UTC)

First Example
Am I not understanding something or is the first example given (dice A, B, and C) incorrect? Die A should roll higher than die C in 66% of samples. It seems like someone conflated the order that the numbers occur on the die faces with how they will appear when two die are rolled together. —Preceding unsigned comment added by 24.186.2.170 (talk) 02:16, 2 December 2010 (UTC)

It'll take too long to explain, but if you want proof, take a probability class. — Preceding unsigned comment added by 204.38.188.254 (talk) 15:28, 13 January 2012 (UTC)

Talk About Meaning?
Can one of you folks who knows what they're talking about explain what this means?

From the first set of three dice:

From what I can calculate, the *average* roll any of these dice will be 5. So, if over a large number of rolls the average will be 5, then for any 2 dice, the average will tend to be 5.

On the other hand, I did what you recommended and worked out the probability trees, and indeed A tends to be higher than B, which tends to be higher than C.

How can they all tend to be higher than one another, and yet perversely have the same average? Magic? -CC.

Well, that's the fun part. But consider another probability situation: Let quantity A be zero with 90% probability, and let it be 100 with 10% probability. The average of A is 90%*0 + 10%*100 = 10. Also, let quantity B be 1 with probability 100%, i.e. always. Then, while A on average is 10 and B only 1, the probability that A is larger than B is only 10%. Is that magic too?--Noe (talk) 20:20, 11 August 2008 (UTC)

... Ok, I got that. Thanks... 209.6.221.208 (talk) 01:19, 12 August 2008 (UTC)CC.

Think about pokemon. It's like that — Preceding unsigned comment added by 204.38.188.254 (talk) 15:30, 13 January 2012 (UTC)

Efron's dice redirection
This page previously redirected to Efron's dice. I have changed this, because the redirection target was only one particular case of dice intransitivity, and the subject warrants more discussion than just Efron's. ~ Booyabazooka 06:59, 24 April 2006 (UTC)

Merge proposal
I'm not sure why we need seperate pages for these. Are there any other examples of non-transitive dice? And as Booyabazooka's most recent edit suggests, this page is mostly just duplicated/redundant info from Efron's dice, yes? Ewlyahoocom 22:52, 10 May 2006 (UTC)
 * I've eliminated most of the overlap. The Efron's dice article is decently-sized already, and I think Nontransitive dice has potential to be longer.  I'm not sure if there are any other well-known examples, but clearly there are infinite possibilities; so in addition to the specific examples, there is also room for more abstract mathematical discussion.  So, I don't see any pressing reason to merge theses. ~ Booyabazooka 23:37, 11 May 2006 (UTC)
 * We can certainly talk about splitting the page up, even creating a category, if and when anyone ever bothers to document another one and we find the combined page getting to be "too big". But until such a time I think there's plenty of room on this page for the one other example of non-transitive dice. Ewlyahoocom 05:58, 12 May 2006 (UTC)
 * I have an article on creating nontransitive dice by George Trepal which demonstrates that there are other examples than Efron's, shows how to create them, and touches on which is 'best' (see below). New to wikipedia so, erm, wrestling with putting it up Whiteknight1066 (talk) 20:33, 6 September 2008 (UTC)
 * Can you give a link to the article? Otherwise it's hard to give you advice. C RETOG 8(t/c) 22:00, 8 September 2008 (UTC)

Best overall die
This section needs some citation. As Wikipedia is not the place for miscellaneous mathematical musings, I don't think it's fitting to include this detail unless Efron or some other notable mathematician discussed it. ~ Booya Bazooka 08:57, 7 November 2006 (UTC)
 * I disagree. This is not a "miscellaneous musing" but rather a quite helpful thing to know about the dice.  Whether or not Efron (or anyone else) discussed it is irrelevent.  Mathematics is not based on authority - it's always been "open source".  A citation is not needed, since details are shown, and the result has been proved by perhaps thousands of people in the course of time. It's actually a common homework assignment in probability courses.  The section could, indeed, be shortened, since one calculation detail is probably sufficient.  Cheers, Doctormatt 17:37, 7 November 2006 (UTC)
 * I just clarified a bit the premise of "best" by stating clearly the "uniform random" choice of opposing die (which is admittedly, perhaps obsessive). I agree that this section is kind of silly; the computations are over-wrought and not particularly illuminating. Perhaps more interesting is that if players choose dice in turn, then the choice of die is irrelevant for the first player. Whatever he chooses, player 2 can always respond by choosing a die to reduce player 1's win chance to 1/3. I have no idea if this is accidental or not. —Preceding unsigned comment added by 64.131.177.208 (talk) 03:10, 21 April 2009 (UTC)


