As part of the preparation for some teaching materials, I was looking something up on the AP Statistics webpage, and I came across the phrase “six-sided number cube,” which is a kind of person-from-Mars description of what in English is called a die. Kinda like if you referred to a cigarette as a “tobacco-filled paper cylinder” or something like that.
I was curious what was going on, as I’d never in my life heard the term “six-sided number cube,” so I googled it and it’s all over, lots of websites where it seems that students are looking for help on their homework. It seems to be something they say in high school statistics classes?
Where did this expression come from? Is it something that some committee put into the AP statistics curriculum and then it just spread from there? I have nothing against the phrase—it’s descriptive and kinda charming, kinda like calling a car a “four-wheeled motorized vehicle”—I’d just never seen it before.
P.S. In comments, Raghu points out that you can just call it a “number cube,” which sounds a lot less science-fictiony. All cubes have six sides (or faces, if you want to use math jargon).
Think of the children!
https://www.quora.com/Why-do-math-textbooks-refer-to-dice-as-number-cubes
Why do math textbooks refer to dice as “number cubes”?
Tracy Cole
Glad you asked! I am aware of this because I am old enough to have been in the classroom when that ruling was handed down to us in the classroom. It was deemed not “PC” to say dice because the dice connotation is gambling and that was not okay for some parents. They didn’t want their kids to be exposed to anything that they considered immoral such as gambling. Number cube disassociates from gambling. It is a neutral term.
“Gaming,” please.
Good God. 🤦🏼♂️
I am old enough to have been in the classroom before this nonsense started.
Well, this is at least better than my guess, which was that “die” was deemed too violent and morbid and traumatic.
That was my guess, too.
I remember a dozen years ago (or so) when the Democrats in Congress objected to the name of a Republican bill to repeal Obamacare. The Republican bill was called something like “Repeal of the Job-Killing Health Care Act”, and the Democrats objected to the terminology…not on the grounds that the title is prejudicial or inaccurate or any of the other things they said about it, but because it included the term “killing” and they argued that this language was too violent.
If this were the reason, sure they could just use “dice” instead.
So if it’s not a six-sided die, then we’d need to refer to it by the specific polyhedron name? Number tetrahedron/octahedron/decahedron/icosahedron etc? (I guess that’s good for learning polyhedron names at least!)
Conchis:
A “cube” has 6 faces so that’s that. If you wanted a polyhedron with equal probability of all faces coming up, it just needs to have some basic symmetry; it does not need to be a regular polyhedron. Any even number of sides is easy to imagine: just start with a regular (n/2)-gon on the plane, then put vertices an equal distance above and below the centroid, then connect the vertices and you’ll have n identical faces.
n-2 or n/2 ?
Typo fixed; thanks.
Lmao christians
This is so ridiculous, but reminds me that I’m israel, in the lower classes at least, we were taught to use a plus sign that didn’t have the lower half of the vertical line. Years later I learned that that was because they didn’t want jewish kids to be drawing crosses.
*in
Sounds like one of them “tortured phrases”.
https://retractionwatch.com/2021/07/19/tortured-phrases-lost-in-translation-sleuths-find-even-more-problems-at-journal-that-just-flagged-400-papers/
“I have nothing against the phrase” — what about the redundancy that cube can’t have any number of sides other than six? Though I suppose that “six sided number polyhedron” is a mouthful.
I hope the answer posted by Carlos Ungil isn’t correct, since that’s just sad. My guess was going to be that too many students didn’t have the language skills to parse “die” not being a verb.
Raghu:
Actually, as a mathematician I’d say that a cube doesn’t have “sides” at all! It has eight vertices, twelve edges, and six faces.
I was thinking about this potential objection, actually, but I decided that referring to the “sides” of a cube was self-explanatory enough — no one would think it means a vertex or an edge. (Hitting the side of a barn is easy, after all…)
“six-faced number cube” sounds like a name of old-forgotten Eldritch God.
Raghu said, “I hope the answer posted by Carlos Ungil isn’t correct, since that’s just sad.”
I’m pretty sure that his answer is indeed correct — I’ve worked with secondary math teachers quite a bit, and have often heard them say that parents object to teachers using dice to help students understand probability since taking about dice is talking about gambling, which the parents consider immoral. I remember one local high school teacher in particular who was very interested in teaching probability and statistics well, and had taken several courses from me. In fact, she sometimes came after school to probability and statistics talks at the university. At one of these, a senior statistician started up a conversation with her before the talk. When he asked where she taught, she said (truthfully) at a Baptist school. The statistician asked whether she was allowed to use dice when teaching in a Baptist school. She replied cheerfully, “We use spinners instead.”
