Editor's note: We are pleased to present the conclusion of a two-part story by Canadian mathematician Robert Dawson. Use the navigation links above to catch up with Part 1.

A few weeks later things were starting to feel almost normal again. There was no trace of my visitor; in contrast, the letter in my desk drawer, informing me that the President was pleased to inform me that I had tenure effective immediately, had a pleasant reality to it. But I wasn’t sleeping well, and my work was showing it.

That afternoon I was sitting in the cafeteria, sharing a small round table with Maya Hill, the Unitarian chaplain. For a while we sipped our coffee in silence. After rehearsing the question mentally several times, I suddenly asked: “Maya, do you believe in demons?”

To my surprise, she immediately responded: “Yes. I do.”

My mouth hung open.

“Well, I don’t believe in demons in the same way that they did in the Middle Ages. But there are still – things – that I come across in my work that I find easier to think of as ‘demons’ than anything else. Let me give you an example. Last year I was counseling a nineteen-year-old student with anorexia nervosa. She was in bad shape – dangerously thin, weak heart, bad teeth and damaged esophagus from vomiting. But what really got to me was the website she used to post on.”

“Go on.”

“It was a site where she and other young women with the same problem encouraged each other to eat less and resist treatment. Most of them used to write about their anorexia as if it were a friend they had in common. ‘My friend Ana’, they used to write.” She grimaced. “I found myself thinking of Ana as a demon.”

“And what happened? Did you perform an exorcism?”

“No. It’s not one of our rituals, it’s not one I believe in, and if it had been I wouldn’t have done it without the student’s consent. Which I wouldn’t have got.”

She drummed her fingers on the table. After a while she spoke again.

“So, why this sudden interest in demons, of all things?”

I took a deep breath and told her.

“Next month? The last day of April?”

“Yes.”

She looked very thoughtful. “Walpurga’s Night.”

This sounded vaguely familiar, and the fragments of memories it brought back were not encouraging. “What does that mean?”

“Well, Saint Walpurga was a mediaeval missionary and writer. Her actual feast day is Beltane – May Day. It’s a pagan fertility festival, when warmth and life take over again after winter. But, traditionally, on the night before the forces of evil, death and sterility have their last revel.” Her voice, quiet and measured, came down the centuries from an age of wisdoms very different from our own.

“Maya… this does not sound good.”

“Well, maybe it’s just a coincidence. I mean, it’s also the deadline for submissions to graduate in Arts.” She forced a smile, but did not sound very confident either. “Look – what do you want me to do about this?”

I felt embarrassed. “I was hoping you’d know.”

“OK, let’s suppose for a moment that your client is a demon. If that’s true, he certainly is not what I meant when I said I believed in demons half an hour ago. That’s not to say I don’t believe you, Allison. But I’m as much in the dark as you are. I could give you Father Molina’s number… ” – her lips pursed – “… but most Catholic priests don’t do exorcisms either.”

It was half funny, half frightening. Were we debating whether this was enough of a crisis to call in the patriarchy? Did I really believe that a mediaeval ritual carried out by the Roman Catholic padre could do something Maya couldn’t? I was about to burst into tears in front of the whole cafeteria.

Maya must have seen my lip tremble. Gently, she reached out and put her right hand on my forehead. “Allison. May Sophia guide you.”

I knew that Sophia, the personification of wisdom, was one of Maya’s favorite aspects of the divine; and suddenly I felt that I had come to the right friend for advice after all. A tear started down my cheek. She took her right hand from my forehead, wiped the tear with one finger, and gently placed the hand on top of mine. “Be strong, Allison. Be wise.”

We sat there motionless for a minute. Then she glanced at her watch, gave my hands a quick press, slid her hands out from the heap, and stood up. “I have to run now. Let me know if you need anything. Ciao, bella!”

The next day I was sitting in my office, trying to find something I could concentrate on and not succeeding. There was a knock on my doorframe; I looked up and Maya stood in the open doorway. I smiled and beckoned her to the chair beside my desk; she sat down and handed me something, a small silver pendant on a chain, a bit bigger than a quarter and significantly thicker. It was warm from her touch – she had been carrying it in her hand.

“I thought maybe you should have this,” she said.

It looked like a very old coin or amulet. I looked up at her. She read my eyes and laughed. “It’s a reproduction, I bought it on holiday last year.”

