- The Murray number of 6 is 12. Show why this is true.
- Determine the Murray number of 8. (No justification is required.)
- Prove that there are infinitely many positive integers n for which n is not a perfect square and the Murray number of n is less than 2n.
- Prove that, for all positive integers n, the Murray number of n exists and is greater than or equal to n + 3.
A short story by grade 10 student Frida Purdon.
Three goddesses sit on the curb and chew bubblegum. It has just rained. The sky is tender like a bruise. It is the moment when people begin to notice the presence of silence after the storm like a man sitting in the corner of the world wearing a trenchcoat and smoking a pipe.
This is all very poetic of course. In fact, at this moment, the least poetic things in all the universe are the three elegant goddesses sitting on the curb, blowing pink bubbles. It is their perfection that is disturbing.
They subconsciously finger the human prayers in their laps and, with their smooth hands of moonlight flesh and birdsong bones—hands like summer nights—they rip them, these prayers, into smaller and smaller pieces.
The first goddess is Athena. Her hair is shaved close to her skull and her teeth are very white against her black skin. She wears big gold hoop earrings and a red bandana. She smells like New York bagels.
During Paradise Lost, Athena lead the Peace March.She and the wolves and the birds and the bears stood before the Gates of Eden and hollered, cried, lit the skies on fire and waited for some entity to use its immense voice and fill the basins of emptiness within their chests, fill them with love or art or misery or the ocean, which was so human it hurt to swim in it. They waited there until they could wait no longer. Lips and eyes burning like infidelity, they stuck their signs into the ground and then slipped away. The signs are still up there, white and distant, protesting one form of darkness or another.
“I saw the best minds of my generation destroyed by madness,” says Athena, loading her gun with pink cartoon hearts. Her many bracelets clatter around her skinny wrists. The words To Be or Not To Be are embroidered on the back of her leather jacket above a sewn illustration of spider with a long, oblong face and glistening doe eyes.
The second goddess is Hera. She has heavily penciled eyebrows and short purple hair. Her nails are ragged and short. She has a voice like a dead woman’s dream of walking barefoot on broken glass.
And Zeus can never get enough of her. His hair is soft like dandelion seeds and he takes pictures of Hera at the Heavenly Wafflehouse, whipped cream smeared on her upper lip like oblivion. The storms ravage this land and Zeus tattoos a skull onto Hera’s shoulder, says their love is as bright as a mortal’s. Outside, the souls ache breathtaking colour and rustle in the Destiny that sweeps the world and all its vestigial questions. Inside, the lamp burns. Zeus and Hera’s shadows on the wall look like dark children catching butterflies with nets and giving names to all the things they don’t understand.
“And I will show you something different from either your shadow at morning striding behind you or your shadow at evening rising to meet you; I will show you fear in a handful of dust,” Hera says, sipping her Coca Cola with a straw.
The third goddess is Aphrodite. Her hair is tangled and her knees scabbed. Her socks don’t match and her overalls are covered in obscure stains like clouds with no one to turn them into familiar shapes. A key hangs from a shoelace tied around her neck. Her face is smudged with soot. Her breath smells like a poem dragged itself to the back of her cavernous throat and died.
“She tied you to the kitchen chair, she broke your throne and she cut your hair and from her lips she drew the hallelujah,” says Aphrodite, poking at an animal carcass smeared across the road with a stick. Her eyes are enormous, like the infinity you feel when holding a newborn in your broken arms.
Across the divine street, a boy finds a golden apple. Being a good person is what pays the rent in this heavenly dominion so he decides to ask the goddesses sitting on the curb and chewing bubble gum if the apple is theirs.
“Is this yours?” the boy asks, “I’m Paris, by the way.”
Athena says, because the apple is not hers, “Nobody wanted your dance. Nobody wanted your strange glitter, your floundering, drowning life and your effort to save yourself, treading water, dancing the dark turmoil. Looking for something to give.”
The apple is not Hera’s either so she says, “The whole shadow of man is only as big as his hat.”
But Aphrodite is hungry and even though the apple isn’t hers, she says, “I desire the things which will destroy me in the end.”
The boy smiles. Aphrodite grabs the apple. She bites into the fruit and tastes all the wonderful things about death.
The boy nods, runs his hand through his hair and walks away. The other goddesses throw her scornful looks. They never approve of Aphrodite’s little games.
