79

Many counting problems start with the assumption that we have a certain number of men and women or a certain number of couples, with the assumption (often unstated) being that that gender is binary (only men or women) and couples are only heterosexual. (See for example this or this problem or think for example of Hall's Marriage Lemma.) Can anyone suggest a good replacement for the concept of couples in these sorts of math problems that better reflects the societal norms of the latest generation of students?

As an example, I'll routinely replace "men and women" with "undergrads" and "grad students" or something similar depending on the class make up. I'm currently racking my brain for two distinct sets of objects S1 and S2, where we might naturally think of pairing objects in S1 to objects in S2, and where there is one most natural pairing (i.e. replacing men and women, where we compare any heterogeneous couple to pairing men and their wives). Anything I've thought of makes for a ridiculously long word problem. Any suggestions appreciated.

Mod note: If you answer the question please answer the question

What's a replacement for "married couples" in combinatorics problems?

Please, do not provided answers that say there is no need to do so. This is beyond the scope of this question.

Community
  • 1
Mathprof
  • 1,205
  • 1
  • 8
  • 12
  • Comments are not for extended discussion; this conversation has been moved to chat. – quid Mar 08 '18 at 17:59
  • 1
    I removed all comments. I first tried to preserve some, but which one's I'd select to stay would likely create more controversy. Whoever is interested in the discussion can visit the room linked to above. – quid Mar 08 '18 at 18:05
  • Customer vs. product? – user541686 Mar 08 '18 at 23:08
  • 3
    I removed a re-post of a comment that was moved to chat. I have nothing against the comment itself (indeed it had made the shortlist of those I considered leaving around), but if I allow one repost, I'll have to allow the next too, and I am sure y'all can see where this would be going. Please do not repost comments. (Otherwise I'll lock the comment thread.) – quid Mar 09 '18 at 09:19

17 Answers17

91

I've been using "pets" and "owners" (as in: possible pet-shelter adoptees) in recent years.

Daniel R. Collins
  • 25,342
  • 70
  • 121
  • 12
    I like this one. The sets are disjoint and the objects are still clearly distinct. (I suppose one has to specify owners have one pet, but that's not so bad.) I'll wait to accept an answer for a bit because I think it's nice to have several options, but thanks! – Mathprof Mar 02 '18 at 23:43
  • 2
    @Mathprof: That's true; and even previously, I would always specify that we weren't in a locale that allowed polygamy. – Daniel R. Collins Mar 03 '18 at 02:23
  • 2
    To add a bit of color to the pet example, you could make the pets really big, like St. Bernards or English Mastiffs, and add a remark like "Obviously each person can have at most one pet" – idmercer Mar 03 '18 at 15:06
  • 16
    But the relationship is clearly inverted when you compare dogs and cats... – curiousdannii Mar 03 '18 at 23:47
  • 35
    @Mathprof, a variant on this which clearly forces pairing would be horses and riders. – Peter Taylor Mar 04 '18 at 20:55
  • 2
    This works as long as you don't need a 1:1 pairing. – Pharap Mar 05 '18 at 10:16
  • 8
    This a great political solution, although the number of owners with multiple pets is probably higher than the number of gay marriages right now, so technically this is MORE ambiguous. – Feathercrown Mar 06 '18 at 17:17
  • Married couples provide for a recursive relationship, so replacing this relation with, say pets and owners doesn't solve the problem that most examples dealing with relations assume also that there are parents and children. You could try to replace humans with some animal and still retain the recursive property. – Reebok Mar 08 '18 at 12:50
  • You can't say "owners" you have to say "guardians"... – Sam Mar 09 '18 at 16:34
  • 5
    our class used "wands" and "wizards". The wizard prefers certain wands, and, as we know, "the wand chooses the wizard"! – bob Mar 11 '18 at 22:40
  • In another century this will probably also be politically incorrect. –  Jan 16 '23 at 04:47
84

In the stable marriage problem, you can introduce the problem as it is. But then you ask your students how things change if you assume there are not only heterosexual but also gay and lesbian people (assuming that a heterosexual person will never marry a person of the same gender, and a gay or lesbian person will never marry a person of the opposite gender). It's actually an interesting problem that requires a completely different solution.