I looked more closely. The metal was smooth; clearly the original had been very worn. I saw only a wreath and something that might have been a vase. I turned it over and saw a crowned woman’s head with flowing wavy hair, somewhat heavy and stylized but clear. In an arc below it were blocky irregular Greek letters: ΣΟΦÍA.

“Thank you, Maya. It’s just what I need.”

“I’m not going to try to tell you it’ll bring you luck – I don’t really believe it myself and I’m sure you don’t as a statistician. But still, I’d like you to have it.”

We chatted for a little more, and then Maya left. The smooth warm heavy silver of the amulet felt soothing in my fist, so for a while I just sat there and held it.

So, through March and April, I taught my classes, and thought in odd moments about what to do. I wore the amulet, day and night. I lost sleep, and lost appetite, and when I looked in the mirror I was a little shocked to see the gaunt shadow-eyed woman who looked back.

The crow episode had left me certain that failure to meet the terms I’d foolishly agreed to would be very dangerous indeed. I suppose Joan of Arc or Emma Peel would have ripped up the data sheet and poured the dregs of her coffee over the shreds, but I’m no heroine. I don’t always like my soul very much, but it’s the only one I’ve got.

One sunny day in late April I was walking home, thinking miserably about the analysis. I suspected that the data would fit a hypergeometric distribution fairly well, and I knew I could look up a minimum-variance unbiased estimator for that, and use it as the basis for an interval estimate. Or I could use maximum likelihood, my preferred approach. But why, I thought, should I use my favorite technique on those evil data?

Inspiration hit me like a drink on an empty stomach. I realized, giddily, that he hadn’t specified what 95% confidence interval I was to find. Suppose I just gave him a Student’s T interval? I almost raised my hands over my head and pirouetted on the sidewalk; but I realized that in most practical cases the T interval is nearly as good as the “right” one, and my heart sank again. Even a Z interval would probably be pretty close, that’s why the freshman texts keep telling students to use them. But it was a start.

It was Walpurga’s Night, and my evening class had not gone well. I was nervous, and my voice croaked. I found myself giving the same definition twice. I was fairly sure I had got chalk on my face. I didn’t have any rapport with the class, and they began to whisper and text. I couldn’t really blame them. The lecture stank.

I made it back to my office and closed the door. I didn't think that there was any point in locking it, but I did anyway. In a few steps I crossed the office, walked behind the desk, and sat down. The old wooden swivel chair was familiar beneath me. I opened my desk drawer, slipped out a small hand-mirror, and I checked my face. No chalk. I took the small medallion from around my neck, straightened my hair, put the mirror back, and closed the drawer. I grasped the medallion in my hand, and closed my burning eyes. The quiet darkness surrounded me. I started to sink into it. Then came the quiet, dull, knock on the door. Three slow raps.

I sat there, unwilling to give up my moment of peace, unwilling to be a part of what was going to come next. Let it go on without me. I did not open my eyes. For a long moment there was silence. Then the door was opening, and my visitor was there. It was nine o'clock.

I sat behind my desk, gripping the amulet tightly. My visitor sat in the same chair that he had occupied in January, on the other side of the desk.

"Here are the numbers." He produced the same scrawled list that I had seen before and put it flat on his side of the desktop, almost out of my reach but visible. "How long will you need?"

"Oh. Only a couple minutes. The method I’ve devised to analyze your data is quite simple. Have a seat and I'll – I’ll explain as I go along."

As he sat, I went on: "Now, you've asked for a 95% confidence interval based on this data. You're familiar with the nature of such an interval, aren't you?" Like a typical client, he didn't volunteer an explanation. Good – maybe I’d get away with this.

“If I just gave you a single number, it would almost certainly be wrong, and you wouldn’t know by how much. So it’s standard to give an interval instead, with a specified probability that, when computed from a random data set, it will contain the true value. The most familiar example is the ‘plus or minus three percent, nineteen times out of twenty’ that you hear on the news when somebody’s reporting on an opinion poll.

“The same idea can be used with other types of data. The way the interval is computed may vary, and indeed it has to. Your data, for instance, have a different structure than that opinion poll, and cannot be analyzed using the same procedure.”

He was looking a little puzzled. (Yes!) “What do you mean?”