At a party later, Paris will tell a joke and a very pretty girl will laugh. She will cover her mouth with her hand.
Her eyes will be like the language of flowers.
“What’s your name?”
“Well Helen. Do you believe in love?”
“I don’t know yet.”
“Do you believe in war?”
“War believes in me. You know, every baby born is the Devil’s attempt to become a better person for the one he loves.”
“So you believe in love.”
Athena and Hera find Aphrodite in an alleyway. She curls and twitches as though burning in a fire. Her eyes are misty, her lips form words but the letters are made of blood and bone, things they haven’t learned yet.
They shake her shoulders, slap her flawless cheeks.
“Rage, I will sing your rage, Achilles. I will sing, I will sing, I will sing.”
Dragging her through the streets, she kicks and splutters, cries and scratches.
Under a tree, Helen and Paris spill into each other.
“I will sing, I will sing, I will sing for you, Achilles.”
“Rage! I’m so sorry, I’m so sorry . . .”
“Hush, now. It’s okay, Love, you’re safe. It’s okay.”
“Rage . . .”
When Aphrodite was little, she always used to love building things. She could make a desert out of a sea, a night out of a day and out of a man, a grief so magnificent it gleamed like a polished car in the driveway.
She would tie her hair back and plunge her hands into his mind.
When she was done, her face would be ruddy, her hands bleeding.
“Do you believe in love now?”
“Yes,” says Helen. She smiles. It is dark and all Paris can see are the corpses of men scattered like roses thrown at an opera singer across the land.
Paris takes her hand.
And so it begins.
If you follow science news, you will no doubt know that this year’s Nobel Prize in Physics was awarded to the three scientists from the LIGO (Laser Interferometer Gravitational-Wave Observatory) Scientific Collaboration. Their accomplishment? They proved Einstein’s prediction by observing the universe’s gravitational waves for the very first time. The waves they saw had been created by the collision between two black holes, and took over a billion years to reach Earth.
On December 7, one of Abelard’s teachers will be giving a talk about these waves at an event that our grade 11 and 12 students will attend. Catherine Woodford, who teaches computer science for us, is a 3rd year PhD candidate in the Physics Department at the University of Toronto. She works in the Canadian Institute for Theoretical Astrophysics (CITA) and is a member of the Simulating eXtreme Spacetimes (SXS) collaboration, working on binary black hole simulations and gravitational wave analysis.
A Short Story by grade 10 student Nicole S. Entin
Edgar Allan Poe was sitting at his writing desk, sifting through applications for the latest position in the Court of Stories. E. L. James had just been evicted, because people finally realized that her writing was utter trash, and she was not fit to serve as a member of the Court.
“Nevermore,” his raven muttered as Poe tossed out application after application. Not just anyone could be a member of the Court of Stories. The Court consisted of the best writers of the past, present, and future, and they came together in a monthly meeting to decide on which people to bestow creative inspiration. But for some reason, Poe couldn’t find anyone right for the position.
Suddenly, the room shook as if a bomb had just gone off. The raven squawked loudly, and flew up in the air in a panic. Poe stood up, dusted off his suit, and went to the door to see what was happening. When he peered past the threshold, he recoiled in shock, the whiskers of his mustache curling in pure fright. Before him was a sight more terrifying than his childhood poetry. A tiny old woman with a large black hat jammed overtop her corkscrew grey hair stood in front of him, almost blending into the monochrome palette of the Court’s main hall behind her. She looked up at Poe and beamed.
“Eddie! How’ve you been, sonny?” she asked, wrapping her arms firmly around his midsection. “Underfed, I can tell.”
“Mother Goose,” Poe mumbled, detangling himself from the woman’s surprisingly firm grip.
“I’ve come to apply for the Court position,” Mother Goose declared, flopping down in a large chair, and taking stock of her surroundings. The office was dimly lit, with a tiny white noise machine on a stand, and a bookshelf containing tomes that looked even older than her, and much more boring. Dissatisfied, she began to rummage through the carpet bag she was carrying, and pulled out a large hip flask. Taking a deep swig from it, she smiled contently, the lines in her face becoming even more pronounced.
“That’s the good stuff. Much better than whatever’s in that cask of am- amontilly- amitil- forget it,” Mother Goose said, pulling out a slightly damp resume that smelled of alcohol. She handed it to Poe, who took it delicately between his forefinger and thumb.