With the advent of Dungeons and Dragons, die with various numbers of sides became common. Thus the need to distinguish for the modern child.
Jim:
Sure, but then why not just “six-sided die”?
In short, “die” is fine (standard English usage), “six-sided die” is fine (clarification and precision), and “number cube” is fine (pure description). The alienating term is “six-sided number cube.” I almost wonder if it appealed to the AP board because it sounds so technical. Just one more term for kids to add to the list of things to memorize in the end-of-chapter review.
I wonder if some state in the US decided that textbooks can’t talk about dice, because dice lead to gambling (or Satanism!) So some textbook publisher did a find-and-replace.
Sean:
Teaching statistics without reference to gambling would be like teaching the Civil War without reference to guns.
Andrew: psh, don’t encourage them! Teaching statistics without reference to gambling is also like teaching biology without reference to evolution, or US history without reference to the construction of race to let rich settlers boss poor whites and gang up on First Nations and Latins, and we know that many people would rather give up teaching biology or history than expose children to those pieces of knowledge.
See my response above about the teacher at a Baptist high school. In fact, parents of public school students often object to using dice in math classes, so there is a lot of pressure on schools not to use dice in teaching. (But the spinner idea is often used instead.) I think many of the textbooks refer to spinners rather than dice, since many schools have had a lot of parental complaints about use of dice in schools.
Martha:
I guess that makes sense. But then it’s not clear how they can get away with assigning homeworks about “number cubes,” as these are just . . . dice!
Andrew,
>Teaching statistics without reference to gambling would be like teaching the Civil War without reference to guns.
you mean without reference to powder-loaded elongated bullet throwers?
I’ve taught probability courses where, to my great surprise, I found out that not all students knew that a standard deck of cards had 4 colours, 13 cards of each colour, and so 52 cards in total.
So what I thought was an easy in-class assignment suddenly became a very exotic one…
I can well imagine some textbook writer taking this idea, and running along with it to the point that *any* phrase which might possibly not be universally culturally accepted needs some form of alternative explanation.
Four colors? Is that actually standard where you are? I’ve always liked four-color decks–when you’re playing poker, it’s nice to see which suits are on the board at just a glance. From the far end of the table, I sometimes have a hard time distinguishing spades from clubs.
But either out of pure traditionalism or for some other reason, four-color decks are quite the rarity here in the U.S. Red and black is still very much the standard.
That was a translation error — I was looking for the word “suits”, in Dutch they are called “colours”.
In Hebrew (perhaps other languages as well) the same word is used for “cube” and “die”. Referring to dice as cubes is not uncommon in Israeli English.
In Russian, “dice” is translated into “bone”, as historically cow bones (Astragal) were used for gambling. “Small cube” can also be used.
Is there a new word for dreidel, too? I am planning to totally clean out my daughter’s Chanukah gelt this year.
I think that it is part of the same trend that caused “Dead End” signs to be replaced with “No Outlet”. A teacher could talk about the probability of a die outcome, and next a student commits suicide or shoots up the school. Or maybe has to see the school counselor. Think of the lawsuits. We have to protect our precious snowflakes from anything morbid.
In mathematics one can for clarity refer to combinatorial 3-cube or a metrical 4-cube. A physical truncated regular square pyramid will not when “tossed” land on each of its 6 faces with equal frequency.
My totally-unsupported hypothesis is that it’s because of the die/dice irregular plural, which may or may not be confusing to children but certainly seems inelegant, since “die” has a more-common verb sense as well as other noun uses (tool-and-die) in contrast to entirely unambiguous “dice”. I’ve seen some board-game players prefer dice as both singular and plural for this reason, and it may be that textbook writers chose number cube as a different solution.
The Hamaker 10 sided die https://onlinelibrary.wiley.com/doi/10.1111/j.1467-9574.1948.tb00359.x
Never say die!
You may think that a “six-sided number cube” is specific enough, but then note that those numbers could be anything, including 42, √2, π, ℇ, and of course complex numbers.
The textbooks should be corrected to use something like “six-sided number cube with distinct integers between 1 and 6”.