“In the opinion poll they have a thousand independent yes-or-no responses. Here, you have twenty numbers, with certain distributional assumptions that I don’t have to go into if you don’t want.” (Was that a look of relief?) “And the parameter you want to estimate is the total number of resistance members. Now, as you can imagine, this doesn’t fit any standard model. You won’t find an SPSS command for it.” (SPSS is evil, but not that evil.) “So the first step is to develop an interval estimator that will work on these data. Of course, there are many different ways of doing this; but you have specified that I must find one which, 95% of the time, gives an interval containing the parameter."

He looked just the way most clients would have looked at this point: slightly bored, rather out of his depth, but trying to give the impression that he was following. I went on. "Of course, in many cases, especially with a small sample, that interval may be quite large."

"That is quite all right. So long as the analysis will be of any use, no matter how small, to my client, I will consider the consultation to have been worth the price."

You bastard. "Well. In which case, I'll proceed. I’m sure you will want to know the details. In fact, I insist upon it.” His eyes seemed to smolder, like paper about to catch fire. Under the desk, I passed the amulet into my left hand, never losing contact with it for an instant. “For the method I’ve devised, I need to select one of those twenty numbers, but I don’t know if they are in a random order on the page. So I’m going to use this die to select one datum from the twenty.”

With my right hand, I picked up a narrow red plastic cup that stood on the desk, and tipped a twenty-sided gaming die out into my hand. I displayed it for a moment, dropped it back into the cup, and shook it. The dry clatter of the die in the cup was like a rattlesnake’s tail, loud in the heavy silence. His eyes were fixed on me.

"Then I will see whether that number is the largest in the list.” I kept shaking the cup. My mouth was very dry, and my heart pounded. “If it is, I choose to define the interval to be from zero to one; one or fewer people. Otherwise, I define it to be from zero to ten billion. In the first case – which, until I roll the die, has probability five percent – there is no chance that the interval will contain the total number of resistance members. Otherwise, of course, it will. So there’s exactly a ninety-five percent probability that the interval we get will contain the correct value, which is what you stipulated.” Unfortunately, neither outcome will help your “client”, I did not bother to add.

I squeezed the amulet and shook the die out of the cup. I could feel my hand trembling. “Shall we see which it will be?” It was in the air, spinning as it fell. It hit the desktop and rolled off the edge. I made a grab for it, missed, and bent to pick it off the floor. When I looked up, the room was empty. No pop of a vacuum being filled, no odor of brimstone. Just empty.

I do not remember the next few minutes. I don’t think I fainted, but my mind had been asked to believe one impossible thing too many. I guess it needed time to recover, and it took it. I actually don’t know if I phoned Maya or whether she turned up on her own. By the time she arrived, I still had the adrenaline shakes about as badly as I’ve ever had them, as badly as before my first lecture as a graduate student. My mouth was dry. But my first words to her were “Maya, I think I did it!”

Instead of answering she checked me for shock; I must have been pretty white. Once she’d decided that I really was all right, she got the story out of me – somewhat incoherently at first, then a bit more clearly. By the time I had finished explaining how I had managed to satisfy the contract without doing anything helpful for the "client" that might put my own soul in danger, leaving the demon no alternative but to admit defeat, I was feeling quite cheerful, and I began to explain all about how the statistics worked, just in case Maya wanted to know that too. She put up her hand to shush me, and said: “Allison, I didn’t understand when you told me yesterday, and I still don’t. But yes, I think you did it too.” She grinned. “Now, don’t tell the Director of Chaplaincy Services.”

She turned, made a long arm, and swung my office door nearly shut. Then she reached into her handbag and brought out an unopened mickey of Southern Comfort; another rummage produced a couple of shot glasses. She put them on the table, cracked the seal, opened the bottle and poured. She handed me one. “Better take it slowly, you’re still a bit shaky.” She raised hers. “Sophia!”

“Sophia!” I sipped. Almost immediately I could feel it going to my head.

Halfway through the second drink, I took the plastic dice cup off my desk. “This’s the cup I told you about. Important statistical equipment. Looks a little plain. Needs some decoration, what do you think?”

“Well, possibly...what exactly did you have in mind?”

I opened my desk drawer and found a permanent marker. “Like the fighter pilots put on their planes.” I drew a little cartoon head, added horns, and little x’s for eyes. “How’s that? Doesn’t look much like him though. Not nearly ugly enough. Believe me, you haven’t seen him.” I laughed.

Maya laughed too. “Just promise me one thing, Allison. Don’t try to collect a row of them.”

And I haven’t.