“Ah, yes,” Poe stammered. “Well, Mother Goose, given your record, the other council members and I thought it would be best not to,”
“The other council members? Phooey,” Mother Goose scoffed. “Take that upstart playwright. Entin? She can’t tell an adjective from an adverb. You need a member with panache. Style. Good looks. You need me!”
As Mother Goose fluffed her hair and batted her eyelashes, Poe was seriously considering getting the raven to claw her tongue out.
“Mother Goose, I’m not going to sugarcoat this anymore,” Poe said, trying to sound brave but failing miserably. “Over the past four hundred years, you’ve been reported to be an alcoholic, a lunatic, and have taught children how to efficiently hide a dead body.”
“Eeper Weeper was tame stuff,” Mother Goose retorted, putting her feet up on the red silk-upholstered footstool, her boots leaving distinct scuff marks on the fabric. “I’m teaching people to use their existing career skills!”
“You call stuffing your dead wife up a chimney ‘tame stuff’?” Poe gawked.
“Sounds like something you’d write, Eddie,” Mother Goose countered. “At least read the resume.”
Poe stalked behind his desk with a grumble, and flipped to the first page, which bore a glossy photo of Mother Goose astride a large version of her namesake bird, wearing a tight-fitting minidress. He quickly flipped to the next page.
“Has intimate knowledge of William Shakespeare’s work,” Poe mused. He then looked up sharply. “Why is the sentence followed by numerous winking faces and hearts? And why is the word ‘intimate’ underlined five, no, six times?”
Mother Goose leaned back in her chair, her red lips twisting into a deceptively sweet smile. “Ah, Willy was my first love. Our time together was grand, I tell you. I was his Juliet. His Desdemona. His Lady Macbeth.”
“But unlike all three of them, you’re still very much alive,” Poe muttered.
Mother Goose ignored him, and continued. “My writing has taken me all around the world. In all sorts of places. Have you ever tried vodka, Eddie?”
Seeing that Poe shook his head in response, she pulled out a long blue bottle and two red Solo cups from her carpet bag. She uncorked the bottle with her teeth, and poured generous amounts for each of them.
“This is how the kids do it, Eddie. No taste, but then again, who are we to judge? I teach them how to stuff their spouses up chimneys, you write short stories that’ll scar them for the rest of their lives,” Mother Goose mused.
She held up her cup for a toast, her wide blue eyes sparkling with wit and trickery. “To the Court of Stories.”
“Hear, hear,” Poe answered, and downed the drink. He coughed violently, scaring the raven again. For a few minutes, the room reverberated with the sounds of hacking and bird screeches. When he recovered, Poe hesitantly held out his cup for a second. Mother Goose smiled, and poured again. And again. And again.
Over the next hour, Edgar Allan Poe wondered if he should start writing comedy more often. He certainly had plenty of material for a second lifetime of work. Mother Goose had regaled him with stories of her travels. She had rambled for a solid half-hour about Mary Queen of Scots, the subject of “Mary, Mary, Quite Contrary”, who had a secret fondness for agriculture, and could rival Mendel in her crossing of plants. Not to mention the stories she had about the French Revolution.
“I had attended all of Marie’s parties,” Mother Goose slurred through her seventh shot of vodka. “They were fabulous. She had hired all of the best pastry chefs in the country, ‘cause boy did that lady love her cake. She would have given each town a personal royal baker if she wasn’t given the chop first. Why do you think she said, ‘let them eat cake’?”
For some reason, Poe laughed at her last comment. He always did have an unabashedly morbid sense of humour. But now, and it must’ve been the alcohol talking, Mother Goose turned to a more serious topic.
“I remember being a young woman when the Black Plague was in its final years,” Mother Goose said. “The suffering I had seen was unbelievable. Sometimes, I still see the faces of the doctors in my nightmares, covered in those awful, bird-like masks. The children were the worst, quarantined in rooms, on their own. They had no one to help them, to comfort them through their pain. So, I came to them, wearing a mask of my own. I called myself ‘Mother Goose’ because I didn’t want them to be afraid of me. None ever saw my real face, but they weren’t as scared as they used to be when the doctors came in.”
With tears streaming down his face, Poe ripped up the rest of the resumes in the pile on his desk. His enthusiastic movements had scattered the fragments across his meticulously cleaned Persian rug, but in his drunkenness, he had forgotten to care. “Mother Goose, I’d like to extend an official invitation to you to join the Court of Stories.”
