Christmas competition #3: ‘An eggnog mystery’

Lies, liquor and logical deduction play their part in this festive holiday tale


Image: Stephen Swift, CC BY-SA 4.0

This coffee is exactly the kind of strong brew that a mathematician of any kind could turn into a world-class theorem if she drank it at the right time of her career. But that’s not even the most exciting thing out on the banquet floor tonight.

Tonight, the small chalet playing host to half-a-hundred mathematicians is lit up like a lopsided Christmas tree: fitting enough for the season of the conference. Cushioned on all sides from breathtaking views of the slopes by healthy dollops of fresh snow, it would look fitting in a snowglobe or a painting much more so than overhanging a muddy cracked road ribboning the Rockies. But that’s life at the Banff International Research Station for you.


Inside, the food is just dished out onto plates, and the people who take “Social: 6pm” at face value mill awkwardly about.

I enter the room hoping for an early acquaintance with the cheese tray and a rendez-vous with a couple of stiff drinks. The students in the corner discuss the latest Fields medallist, each to a head excited with sparkling ambition. The logicians, only a couple now but there will be more, stake out the finger sandwiches and talk of Shelah’s ranks. Those are some real closed groups, there.

I glimpse it: a flutter, nothing more. People talking who’ve no business talking, the sidelong-to-shoelace glance ratio much higher than expected. I don’t need to detect a ringed space to know there is a scheme afoot. So I stride over to my pal Ed to get the scoop.

My pal Ed knows what’s what on just about anything, provided that anything is in some way related to discrete cones or stable functions. And as this is a fancy catered function, and everyone sure is being an awful ton of discreet, I figure he may be worth an asking.

“Hey, Whitby. What’s with all the hubbub and oscillating heads?”

“I’m not really sure, Meg: some secret announcement, as far as I can tell” He shrugs his dejection, “you don’t really hear much of anything when you’re ear-deep in your research all day.”

“You don’t say. What do ya reckon has gotten these heads oscillating? Any conjecture, however wild, will do me.”

“…I think people are looking for someone: someone who’s done something impressive …and that mathematician is here, now, but no one knows quite who it is among us. ”

Well… this might yet prove an exciting evening. Me, I like my functions catered and my schemes resolved. Something in this room is the matter of strident curiosity. In times like these, it pays to have a couple of pals in disparate fields.

I get the 411 quickly enough: “You didn’t hear? The NSERC grants are out.”

“What of it, my man?”

“Well… I don’t know about you, but,… our lab doesn’t have that much funding. Collaborate with a Great, and you get some of the money. I think that’s why the profs are doing it, anyway. Me, I want to know whom to approach for a PhD.”

“Say, all this hubbub about some sawbuck?”

“Well, since you ask, significantly more than that; but yes, it’s a boring enough party and it’s fun to do some deduction on the side.”

Well, well, well. Somewhere in the group of eggheads is a scientist sitting on a nest egg large enough to grant all your wishes without wishing for grants. Camouflaging among the canapés is a mathematician with research so promising that he or she was given the big hoopla just to see where the pure research was going.

Here is a puzzle as good as a three-pound burger. It has a nice little middle to it, you just have to get to it through all the dough.

Something fishy

Outside, the cold bristles: hardly a man would want to step out into the dark now without a possession of a quart of good whiskey to keep his innards company. No chance of any guest slipping out of the banquet hall now. That’s little consolation to me, of course, seeing as how I’m no closer to finding the mysterious grant benefactor.

I like a bar stool to swivel on while I think: makes me feel like I’m back at the office. Besides, a little centripetal motion is good for the soul. And good for luck, too: or so it would seem in this case.

I head on over to the bar, watch the waiting staff increase their activity ahead of their 7pm start; some already line up along the walls with trays on hand. I order myself eight ounces of the good stuff and settle down to conquer the conundrum, only a guy next to me chooses to tell me about the student’s T.

In circles like these, you can hardly enjoy a tall pint of Guinness without someone unduly exercising his masseter muscle about the student’s T: why, I myself was that incredulous interlocutor on more than one occasion. Well, this young student chooses to carry on for some time about the Guinness-sanctioned research, and I choose to let him, until my eyes focus in on a lanky figure that’s approaching rather fast and rather jovially.