Add bisexual people, and your student's heads will explode - hopefully not because of the bisexuality, but because suddenly the problem gets a lot lot harder. On the positive side, people being transgender doesn't affect the problem at all.

gnasher729
  • 1,197
  • 7
  • 5
  • 27
    +1 for the inclusive answer that allows us to keep posing the same problems when needed to teach combinatorics. – Pere Mar 03 '18 at 19:09
  • 18
    You really propose to open this can of worms in a math class? – Robert Harvey Mar 05 '18 at 22:19
  • 21
    @RobertHarvey, it's a math problem. And actually an extremely interesting one. I dare you to solve it. (This has nothing to do with your social views at all, just standard graph theory and combinatorics.) – Wildcard Mar 06 '18 at 02:42
  • 1
    @RobertHarvey Unless your students are denying that homosexual relationships actually exist at all, I don't see the problem. Maths isn't about setting social rules. It's hardly the professor's fault that they exist. – user253751 Mar 08 '18 at 22:13
  • 14
    @immibis: There are many, many other examples to choose from that are far less controversial. The only reason I can think of to choose this particular one would be to make a political point. Respectfully, I don't think that aligns with the primary purpose of a math class. – Robert Harvey Mar 08 '18 at 22:26
  • 1
    @RobertHarvey: Do you have any specific alternative suggestions that start out with bipartite graphs and then move to more general graphs, as this answer does? – Kevin Mar 09 '18 at 03:22
  • 1
    It's not really a "political point", other than taking the wind out of the sails of any "politically correct" complainer. – gnasher729 Mar 12 '18 at 00:24
  • If this is meant to make a political point, I don't know what it is. Stable matchings need not exist in what is usually called the "roommate problem." – Michael Greinecker Sep 24 '18 at 17:09
  • 1
    -1 This does not answer the question given. (a) OP specifically asks for a replacement to married couples. (b) OP specifically asks for an application of bipartite matching (not for something that leads to changing the problem to one of non-bipartition). – Daniel R. Collins Apr 25 '19 at 15:20
68

A few possibilities off the top of my head:

  • Students and chairs. How many ways are there for $n$ students to sit in $k$ chairs. The game of musical chairs might be fun to play around with. One can also consider natural restrictions, such as myopic students who need to sit near the front.
  • Replace "men" and "women" with faculty from different departments. For example, you might need a curriculum committee made up of mathematicians and physicists.
  • In the senate, there are two senators from each state. How many ways are there of making a committee of 6 senators, no two of which are from the same state? This really only works in the US, but I am sure that there are similar phrasings that could work elsewhere. One could also consider party affiliation here, which could lead to more interesting problems.
  • True story: My daughter, who is in kindergarten, has a 5th grade "reading buddy." If there are $n$ kindergarteners and $k$ 5th graders, how many ways are their of pairing up students? If you want to appeal to an older audience, maybe we are pairing up tutors with students for individual instruction?
  • This is totally not what you want, but still an interesting problem: how many injective functions exist from a set with $n$ elements to a set with $k$ elements? How about surjective functions? How does this change with $n$ and $k$?
  • This question on MSE uses 9th and 10th graders. "Juniors" and "Seniors" could easily replace these if you wanted a college aged example.
Xander Henderson
  • 7,801
  • 1
  • 20
  • 57
  • 8
    +1 I like the students and chairs example. For variety, could also be cars and parking spots. Or ships and berths. Or trucks and loading docks. Or members of a sports team and numbered jerseys. – Dan Christensen Mar 05 '18 at 14:32
  • 4
    @DanChristensen Oh! I like sportsball players and jerseys! That is actually something kind of fun there, too, since teams often retire the numbers of famous players, which means that entirely arbitrary restrictions can be justified (for equally arbitrary reasons). – Xander Henderson Mar 05 '18 at 14:42
  • 2
    I've seen the students and chairs one in the wild and it's been quite easy to understand. – Cubic Mar 06 '18 at 11:59
  • 1
    And at the lower levels, you can put a bunch of chairs in the front of the room and have students come and sit in them, and demonstrate the problem in the classroom. At least until everything goes online. – Guy Schalnat Mar 06 '18 at 17:12
  • 1
    @DanChristensen As stable marriages have been discussed, I wouldn't understand a situation where chairs have preference over students to sit on them. – JiK Mar 09 '18 at 10:31
  • @GuySchalnat This is an oddly prescient comment to read in 2021 ... – Azor Ahai -him- Aug 04 '21 at 19:45
37

The issue is not making problems about heterosexual married couples. The issues are:

  • Implicitly making the assumption that all married couples are heterosexual.
  • Making problems about heterosexual marriages but not about other kinds of couples.