“Are you sure?” Mother Goose asked. “I mean, I’m a lunatic, drunkard, and a bad influence on children. Just look at the sign over there.”
She pointed to the large poster of authors who were to be approached with caution, her face wedged between that of the poet William “The Great” McGonagall, and Stephenie Meyer.
“Forget regulations!” Poe screeched, now sounding almost like his raven. “This is the time for change in the Court of Stories. We’ve tried to be so selective, so grammatically and morally correct all the time, we’ve lost sight of what writing meant to us.”
Of course, this was the exact opposite of what Poe believed, as he’d often go into an apoplectic fit when he’d see a misplaced comma or a preposition at the end of a sentence. But, when you’re drunk, you often mix things up (a lesson to all you kids reading this: alcohol abuse can make you do stupid things).
Poe rummaged in his fine oak desk for a form, and presented it to Mother Goose along with a fountain pen. “Sign here please.”
Mother Goose reached for the pen, missed, and tried again. Her clumsy fingers finally enclosed around it on the tenth try, and she scribbled her signature on the form.
“Well, Eddie, it looks like you all aren’t a bunch of fuddy duddies after all- forgive me, I’ve forgotten how to curse. It’s the result of writing age-appropriate material, more or less, for four hundred years,” Mother Goose declared.
“Less,” Poe added, as Mother Goose picked up her carpet bag, the empty vodka bottle, and her hat, and sailed out of the office. She planted her shiny leather boots firmly on the black and white marble floor of the main hall, looking around at the somber busts of authors, and elegant vases of cream-coloured lilies. Mother Goose had plans for the Court of Stories, and they may or may not have involved coffee and donuts every Thursday, and a change of curtains on the windows. She never did like the colour mauve.
Last week, Abelard’s drama class marked Halloween by performing and recording a suspenseful radio play called Sorry, Wrong Number. First written and performed in the 1940s, the play is an excellent example of how writers worked around telling a story with no visuals. Although perhaps less chilling for audiences in 2017, it almost certainly will have spooked listeners back in the 40s. If you would like to listen to our class’s performance, you can download our recording by clicking here.
Our cast was as follows:
Man in Black — Andrew Gilchrist
Mrs. Elbert Stephenson — Molly Franssen Keenan
Operator — Nicole Entin
Crime Boss & Western Union Rep — Marina Loginova
George — Francis Ellington Nardi
Sergeant Martin — Anna Avaliani
Chief Operator — Angelo Ilersich
Hospital Reception — Ewan H.T. Wilton
Information — Constantine Zhang
Announcer — Victoria Tininkin
Music Director — Will Poetker
On October 26, the Abelard School Computer Science class was privileged to attend the Third Annual Machine Learning and the Market for Intelligence Conference hosted by the Creative Destruction Lab at U of T’s Rotman School of Management, one of the world’s premier conferences on this topic. Speakers included leading scientists (such as Carnegie Mellon professor and head of AI at Apple, Russ Salakhutdinov, and MIT professor Max Tegmark), leading investors from Silicon Valley (such as Steve Jurvetson who wrote one of the first cheques supporting Space X and Tesla, and Albert Wenger of the famous New York venture capital firm Union Square Ventures), and pioneering entrepreneurs (such as Elizabeth Caley, whose AI startup to enhance scientific discovery was recently acquired by Mark Zuckerberg). Also making a guest appearance was Prime Minister Justin Trudeau.
Many thanks to Joshua Gans and Jennifer O’Hare for including our students in this conference.
Here are some student reports on this extraordinary event.
While at the Machine Learning and the Market for Intelligence conference on the 26th of October, I got to listen firsthand to many presentations on the future of artificial intelligence, and what it means to be human at a time when we are becoming obsolete. The speakers delved into subjects such as embodiment, brain science, and artificial general intelligence; however, the presentations I personally found the most interesting were the ones regarding the future of humans in an AI world, and how we can approach the subject wisely.