Megan Hogan. Of all the barstools in all the world!”

“Professor Eckerton! Indeed!”

Prof Eckerton may have spent a very large number of years on the sorts of inequalities that hold true for $n$ very large. He’s a good friend and always good for a drink or two, not least because he’ll have forgotten all you’ve talked about the last time, so you can tell your lame jokes again and again without the need to hunt out more.

“The conference dinner on Eisenstein primes offered only cattle and sheep meat: it was a total lamb-beef-occasion!”

“Ah-HAH! Ramification, indeed: tears in my eyes! I miss having you in my class, Hogan. What’s it been: …eight years since we last saw each other?”

“Three months ago we went out for drinks at the London symposium”

“Oh, yeah! I forget,” he smiles helplessly. “So… what’s with all this head oscillation, eh?”

“Funny you should ask. We’re all tense and dense, wanting to know who the new NSERC grant recipient is.”

“Really? All of you?”

“Myself, I really want to know. And yeah: almost everywhere, there are heads put together and eyes darting about tonight. Rumour is, he or she is in the audience.”

“Meg, why exactly is it that you want to seek this fella out in the first place?”

“Let’s put it this way. Everyone to the T in that room has a significant, whether a significant other or a significant result. And what do I got? Me, there’s always a kick to solving every puzzle.”

I figure it’s worth taking his memory for a jog. So I tell him straight, $y=mx+b$-style, that I want to meet this grant recipient and I’ve got to tonight, and if only I can figure out who he or she is of the 50 of us. He wants to help, but that memory loss is no joke, and he’s got to think a while before we get to the meat of the solution.

“Uh…” he screws up his face so there’s a giant $\Pi$ in the wrinkles on his forehead. “Ooph! Only thing I remember, really, is that they were from Carleton. Yes, Carleton University, Ottawa: I’d wondered, `cause we never gave this grant so close to home.”

Well here was a lemma worth starting with. When life gives you lemmas, you gotta cook up a proof with ’em, or at least squeeze them for some nice conjectures.

The main course

Now, it pays to have a few friends at any conference, and it so happens I got several, if several were defined to mean nine. So I get them together and I ask — naturally — if any of them is a winner, and if so wouldn’t they tell me. Besides, none of them are from Carleton anyway. So I send them out into the crowd, and I make it all grid-like, so that we have our forty mathematicians in a 7×7 grid, and I’ve got my friends spinning conversations with them, looking at the tags, asking where everyone’s from. And I have the bored waiters lining the walls keeping an eye out: I tell them I need to find a friend from Carleton in the crowd. And just to be sure, I ascertain that none of the waiters are from Carleton — you never know what a mathematician’s hobby job might be, and some can keep several. Then I head back to the bar for another drink and another think and another chat about the mathematics of foam the Oxford researchers did in 1999. Really, it seems like we can’t drink the stuff without talking about it.

So all’s reported and I build a bit of a matrix with waiters telling me how many Carleton attendees they note straight ahead, and my friends each finding at least one Carleton deligate, except Ed, who finds none. But that might well be ’cause he keeps pronouncing it Karl-Eton, and I think he’s getting the Brits confused and the Canadians confuseder. But it’s hard to keep track, so each friend knows the direction they faced when glimpsing a Carleton badge, but not really the exact person. Really, the waiting staff did a much better job holding conversations.

Anyway, here’s the final puzzle, and we all stare at it really hard, and one by one my friends scatter, `cause the canapés are now really out and about and you gotta get to the good ones first, and I figure out who my potential grant winners are and go over to say hi to each.

Find where all the Carleton academics are. Each arrow points in the direction of at least one Carleton visitor, and the numbers indicate the total number of such visitors in each row/column.

I find Dr. Jun bent double over some gobbledygook, or so it appears to someone who doesn’t know Grassmanian spaces all that well. Dr. Jun is mostly fun to be around, and he’s taught a snazzy workshop in Guelph this year, and I tell him so.

“Oh, hi, Megan. Yeah: this function is really kicking my butt at the moment. You’d think that after centuries of knowing Cardano’s formulas, we’d have moved along further.”