Both points are unengaging for people following other types of marriage, but they can easily be solved while keeping the problem clear and interesting. We just need to make all assumptions explicit (thus acknowledging that they aren't universal while making the statement unambiguous) and making a wider array of problems (which can be useful to teach different tools).

An example

There are 100 single people in a village. How many different marriages can be performed among them in the following situations? Please beware that situations depicted are just simplifications to keep this problem easy and solvable.

2018 situation

All people are free to marry any other villager of their choice.

1990 situation

There are 55 men an 45 women in the village. A valid marriage must include a man and a woman.

1940 situation (somewhere)

There are 30 white men, 28 white women, 21 black men and 21 black women in the village. Local laws only allow a person to marry another person of the same race and different sex.

Yanomamo kinship system

There are two clans in the village: clan A (30 men, 28 women) and clan B (21 men and 21 women). People are only allowed to marry somebody of different sex and different clan.

Caste system

There are four castes in the village: caste A (15 men, 15 women), caste B (15 men, 13 women), caste C (10 men, 11 women) and caste D (11 men, 10 women), being A the highest caste and D the lowest caste. A man is allowed to marry a woman of his caste or any lower caste, while a woman is allowed to marry a man of her caste or any higher caste.

Arthur C. Clarke's Rendezvous with Rama system

There are 100 people in the village. Any arbitrary group of people of any size can form a marriage if all they agree.

End note

Please notice that the examples don't endorse any marriage system, and make explicit all assumptions. For example, the 1940 system problem with explicit assumptions is just historical, but the same problem with implicit assumptions would be plainly racist.

Community
  • 1
Pere
  • 1,017
  • 7
  • 9
  • 4
    Definitely a more interesting and fun exercise in your proposed cases. – Carl Witthoft Mar 05 '18 at 20:49
  • 2
    I like this answer very much. It is not the problem itself, it is making the implicit assumption that of course every couple is heterosexual. In mathematics, we often make simplified assumptions - even more in natural sciences (point-like masses, no friction etc.) - that make a problem more accessible. There is nothing inherently bad about this. Just state that it is part of the assumption, and everything is fine. – Thern Mar 06 '18 at 08:43
  • 8
    “Both points are unengaging for people following other types of marriage” What? Does that mean I (not a pet owner) should not find "engaging" a problem that talks about pets? This doesn't make sense. Analogies in math problems are there to state a problem to solve, not to "engage" people. – Andrea Lazzarotto Mar 06 '18 at 17:18
  • 2
    @AndreaLazzarotto Motivation and engagement is an important point of any exercise - in fact, of any learning activity. Relating statements to elements interesting for the learner improves motivation. For example, we teach children to count by counting sweets and pets, not by counting bolts and nuts, or bricks and beams, although both analogies can state the same problem. – Pere Mar 06 '18 at 18:18
  • 4
    It is still painful to people of color, who continue to be discriminated against, to introduce problems like the 1940 version, just for the sake of the math. I advise against problems like that and the caste one. – Sue VanHattum Mar 08 '18 at 00:37
  • 1
    "All people are free to marry any other villager of their choice". True, but most restrict themselves voluntarily to about 50% of the population. You'd try to take advantage of this to make the solution easier. – gnasher729 Mar 08 '18 at 23:45
  • 1
    The Yanomamo kinship system is easy to solve: It's just two instances of the classic stable marriage problem, one with men of clan A and women of clan B, and the other is men of clan B and women of clan A. – gnasher729 Mar 08 '18 at 23:47
  • -1: Does not answer the question (replacement for "married couples"). Note that the real goal is to find an application for the mathematics of bipartite matching, not to explore the permutations of marriage itself. – Daniel R. Collins Apr 25 '19 at 15:24
34

Try objects that often occur in pairs but are distinct from each other: forks and spoons (or forks and knives), left and right shoes, salt and pepper shakers, and so on (where each fork has an obvious partner spoon, perhaps sharing the same color or design, and so on).