 * I've removed the final section, which is unclear and perhaps inaccurate, for review here. Is it valid? Can it be clarified?
 * "In this case, this increased chance is reflected by comparing the sums the numbers on every face of each die. However, if one face of die B is changed to 100, one 4 on A to 101, one 5 on D to 102 and one 6 on C to 103, the relative strengths of the dice remain unchanged with C as the most likely winner, yet the highest average result will be the B die." hgilbert (talk) 10:07, 6 May 2009 (UTC)

First Sentence
The first sentence is incomprehensible to everyone where I am...any suggestions for re-working? 76.1.192.45 03:47, 18 March 2007 (UTC)


 * How it is incomprehensible? It says what it is and links to meaning of transitive... It is clearly saying that because one dice beats another which beats another, the first does not have to beat the last. —The preceding unsigned comment was added by Dacium (talk • contribs) 04:06, 17 April 2007 (UTC).

"Dettinger's Dice"
I have removed this bollocks. It is completely unsourced, the only Google hits being this article itself (and copies of it). It was added by Dettinger himself, which is a massive conflict of interest, if not actually unethical. These supposed new dice are simply an extension of Efron's formula and are not notable as per Notability. Perhaps I'll "invent" a set of 13 eleven-sided dice, name them after myself, and then add a whole section about them to this article.

Furthermore, all of the assigned probabilities were incorrect; eg Die B will beat Die A precisely 5/6 of the time, not 13/18 as stated. This is precisely why information of this nature should only be added to Wikipedia from peer-reviewed publications not from a webpage you created yourself. 203.206.43.191 (talk) 09:00, 22 February 2010 (UTC)

Just to clarify, Dettinger would be allowed to add a section about Dettinger's dice under the non-controversial edits clause of the Wikipedia conflict of interest guidelines by only making edits that have been agreed to on the talk page. Getting that agreement is another matter, given the lack of citations to reliable sources and the lack of notability, but in general such behavior is allowed if done correctly. Guy Macon 20:54, 20 May 2010 (UTC)

"Dead link"
Trying to replace a dead link to sciencenews.com with a current link to http://singingbanana.com/dice/article.htm Article contains lots of information, video and three player games. If you prefer you can wait for this to be published, in a month or two, but the article is there now. I'll also add, some the information in dice finder link is not accurate. —Preceding unsigned comment added by 90.208.51.25 (talk) 11:23, 10 August 2010 (UTC)


 * When I tested the link, it was not dead. I just rechecked the science news link -- still not dead.  They linked to different content about Nontransitive dice.  The singingbanana.com ink may very well be a good thing to add to the article, but not as a replacement for the sciencenews link.


 * So, let's discuss whether http://www.singingbanana.com/dice/article.htm should be added to the external links on its own merit. Please refer to http://en.wikipedia.org/wiki/Wikipedia:Manual_of_Style_%28linking%29 when discussing this.


 * BTW, are you by any chance the author of the singingbanana.com webpage? If so, please see http://en.wikipedia.org/wiki/Wikipedia:Conflict_of_interest and http://en.wikipedia.org/wiki/Wikipedia:No_original_research for guidelines.  Guy Macon  13:49, 12 August 2010 (UTC)

Theoretical dice with less faces
What are the results for theoretical 'dice' with less than six faces? What is the set of 'dice' with the least faces overall? — Preceding unsigned comment added by 82.22.8.106 (talk) 15:34, 16 August 2011 (UTC)

logically, three. 204.38.188.254 (talk) 15:36, 13 January 2012 (UTC)

File:Intransitive_dice_2.svg
The numbers on each die are those from reading the rows or columns of the unique order-3 magic square (Lo Shu square). AnonMoos (talk) 13:36, 12 November 2011 (UTC)

Freivald's Investigation
The result mentioned was proven theoretically by S. Trybula in 1965 in the paper that started the phenomenon. References; The Paradox of Nontransitive Dice, Richard P. Savage, Jr. The American Mathematical Monthly, Vol. 101, No. 5 (May, 1994), pp. 429-436 http://www.jstor.org/stable/2974903 and; S. Trybula, On the paradox of n random variables, Zastos. Mat. 8 (1965), 143-154. — Preceding unsigned comment added by 90.193.168.4 (talk) 09:40, 18 May 2012 (UTC)

"Even stronger property" in opening section?
The opening section states, after a description of nontransitive dice:


 * It is possible to find sets of dice with the even stronger property that, for each die in the set, there is another die that rolls a higher number than it more than half the time.

Isn't that just a re-statement of the standard properties of transitive dice?