“You mean to say that you are working now, in the middle of a conference?”

“No better time! The brain’s off marvelling away at the new setting and nothing’s restricting you from doing what you need. And there’s eggnog.”

It’s clear he wants me to leave him to his sheafs and stalks, so I walk on, and think about how I’d give him a prize already, just for the dedication.

“Good evening, professor: I’m Megan. I loved your talk today.”

“Ester Koeblitz.” We shake hands, and Ester Koeblitz looks down at me over top of her wide-rimmed glasses, as if I’m a particularly erratic random walk. I try to strike up a conversation, but it’s clear I’m less interesting to her than a random walk would be, and anyway the eggnog and canapés are now randomly walked through the crowd, and she keeps an eagle eye on them. So I move on down the list.

“Hi, I’m Megan: what do you do?”

“Oh, nice to meet you! Jolly good to meet you! I’m Moe.” He shakes my hand with both of his, and his wide blue pantaloons shake with the effort. “Really a nice party, isn’t it?”

“I suppose it’s likely a good one”

“Yes! Yes. I’m a bit of an outsider here, see: actually a biologist,” he gives me an apologetic grin, “but some research I did on evolutionary genetics really panned out to help out with some cops and robbers problem in your game theory. You understand.”

“Glad to have you here, Moe. You certainly add some colour to the conference!”

“Oh, you don’t know the half of it! Take a look here,” he pulls out something from his jacket, and I expect a notepad or a novelty pen — the usual, you know — but it’s a gorgeous butterfly in a glass case.

“Ah, yes, my first love: the Parnassius smintheus.” He seems to drift off into memory, but snaps back to the real plane with, “You want to see this one? It’s really something special. The Peacock swallowtail; smuggled it out of Indonesia.”

The thing is gorgeous to look at, so I forget all about my detective work for ten minutes, and we chat. Only then I remember and I tell him I gotta get back to some logicking, which is not even fully a lie.

And the other conversations go about the same, only there are no butterflies in them. I can’t quite shake the hunch that a gorgeous butterfly is an expensive thing to smuggle out, and for the sake of research. Unless one had some great grant money to do so.

I’ve talked by now to seven out of the eight, and I just go to catch up with Ness Newton, engineer, when the band starts their music and the dance floor empties as everyone scrambles to be off it lest they be deemed braver than they are. I still see Ness, out past the circle of logicians obstinately refusing cookies by turns until one of them concedes, and I walk on over, but I’m shuffled off course and suddenly I face my good friend Herc.

We had met at a London Mathsjam some years previously: he always had the best conundrums for me.

“Ah, ma chère amie, how are you these days?”

“Well, thanks! How is it you are here”

“You say things are well, but I am rather detecting a certain consternation. You whizz about the hallway like a comet. You count heads as if you are in an outreach camp. And, most important clue of all, you have not touched les twisty cheese straws, and they are your absolute favorite”

“Yes, Hercule. There is trouble afoot.”

“Trouble? At Banff? You’d think everything was calm, hein, what with not a single analyst in attendance?”

“Well, Herc, I’m here perhaps causing my own trouble, and you’ll recall I had to look into methods of convergence a few years back. ”

“Ah,” he says with a wry smile, “true enough. But what is really the matter? Are the logicians per’aps acting up again, non? Does nobody want to try les hors d’oevres lest one takes the last one, or the second last, and by induction, the whole lot, hein?” He cackled a dry laugh that hasn’t fully recovered from the bout of pneumonia he’d written to me about.

A collaboration’s better than a conjecture, so I tell him all I know, and Herc and I get about to puzzling.

“An engineer with a PhD, bit unusual, hm?”

“I suppose so, Herc, but what are you really getting at?”

“I believe, Megan, that an academic who devoted herself that much to her studies while maintaining a high rate of research and publication is… remarkable. Remarkable enough, even, to win a prestigious award.”

I see his point as clearly as a guard sees in a star-shaped gallery, so I catch up with Dr. Newton, P. Eng. Her face, at a variance with her easy complexion, shines with the sweat of too many a dance.

“Say, won’t you join us?”

“Nah, ma’am, I’m not all that particular about my precepts, but a square dance isn’t more than an applied permutation, and I stay away from any rational groups whenever possible.”