JRN
  • 10,796
  • 2
  • 37
  • 80
28

When I taught a class about the stable marriage problem last week, I replaced "men" and "women" with "medical students" and "hospitals": the classical instance in which the Gale-Shapley algorithm is used in real life.

In addition to the gender issues already mentioned, this has the benefits that:

  • We avoid envisioning a dystopian future where everyone's preferences are fed into a computer and the computer tells everyone who they should marry. That's just kind of weird.
  • The very similar-looking "men propose" and "women propose" versions of the algorithm now get very different flavor text to distinguish them. In one version, medical students all travel to their next top hospital, and each hospital tells all but one of the visiting students to go home, game-show style. In the other version, hospitals send offers to their favorite student, and each student rejects all but one offer, provisionally accepting the best.

I also took this opportunity to give the participants of my examples the names of characters and hospitals from "House" and "Scrubs". This amused me but appears to have flown over the heads of today's undergrads. Kids these days.

Misha Lavrov
  • 821
  • 6
  • 15
  • 3
    That’s great! I couldn’t think of a non-gendered version where each of the two groups have preferences for choosing members of the other group, as in the stable marriage problem. This fits the bill perfectly. – pjs36 Mar 06 '18 at 04:32
  • 5
    I'm okay with the fact that no one got my Spice Girls reference last week, but if they don't get Scrubs and House, we are all doomed! – Xander Henderson Mar 06 '18 at 14:41
  • 1
    This answer is good. The original problem of pairing couples is somehow artificial anyway. The context of matching in employment or admissions is not as it's done widely in practice. – Dan Fox Dec 28 '21 at 10:34
14

  1. Protons and electrons (form hydrogen atoms)

    • Or cations and anions (form salts), e.g. Na+ and Cl-

  2. Pens and pen-caps

    • Bottles and bottle caps, etc.

  3. Textbooks (for the course being taught) and students

  4. Light bulbs and light sockets

  5. Power cords and electrical outlet sockets

  6. Cars and parking spaces

  7. Seats and attendees, e.g. students in the classroom and classroom seats

  8. Mimes and invisible boxes

  9. Numerators and denominators

  10. Boxes of cereal and toys-that-go-at-the-bottom-of-boxes-of-cereal

I'm currently racking my brain for two distinct sets of objects S1 and S2, where we might naturally think of pairing objects in S1 to objects in S2, and where there is one most natural pairing

Just for fun, $\TeX$'d up a table:

$$ {\begin{array}{c|c|c|c} & \textbf{S}_\textbf{1} & \textbf{S}_\textbf{2} & \textbf{Natural Pairing} \\ \hline \small{1} & \text{proton} & \text{electron} & \text{Hydrogen} \\ \hline \small{2} & \text{anion} & \text{cation} & \text{salt} \\ \hline \small{3} & \text{pen} & \text{pen cap} & \text{capped pen} \\ \hline \small{4} & \text{bottle} & \text{bottle cap} & \text{capped bottle} \\ \hline \small{5} & \text{student} & \text{textbook} & \text{prepared student} \\ \hline \small{6} & \text{student} & \text{seat} & \text{seated student} \\ \hline \small{7} & \text{power cord} & \text{power outlet} & \text{plugged-in cord} \\ \hline \small{8} & \text{light bulb} & \text{light socket} & \text{light} \\ \hline \small{9} & \text{car} & \text{parking spot} & \text{parked car} \\ \hline \small{10} & \text{mime} & \text{invisible box} & \text{boxed mime} \\ \hline \small{11} & \text{screw} & \text{screw hole} & \text{inserted screw} \\ \hline \small{12} & \text{left glove} & \text{right glove} & \text{gloves} \\ \hline \small{13} & \text{numerator} & \text{denominator} & \text{fraction} \\ \hline \small{14} & \text{Na}^{+} & \text{Cl}^{-} & \text{NaCl} \\ \hline \small{15} & \text{thing} & \text{name} & \text{named thing} \\ \hline \small{16} & x & y & \left(x,~y\right) \\ \hline \small{17} & \text{longitude} & \text{latitude} & \begin{array}{c} \text{possible site of} \\ \text{buried treasure} \end{array} \end{array}}_{\huge{.}} $$