Tamás:
No, once you say “cube,” you don’t need to say “six-sided”! But good point regarding pi etc. For that matter, every face of the cube could just have a 0 on it.
Old-school gaming grognards should be aware of the ‘averaging die’, which is labeled ‘2, 3, 3, 4, 4, 5’. You swap it in when you want to push results closer to the mean. I.E., and experienced combat troop may use it, while a newly recruited unit may use the more chaotic standard die.
“Six sided number cube” is the price native English speakers pay for having English as the default international academic language. The overly descriptive phrase likely owes to a combination of the fact that ‘cube’ and ‘die’ are interchangable in some languages (e.g. Hebrew and German (i think) and probably others), that die/dice has an irregular singular-plural combination, and that the very different other meaning of ‘die’ is learned earlier by non-native speakers.
Fafa:
Again, though, no need for the “six-sided” thing. Just say “number cube.” There is no way that the AP curriculum is doing this for the benefit of non-English speakers! I think that adding “six-sided” is just an example of bureaucratic hyper-precision, and it sounds like the avoidance of the word “die” is coming from some pressure from anti-gambling groups.
After some light googling, it appears that “die” is not always equal to “number cube”. The former have dots, while the latter can have numerals. So number cubes were more often used in early elementary lessons, and it seems like the phrase may have filtered up the grade levels.
You might be right about the redundant “six-sided” addition being bureaucratic BS, but also might have been added to make the math games and problems easier by explicitly prompting with the ‘divide by six’. I also wonder whether educational product catalogs had some influence. Perhaps marketing BS instead of purely Ed School BS.
Based on Google Books, it looks like “six sided number cube” started slowly in the late 80s/early 90s in Math books.
Perhaps everywhere it says “six-sided number cube” previously said “six-sided die/dice”. Then they substituted “number cube” without dropping the now redundant “six-sided”.
Dumb question: As Andrew has pointed out you can make a fair n-sided die for any even n>2. (His method works for n>4 and tetrahedra handle 4.) Is there a theorem that it’s impossible to make one for any odd n? Or a counterexample?
Look at:
Grünbaum, B. “On Polyhedra in E^3 Having All Faces Congruent.” Bull. Research Council Israel 8F, 215-218, 1960
If you aren’t restricted to polyhedra you can have any number of faces you want, although “my” method/invention probably has a practical limit around 20:
Make a football with N identical curved faces.
A die may not be stable on a face on which it lands:
https://en.wikipedia.org/wiki/Monostatic_polytope
The shape proposed by Phil is obviously stable on every “face” given its symmetry.
If you don’t need all sides to be congruent, the intermediate value theorem from freshman calculus can point you to a fair die for any n > 4: Start with a regular (n-1)-gon in the plane and pick a single point above the center of the plane to be the nth vertex. Now raise and lower that added vertex. When it’s really close to the plane the flat side has probability ~1/2, and when the added vertex is really high above the plane the flat side has probability ~0, so somewhere in-between it must have probability 1/n.
Again, that’s if you don’t need the sides to be congruent. Apparently that Grunbaum paper will help you if you do need congruent sides. :)
> somewhere in-between it must have probability 1/n
What does it mean that it has probability 1/n? I’d say that the probability is not really a property of the side because it depends not only on the geometry of the “number generating polyhedron” but also on the details of the throw. Only when there is enough symmetry one can justify that all the probabilities are equal unconditionally with minimal assumptions.
There are many subtle issues involved in designing “fair dice.” Here is one place to get started:
https://statweb.stanford.edu/~cgates/PERSI/papers/fairdice.pdf
The last section there proposes the same construction method given by Josh Rushton. However, as they note, this “fair by continuity” argument depends on the mechanical properties of the die and the table. Fairness is not a geometric property when it’s not a consequence of symmetry alone.
Studying dice, polyhedra and more general surfaces, and “rolling” shapes leads to interesting objects and a variety of new mathematical questions, ideas, and “art.”
https://en.m.wikipedia.org/wiki/Gömböc
An example of a number cube: 2^3.
At least “six-sided” clarifies that it’s an object.
Since coins can be used for gambling too, do they use two-sided face disks?
We should thank gamblers for the notion of probability (see Ian Hacking’s book about The History of Probability; while some critics have disputed some of his claims it’s an interesting read). Alea means die in latin, or game of chance, or risk.