I'm sure I'm missing something, but what? Doesn't saying that "a set of dice is nontransitive if its "rolls a higher number than more than half the time" relation is not transitive" imply exactly that "there is another die that rolls a higher number than [a particular die] more than half the time"?

Is the exception to the stronger property when C rolls more than A exactly half the time? 212.9.31.12 (talk) 11:59, 9 December 2013 (UTC)


 * Yes. A>B, B>C, C=A would be non-transitive but not have the stronger property of A>B, B>C, C>A. PrimeHunter (talk) 13:49, 9 December 2013 (UTC)

Probabilities in the first example
In the first example we are told that:


 * The probability that A rolls a higher number than B, the probability that B rolls higher than C, and the probability that C rolls higher than A are 21/36, 21/36 and 25/36 respectively

Then goes on to say:


 * [One] can always find a die that will beat [another] die with probability 5/9.

Where does the 5/9 come from? Isn't it the case that one can always find a die that will beat another with probability at least 21/36? 212.9.31.12 (talk) 12:04, 9 December 2013 (UTC)


 * You are right. The problem originates from a change [//en.wikipedia.org/w/index.php?title=Nontransitive_dice&diff=575547840&oldid=574109874] away from the example in the image. It's confusing that the text and image don't have the same example so I have reverted [//en.wikipedia.org/w/index.php?title=Nontransitive_dice&diff=585274325&oldid=579350854] to the image example where 5/9 was true. PrimeHunter (talk) 13:38, 9 December 2013 (UTC)


 * Ah, thanks. Those original numbers do make for an easier to understand example. David (talk) 18:02, 10 December 2013 (UTC)

Representation of Efron's dice 'picture'
Is it my browser or does the last die in the picture (magenta color) have a missing number? The top face appears to be blank, is there a reason for this? Thanks 92.28.144.152 (talk) 16:31, 21 September 2019 (UTC)
 * It's intentionally blank to indicate the same chance of 1 and 5. This cannnot be done by showing three sides. The description was wrong. It has sides 5, 5, 5, 1, 1, 1. I have reverted the error.[//en.wikipedia.org/w/index.php?title=Nontransitive_dice&diff=916989534&oldid=916484775] Thanks for reporting there was a problem. PrimeHunter (talk) 18:52, 21 September 2019 (UTC)

Requested move 28 February 2021

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion. 

The result of the move request was: Moved User:Ceyockey ( talk to me ) 01:58, 8 March 2021 (UTC)

Nontransitive dice → Intransitive dice – The term "intransitive" is more commonly used than "nontransitive" according to Google Ngram Viewer, which is likely to make "intransitive" sound more natural. The query "intransitive dice" (in quotes) gets almost twice the Google search hits of "nontransitive dice". I do think the current title is also recognizable, precise, and concise, making this move not urgent. Ilzolende (talk) 20:56, 28 February 2021 (UTC)
 * Support as the main article title is located at Intransitivity. Rreagan007 (talk) 22:30, 28 February 2021 (UTC)

More simply, A normally beats B, B normally beats C, but A does not normally beat C.
From the lede I would like to verify that this is actually a necessary and defining property of intransitive dice, rather than a common property. I don't think this matches with the section Intransitivity but I am out of my depth here.

From the article today: "Buffett’s four dice (call them A, B, C and D) formed a pattern reminiscent of rock-paper-scissors, in which A beats B, B beats C, C beats D and D beats A. Mathematicians say that such a set of dice is 'intransitive.'" —DIYeditor (talk) 22:18, 22 January 2023 (UTC)


 * The intranstivity#cycles article says the equivalent thing. A to B, B to C, but C to A (rather than A to C, which would make it transitive and not a loop). In short, yes it is a requirement and not a dubious statement. 73.19.55.128 (talk) 21:44, 23 January 2023 (UTC)


 * Ahah, indeed, I misread that entirely and thought it was saying that C does not normally beat A! Thanks for double checking. —DIYeditor (talk) 13:27, 24 January 2023 (UTC)

Comment regarding the equivalency of intransitive dice
Regarding this section, please explain:

1) The section is unsourced. Who is the author of this comment? If it is a Wikipedia editor, no matter how knowledgeable, it is original research (WP:OR) 2) Why is the comment made? Please explain the point to a non-specialized reader. More generally, please argue for its value as if someone suggested it to be removed.

Regards, CapnZapp (talk) 08:07, 14 November 2023 (UTC)

Very difficult to parse
The section Comment regarding the equivalency of intransitive dice begins with a ridiculously long sentence that is virtually impossible to comprehend.

I hope someone knowledgeable about this topic can break that sentence into smaller ones that together make sense to the reader.

Perhaps a better idea might be to remove this incomprehensible section entirely.