Nat
  • 377
  • 2
  • 10
  • 19
    Well, to nitpick: fundamental particles like protons and electrons are indistinguishable, so if you ask "How many distinct Hydrogen atoms can be made by pairing from a group of 6 protons and 8 electrons?" the answer might be 1. :D – N Unnikrishnan Mar 06 '18 at 10:34
  • @NUnnikrishnan Hah just tell the student that even fundamental particles are distinguishable by their other properties, e.g. location. This is one problem where the implicit sub-typing of objects is important! – Nat Mar 07 '18 at 00:38
  • 3
    I'm a chemist. I agree with N. U. and would avoid this type of example. teaching wrong thing and bringing a confusion in, if particles are indistinguishable. – guest Mar 07 '18 at 04:28
  • 1
    @guest Naw, there's nothing wrong about it; pretty sure that they were just cracking a joke by subtly misapplying the concept of indistinguishability. – Nat Mar 07 '18 at 05:01
  • @Nat But once they are paired their location changes (specifically they must now be adjacent to each other). – user253751 Mar 08 '18 at 22:15
  • @immibis I dunno; I'm mostly just a tad worried that this might not be a joke. I mean, for starters, the concept of indistinguishably is misrepresented above; it's a property of our models as opposed to nature, much like a model that ignores isotope differences in chemistry. We may as well say that all mimes and invisible boxes are effectively indistinguishable, such that there's only one combination. But then, the problems we're talking about aren't even about "distinct" combinations in the first place, in the physical sense, but rather in the combinatoric sense. I hope it's a joke. =/ – Nat Mar 09 '18 at 02:05
  • The internet gets me on junk like this. Like, I really think @NUnnikrishnan's comment was funny in a silly sense, and I love jokes! But then people start talking about it as though it were serious, and then it's like... what do ya even make of that? – Nat Mar 09 '18 at 02:07
  • 1
    Further to car//parking spot, any vehicle generally and its storage location: aircraft and gates on the ramp, boats and piers, trucks and loading docks, cars and fuel pumps/charging ports, etc. – Nij Mar 09 '18 at 04:35
  • @Nat Except that we have available accepted models that incorporate isotope differences, while for the case in question, there are NO established physical models (am I mistaken?) that incorporates a way of distinguishing truly elementary indistinguishable particles. Yes, the original comment was partly in a lighter vein and partly serious. I am not a physicist, after all. – N Unnikrishnan Mar 13 '18 at 07:26
  • @NUnnikrishnan The issue's that our lack of a model at whatever the current level of understanding is isn't a physically meaningful thing. For example, it'd have been inappropriate for folks not too long ago to claim that atoms of an element are fundamentally indistinguishable (though many probably thought that). Likewise, there's no reason to teach students to think of protons or electrons as fundamentally indistinguishable; that'd be a misconception, just as the isotope thing would've been, and prior elemental models before that (e.g., the earth/wind/fire/water stuff). – Nat Mar 13 '18 at 07:51
  • For an example of someone who mistook their own ignorance for truth, there's Nikola Tesla: "Tesla disagreed with the theory of atoms being composed of smaller subatomic particles, stating there was no such thing as an electron creating an electric charge. [...] Tesla believed that atoms are immutable—they could not change state or be split in any way.". Obviously, we know that that's very wrong, but more over, he never had any reason to believe that in the first place; it was a misconception. – Nat Mar 13 '18 at 08:09
  • Sorry to resurrect an old comment thread, but electrons are not "distinguishable by their other properties, e.g. location". The quantum mechanical wave function for a multi-electron system is required to be antisymmetric under interchange of any pair of electrons, which means the indistinguishability is built in to the theory. Many familiar properties of matter, including the shell structure of atoms, the electronic properties of conductors, semiconductors, and insulators, and the structure of atomic nuclei stem from this. – Will Orrick Dec 27 '21 at 16:01
  • Likewise for systems of bosonic particles, the quantum mechanical wave function is required to be symmetric under the interchange of any two particles. Bose-Einstein condensation occurs because of this. These systems would behave in a vastly different manner of the the bosons were distinguishable. – Will Orrick Dec 27 '21 at 16:03
12

An idea I find interesting is to use abstract objects: "given X squares and Y circles, in how many ways it's possible to pair one circle with one square?"

