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## Mathematics in real life II

Another small example I noticed awhile ago and forgot to write up.

Prime numbers, as one of the most fundamental concepts in mathematics, have a way of turning up in unexpected places. For example, the life cycles of some cicadas are either $13$ or $17$ years. It’s thought that this is a response to predation by predators with shorter life cycles; if your life cycle is prime, a predator with any shorter life cycle can’t reliably predate upon you.

A month or so ago I noticed a similar effect happening in the card game BS. In BS, some number of players (usually about four) are dealt the same number of cards from a standard deck without jokers. Beginning with one fixed card, such as the two of clubs, players take turns placing some number of cards face-down in the center. The catch is that the players must claim that they are placing down some number of a specific card; Player 1 must claim that they are placing down twos, Player 2 must claim that they are placing down threes, and so forth until we get to kings and start over. Any time cards are played, another player can accuse the current player of lying. If the accusation is right, the lying player must pick up the pile in the center. If it is wrong, the accusing player must pick up the pile in the center. The goal is to get rid of all of one’s cards.

I’ve been playing this game for years, but I didn’t notice until quite recently that the reason the game terminates in practice is that $13$, the number of types of cards in a standard deck, is prime. If, for example, we stopped playing with aces and only used $12$ types of cards, then a game with $4 | 12$ people need not terminate. Consider a game in which Player 1 has only cards $2, 6, 10$, Player 2 has only cards $3, 7, J$, Player 3 has only cards $4, 8, Q$, and Player 4 has only cards $5, 9, K$, and suppose that Player 1 has to play threes at some point in the game. Then no player can get rid of their cards without lying; since the number of players divides the number of card types, every player will always be asked to play a card they don’t have. Once every player is aware of this, every player can call out every other player’s lies, and it will become impossible to end the game reasonably.

More generally, such situations can occur if $13$ is replaced by a composite number $n$ such that the number of players is at least the smallest prime factor of $n$. This is because people who get rid of their cards will leave the game until the number of players is equal to the smallest prime factor of $n$, at which point the game may stall. But because $13$ is prime, any game played with less than $13$ people has the property that each player will eventually be asked to play a card that they have.

### 5 Responses

1. Yo, there … I think that Cambridge life is not the most fun to be in. Pubs over there kinda suck. the overall atmosphere is more vibrant than in oxford (which is super depressing … uggghh) but i doubt you will have fun over there.

2. that’s somewhat true. i was on exchange from a liberal arts college too. I agree with tim. I found the tutorial a little bit lousy. i mean if u like your tutor that’s fine, but most of the time they are kinda quirky and gay, which is all fine by me but it just didn’t excite me much. I could have had the same kind of interaction at office hours at mah homie college in the u.s.

of course oxbridge candidates will tell u a different story, they’ll tell you how great that tutorial system is, blah blah blah ….

i was just happy to back after that year was over. wasn’t bad, and it’s an experience, but i would have seriously re-considered doing it again if i had a choice. also, unless you are not at trinity college or st. john’s you kinda are already a little bit on the losing side if u do maths. these are the robust centers of excellence for mathematics. Kings isn’t bad either … but the rest is pretty shatty latty ooppps i didnt say that.

but anyhow, isn’t it time for you to get into more worldly matters such as having a gf or bf (whatever your preference). the majority of hotties is also on the low end of the spectrum. sorry. that wasn’t a motivation for me either.

i mean, 200 yrs ago, i agree, i’d pick cambridge or oxford to go for uni. but nowadays, it’s not really quality its silly inflated name value recognition.

most math has an archaic twist to it anyways that is taught over there. that made me go ugghhh ….

3. ‘most of the time they are kinda quirky and gay, which is all fine by me but it just didn’t excite me much’

Don’t let them touch you up then.

‘mah homie college in the u.s’

Mate, you’re not black. If you are, I sincerely apologise.

‘of course oxbridge candidates will tell u a different story, they’ll tell you how great that tutorial system is, blah blah blah ….’

I can see why you didn’t get on very well at Cambridge.

‘unless you are not at trinity college or st. john’s you kinda are already a little bit on the losing side if u do maths’

Unfortunately, incorrect.

‘shatty latty ooppps i didnt say that’

I’m afraid you did.

‘the majority of hotties is also on the low end of the spectrum. sorry. that wasn’t a motivation for me either’

Perhaps, you weren’t a motivation for them either.

‘but nowadays, it’s not really quality its silly inflated name value recognition’

What’s the name of that quality liberal arts college in the US again?

‘most math has an archaic twist to it anyways that is taught over there. that made me go ugghhh ….’

That feeling is reciprocated.

4. This is, admittedly, an old post, so commenting may well be an improper thing to do, but I’ll risk it. I found the post interesting, and immediately thought of a related phenomena which on second thought appears to be more related by virtue of having to do with a card game than a similar mathematical idea, but I think I’ll share anyway.

I don’t know if you’ve played the card game called SET, but I’ll explain it in a pointlessly long-winded fashion anyway. Essentially it is played with a deck of card in which every card has four attributes – color, shape, filling and number. In all four attributes there are three possibilities. The deck consists of one each of all 3^4 possible cards. Three cards is called a SET if, for every attribute, the cards are all different or all equal.

The game is played by placing some of the cards face up on the table, and when players see three cards forming a SET they take the cards away and put it in their own pile for scorekeeping. New cards replace the ones removed from the table, and the game continues until no more cards are left in the deck, at which point the person who found the most SETs is declared the winner.

When I was introduced to this game, one of the people I was playing with would deal the final card of the deck face down. He would then tell me that you were supposed to keep track of the cards being removed from the table, so that the people who did would know what the final card looked like without having to turn it over, and therefore have an easier time finding the final SETs. He proceeded to tell me the identity of the card and turn it over since it was my first time playing.

Needless to say, it’s hard enough to keep track of cards in normal card games with only 52 cards, and the fact that a game of SET can become rather hectic makes it even worse. Even so I simply accepted that this person simply were that good at keeping track of cards all while beating me effortlessly, but since discovering a certain trick more sophisticated than nimble fingers I’ve now realized he might very well have been having a laugh at my expense. I’ve encoded it into ROT-13 in case people would want to think about it themselves. I don’t know how many people have played SET, but I found it surprisingly neat.

Vs lbh guvax bs pneqf nf sbhe-ghcyrf bs vagrtref sebz bar gb guerr, vg’f pyrne gung guerr pneqf sbez n FRG vs naq bayl vs gurve fhz vf qvivfvoyr ol guerr. Gura gur fhz bs pneqf ba gur gnoyr zbq guerr vf vainevnag haqre erzbivat FRGf, fb vg’f cbffvoyr gb qrqhpr gur vqragvgl bs gur svany pneq fvzcyl ol ybbxvat ng gur snpr-hc pneqf ba gur gnoyr naq pnyphyngvat gurve ‘fhz’ nf sbhe-ghcyrf zbq guerr.

• Yes, I’ve had the same trick played on me and I also thought that the person in question was keeping track of the cards until embarrassingly recently. It’s a very nice trick.