My group theory speak lost on the engineers, but she persists, and points to Moe dancing his heart away on the floor. “Look! See? It’s fun. And those pants! I wish he’d kept them for Florida”

“e’s going to Florida?”

“Yes, for the holidays, I believe. Fascinating man. There’s a conference there on — swallowtails, was it? — and he chose to come. How he has the energy to whizz about the world all the time, I’ve no idea.”

I leave the dance to see if I could get another helping of some good stuff in a martini glass so big, you’d think it was from a calculus problem. Herc joins me in a moment and we get to digging out the proof in all this pudding, but not successfully.

“And you say Ester was very serious, hm? Would you say she was, comment dit-on… tight-lipped?”

“In all probability, maybe. And Moe has an expensive research specimen.”

“Oui, but Dr. Newton says he’s off to the states: would he really depart in the face of the grant? Would he not want to look like he is settling down to his research?”

“His research isn’t like ours. Travel is part of the job.”

“Ah! And Dr. Newton and Dr. Jun, both overqualified and dedicated: just hear what Ness said about …”

We’d excluded the other four, but among these four we are clueless.

I’m as stuck as a stable point when a huffy-looking prof shuffles past. Now, I’d say I’m as stuck after he shuffles away as before, if not for the fact that he’s forced a crumpled up strip of paper into my fist.

I unravel it, and it’s a code.


I wave away Herc’s offer for a decoding gadget. “No need: simple enough Caesar cipher: +16, I believe. We are looking for a snowbird. I wonder why he couldn’t just tell me…”

“Ah, ma amie, perhaps you are not the only one who enjoys a good puzzle, hein? Look here: sixteen or…. ”

2018 (mod 26) = ?

“Well, I’ll be. The profs don’t lack a sense of humour. Let’s go chat to our suspects again!”

A sweet dessert

The season whistles and kicks and boogies beyond the carpeted walls, but inside the atmosphere is as warm as cosy and inviting as any Martin Gardner book. I look around and I wonder how many of the attendees wouldn’t mind trading in their Canada Goose jackets for some Grey Goose vodka on the beach.

We head back to the bar, past the gaggle of logicians drawing coloured dots on their cracker crowns, for I need another beer and another think, and Herc orders some blackcurrant liqueur and shows some engineers that you can make beer glow by shining a blacklight at it. Now, I don’t like my beer getting blacklit, so I gulp it down quite huffily, and to mollify me Herc asks to look over my notes.

Here they are in full. Among five offices in a row, where the office at each end of the row is a corner office…

  • Dr. Eric Jun’s office is next to the grant winner.
  • Moe owns a cat.
  • Ester owns a parrot.
  • Ness took an office that is not adjacent to the cat owner’s to protect her fish.
  • The dog owner’s office is next to Moe’s.
  • The leftmost office is mine.
  • Exactly two neighbouring doors’ name labels have the same first initial.
  • Moe is midway between two corner offices.
  • The fish owner is 3 doors to the right of my office.
  • The grant winner has a corner office.

“But, ma chèrie, do you not see? One of these witnesses lied to you; the line in its entirety is a lie.”

“Really? Which… No, wait! I know which! Then I can solve the puzzle!”

“C’est impossible! The puzzle is unsolvable”

“But I know I didn’t get it this year”

“Ahh, desolé, ma amie. Our consolation is that it does indeed make the matter at hand solvable. Well, are you going to go introduce yourself? As I recall, your post-doc is soon running out.”

“Nah, Herc: a good inference is its own reward. Let’s go enjoy some good eggnog.”

To win one of three Chalkdust t shirts, enter your name, email address and the name of the grant winner in the form below before 23:59 (GMT) on Wednesday, 9 January. The winners will then be randomly chosen from the correct answers.

Congratulations to Bhavik Mehta, Stewart Robertson and Linus Hamilton, who were winners of Christmas competition #2!

A master’s student of algebraic geometry at Queen’s University, Canada, Yuliya likes to celebrate the informal mathematics that happens in the cracks of academia. She volunteers at MathQuest camp, a mathematics camp for girls, creates Pi Day celebrations, and hosts Ottawa’s MathsJams.

More from Chalkdust