If the students are at kindergarten or primary school level, maybe you can give the children squares and circles made with paper, plastic or wood, each with a different color, so they can try these combinations with the objects:

enter image description here

Personally I find this way easier and less distracting than using real life examples. Of course in some exercises you will want the student to be able to examine a real life situation and transform it in a mathematical problem, so in this case abstract objects will not work so well (and there are many good answers with interesting suggestions on meaningful pairings), but when you want to focus only on the mathematics involved, this can be useful.

However, I recommend you to avoid using topics which attract controversial and hot discussions (as gender/sexuality issues) as examples in a math class, as some answers suggested. Firstly, these kind of topics can be confusing to most small children and very distracting for older students, who possibly already have political opinions. I would be very distracted if I feel that the teacher is trying to punch "these damned X" inside a math problem - specially if I am one of these "X"! Also bringing a political topic can even reduce the ability of properly solve mathematical problems! Besides that, parents with a different political leaning may not like to see a teacher injecting political topics on non-political subjects, and if the example implicitly supposes a side is the correct one, this only makes things worse. These parents can cause a problem to you, as trying to process you or the school you teach, for example (if they would be right or wrong on doing that is another question, but I think you want to avoid problems). Basically, just be sure that your replacement is socially acceptable - not only to you, but to other people too!

Brian Hellekin
  • 221
  • 1
  • 5
10

I like several of the answers which found clever sets that avoid assumptions about genders and sexes. Just to provide an alternative, what about keeping the assumption about sexes, but changing species and turning it into a free biology lesson as well. Many birds mate for life, such as bald eagles.

It would also open up the door for some other interesting math problems. Emperor Penguins are serial monogomists. They pick a mate and stay with them for a whole year, but rarely pick the same mate year after year (15%), so you could put together some interesting combination problems with that! They also have the neat fact that the male takes care of the egg, balancing on his feet as the males huddle in brutal cold winds. The females return later, when the eggs are hatching. This gives the penguin chicks a head start over other creatures which weren't able to manage this brutal feet. Mind you that might not help the math any, but I think it's freaking awesome.

You might also be able to play some really interesting combinatorics with bees. Bees have fascinating caste systems, especially regarding how new queens are dealt with. (The queen has to make sure nobody usurups her, but if the queen dies for any reason, the hive needs to vote a new queen into office by feeding her royal jelly).

There's also the "other" sex systems. We are used to XX/XY, because that's what humans have. Many of the arguments regarding not accepting the other genders or sexuality stem around the universality of this system. Well it's not the only one out there. Bees use X0/XX. Many lizards us ZZ/ZW. A quick biology lesson might start minds turning. Evolution has had several goes at this concept. While they, themselves, are not going to be straightforward substitutions into combinations problems, they are a fascinating chance to learn something other than math in math class.

Cort Ammon
  • 1,225
  • 6
  • 8
  • Except that even mate-for-life birds get a lot of action on the side. DNA studies have proved this. – Carl Witthoft Mar 05 '18 at 20:50
  • @CarlWitthoft Which is not really all that different from marriage of humans (I don't think I need DNA studies to prove this =) ) Apparently for Emperor penguins the infidelity rate is 15%. – Cort Ammon Mar 05 '18 at 21:04
  • 1
    Interestingly though, I'd stay away from the penguin one. I think people have found penguins can be gay, so it's not as perfect as it seems. – Feathercrown Mar 06 '18 at 17:29
  • @Feathercrown The point isn't to be accurate, the point is to be inoffensive. People who object to heteronormative pairings with humans tend not to do so when talking about the mating of animals, so the solution still works. – Please stop being evil Mar 08 '18 at 00:11
  • 1
    It might be more than penguins that 1. mate for life, and yet 2. can form same-sex pair-bonds. For a children's story that comes from a real-life incident, check out And Tango Makes Three. – Sue VanHattum Mar 08 '18 at 00:44
  • @thedarkwanderer True, but there are slightly better examples on this question because of this answer's drawbacks. – Feathercrown Mar 08 '18 at 05:46
8

I'm just thinking out loud here, but one possible option is to keep it as men and women, but have it be about ballroom dancing, or figure skating, instead of dating or marriage.

That way, it might have fewer undertones of "everybody in the world needs to be married off to someone of the opposite sex", and might have more connotations of "this is a specialized hobby where we want to pair men with women for aesthetic reasons, but we're not claiming that the world as a whole needs to be this way."

Such examples could begin with "In a ballroom dancing club, there are 12 women and 10 men..." That wording, at least ideally, leaves open the possibility that the larger world could have nonbinary people. It's more like "this particular club has 12 women and 10 men."

idmercer
  • 345
  • 1
  • 5
  • 7
    I don't think that this answers the question. The difficulty is that gendered language is a problem in the first place, in that it implicitly ignores the existence of gay folk, trans folk, ace folk, and so on. The goal is to come up with problem statements that don't exclude people. – Xander Henderson Mar 03 '18 at 04:26
  • 4
    @XanderHenderson: What's the problem with a gay guy doing figure skating with a transwoman? How does this answr exclude anyone? – Eric Duminil Mar 03 '18 at 14:43
  • 15
    @EricDuminil Assuming good faith, consider this problem: "We have 4 black people and 6 white people. How many ways can we sort them into married couples such that no black person marries a white person?" You would never use this example, and it is somewhat nightmarish to even think about presenting it. The reason is that it is exclusionary in a way that shows a lack of understanding about current society. If I changed it to "How many ways can you create pairs of penpals such that no white person writes to a black person," I would have missed the point: why exclude actions based on race at all? – Chris Cunningham Mar 03 '18 at 15:49
  • 4
    @ChrisCunningham: My question was a genuine one. I understand the problem in wanting to only marry men and women together. I don't see any problem in asking men and women (who both could be trans- or cis-) to skate together. Also, your point with black and white people isn't a good example IMHO. Allowing only black or white couples would be the same thing as forcing people to marry the same gender. Anyway, the "students and chairs" is a much better, less controversial, example. – Eric Duminil Mar 03 '18 at 16:56
  • 2
    The problem in asking men and women to skate together is exactly analogous to the problem of asking white and black people to write to each other; I worked pretty hard on the analogy. Cheers and your photography looks beautiful! – Chris Cunningham Mar 03 '18 at 19:43
  • @ChrisCunningham: I completely agree that I would never use an example where no black person can marry a white person.

    But things can be less obvious. Consider "There are six black people and four white people. We must choose a black person to play Magic Johnson and a white person to play Larry Bird. In how many ways can this be done?"

    I probably wouldn't use this example either: it highlights people's racial background when it's perhaps not necessary. There are other examples we could use instead. But I'm not sure it would be offensive; it's just a toy scenario that mentions race.

     – idmercer Mar 03 '18 at 21:16
  • 3
    @idmercer Why must Magic Johnson be played by a black actor and Larry Bird by a white actor? Have we learned nothing from Hamilton? – Xander Henderson Mar 04 '18 at 02:40
  • 1
    I would definitely never use an example where black and white people cannot marry. I would probably not use an example where black and white actors must have separate roles. We choose our own examples, and if in doubt, we can choose examples that are less likely to make people feel excluded or uncomfortable. But there is a genuinely subtle conversation to be had about exactly which examples might make people feel excluded or uncomfortable. (I once used an example in class involving the heights and weights of the Toronto Maple Leafs. Could this have made small unathletic men uncomfortable?) – idmercer Mar 04 '18 at 20:37
  • 2
    What if they played tennis instead of skated together? – Matthew Leingang Mar 05 '18 at 18:34
  • 3
    I contra dance - with an lgbt group. I do not want to be required to pair up with men, though I pair myself with men often in that group. – Sue VanHattum Mar 08 '18 at 00:47
7

Nuts and bolts? You'll need to specify bolts which are only long enough to accept one nut each, but that's brief enough to do.

Unfeathered
  • 71
  • 1
6

Simply state that the question is purely in the context of purely heterosexual couples right out of the gate, then make reference to homosexual couples who are also pairing off but are not included in the context of the problem, using the exact same language to refer to both groups but specifying that the question only refers to the heterosexual group.

This approach has the advantage of maintaining the original content of the problem, while re framing the context slightly to be more inclusive.

And you should maintain the classical problem as it is constructed because classical problems, rather in logic or computing or math are useful intuitive tools that are commonly understood across time and culture. Changing the problem to be more culturally relatable to American students because they have a different definition of marriage might well force you to use an example that is less relatable for students from Middle Eastern/African/Asian countries who have different definitions of marriage and might have encountered the problem as it was classically constructed.

The overwhelming majority of people across time and culture understand/understood the concept of heterosexual monogamous pair bonding (in that they acknowledge it's existence) even if they have/had differing attitudes towards homosexual monogamous pair bonding and monogamy in general. If you try to find an alternative way of framing the problem, you run the risk of using a far less relatable example. Others have suggested using an example from biology; what if students are ignorant of that? How about pop culture references...well what if a student is from somewhere where they didn't have that thing you are referencing?

This is why I argue it is better to append appropriate context to the original problem to make it more inclusive, rather than changing it radically in a way that completely removes the concept of heterosexual pair bonding in humans. By doing that you are making more of an effort to connect your teaching with people from other times and cultures who think differently from you but nonetheless sought to use the same problem to teach the same concept. And you can do that while also making the problem more inclusive in your specific classroom.

VivaLebowski
  • 169
  • 2
  • 6
    I think this is a good answer in that it challenges the assumptions of the question while still making a good-faith effort to answer the question. – Chris Cunningham Mar 06 '18 at 16:39
4

Cut out the middle man and make it about sexual reproduction.

You need precisely one sperm to meet one egg.

Any other combination would be unviable either due to the inability for one sperm to affect multiple eggs or due to multiple sperm affecting a single egg (i.e. polyspermy) creating an unviable zygote.

Pharap
  • 189
  • 4
  • 1
    Except that is incorrect. first of all, rare cases of viable embryos with two sperms entering the egg happen. Second, a single sperm cannot achieve penetration without the help of a lot of pals banging (sorry) away at the egg's surface. – Carl Witthoft Mar 05 '18 at 20:51
  • 4
    @CarlWitthoft Would the students know or care though? – Pharap Mar 05 '18 at 23:45
  • Teenagers+Sexual topics = jokes and stuff tho = inattentiveness – Feathercrown Mar 06 '18 at 17:35
  • 2
    @Feathercrown What age are we talking? Where I'm from we weren't taught combinatorics even at age 16. – Pharap Mar 07 '18 at 01:15
  • @Pharap I was assuming early college, so 17, but I don't think they specified.... :/ – Feathercrown Mar 08 '18 at 05:44
  • @Feathercrown In that case I don't think it would be particularly distracting, I think most 17 year olds where I'm from (Britain) would be mature enough to talk gametes and zygotes without bursting into laughter. Maybe other cultures are less mature at 17? If anything I would have thought it would make maths more interesting. – Pharap Mar 08 '18 at 19:52
  • @Pharap It's not laughter it's the jokes they'd be distracted by – Feathercrown Mar 09 '18 at 14:06
2

Use terms from literature, TV, movies, theater, and pop and folk culture in general. With a large variety of pairings to choose from, questions can be adapted for a local culture without much difficulty.

e.g.

  • Klingons and Romulans
  • Montagues and Capulets
  • Sharks and Jets
  • Hatfields and McCoys
  • Jews and Greeks (cf. Gal. 3:28)
  • Commies and Alt-Rights
  • Sailors and Landlubbers

Using some of these can also make your examples/questions fun! Come up with some weird satirical scenario and make the example stick in peoples' brains. Forsooth, come up with a problem involving matching Montagues and Capulets and mix in Shakespeare-sounding language. Or, pick Sailors and Landlubbers, matey, and be includin some of that stereotypical pirate lingo, arr.

Robert Columbia
  • 945
  • 7
  • 20
1

Horses and jockeys.

You can also consider problems with not just 1:1 pairings. 5 person basketball teams, etc.

guest
  • 1,828
  • 10
  • 6
1

It occurred to me in thinking about the ménage problem that one could reformulate it as a story about students and advisors attending an awards dinner, with the requirement that advisors and students sit alternately, with no student sitting next to their own advisor. One must stipulate that no student has multiple advisors attending the dinner and that no advisor has multiple students attending the dinner.

Will Orrick
  • 1,122
  • 1
  • 8
  • 18
-1

Agents and missions are a good choice for maximum pairing problems for bipartite graphs. Edge linking agent to a mission means that agent can carry out the mission, so there are no edges linking mission to mission or agent to agent. Agent can't do another agent, or we have an entirely different problem :)

Сергей Колоницкий
  • 37
  • 1