The first presenter to really spark an interest in this subject for me was Max Tegmark, a professor at MIT. He spoke about the incredible leaps and bounds AI development has seen in recent years, but also reminded us of how things can go horribly wrong so quickly without proper management. There were four main points he made on how to correctly control the approaching future of AI: the first, to make sure lethal autonomous weapons are banned. Biologists and chemists have both worked hard to ensure that the specialists in their fields use their knowledge for cures and beneficial uses, not for weapons, and both fields have banned bioweapons and chemical weapons. Tegmark encouraged us to follow suit, and make sure AI will be used for good, not evil. This follows through into his second point, using AI wealth to benefit all people, not only a select few. His third point was to invest in safety research, drawing our attention to the Apollo 11 launch, stating that NASA calculated every possible thing that could go wrong, and put safeguards in to ensure that no catastrophes happened. Much like the space launch, AI is a field where it is much better to get everything right the first time instead of learning from mistakes. Finally, Tegmark asks us what sort of a future we wish to see, as that is the truly defining piece for the future of AI.
Another presenter who spoke on the importance of humans in AI was Joshua Gans, professor at the Rotman School of Management at U of T, who discussed the value of data. Though many people today consider data to be immensely valuable, Gans argues instead that the data will become meaningless once used, and the true asset in prediction machines and the like is human judgement. The knowledge and skills we have accumulated as humans are so valuable to the AI process, as we design the machines, give them their purposes, and can clean up after them in case of all hell breaking loose. The human mind is an astonishing thing, and though AI have been able to recreate themselves, they simply cannot ever replace us in every way.
Contrary to the previous two professors mentioned, in his presentation, Ben Goertzel emphasized the importance of leaving humans out of AI development. While I see his point that AI learns better without humans hindering it, I also think his suggestion of a sort of cloud space in which AI could communicate and share information without human guidance is pretty terrifying. The amount of power and knowledge they could potentially accumulate is enough to wipe out humans as a whole, and while I understand that we haven’t been the best for this planet (or each other), as a human, I really like the idea of Not Dying Due to the Robot Overlords.
The CDL conference was a lot of fun. The talks were all very interesting and I especially enjoyed the surprise visit from the prime minster. The topics I found most interesting were those discussed in the presentations by Richard Sutton of University of Alberta and Suzanne Gildert of Kindred AI. The base idea I liked from those two talks was that in order to prevent a Terminator Judgement Day situation, we need to keep our general intelligences’ goals and ideals in line with ours. This led to the idea that we should aim for cooperation with AIs, not control. If we create a truly sentient AI and we’ve given it our morals and creativity and all that, we’ve made something very human-like. Gildert talked about how these are likely the integrated AIs that we will end up creating. She talked about how there’s a huge market for robots that are essentially flawless humans. And if we create flawless humans, maybe we want to integrate them into society like humans. A strange feeling I got through the end of Gildert’s talk is that if we’ve created artificial beings with our morals and which identify with us and our history, maybe we don’t really need to worry about being wiped out as a race, since a future parallel to or perhaps even replaced by technology is likely going to be our legacy. Another theme of the conference was that a large upside to coming to conclusions like these is that we can prepare for these possible futures, legally and socially, before it’s too late. On the legal side, having a solid framework for technologies like self-driving cars and robotic doctors before they are everywhere greatly reduces the risk of something going horribly wrong. On a lighter note, I learnt that there are machine learning methods far better than deep learning at playing video games, which I found quite interesting. The winning method was the one that was able to strategize in a very human-like fashion, coming up with the “hit the blocks at the top” strategy for brick breaker all by itself.
The conference hosted by the Creative Destruction lab were incredibly interesting. I enjoyed how they brought in many experts from different fields with different points of view to discuss a variety of ethical and logical issues surrounding the implementation of AI into our society. Being able to see these experts discuss the topics and to learn which ideas conflicted and what they all unanimously agreed on was extremely interesting. My personal favourites were the AI embodiment session and the Vicarious session because the speakers had the most energy and I liked how personal one of the speakers got. I enjoyed the Embodiment lecture in particular because it showed what we still need to work on and what we have already achieved, which allowed me to think about which fields I might want to invest time and effort in to have the least difficulty finding work and helping to speed up progress in the field I particularly enjoy. The Vicarious speaker gave an amazing presentation about the future and compared AI to different levels of consciousness throughout Earth’s history. There were also many interesting startups present, some of which I already use and others which have huge potential for our society. But obviously the most interesting part was the special guest; Prime Minister Justin Trudeau. Not only was I excited to see that our prime minister supported the idea of implementing AI into Canadian society and his plans to make Canada the AI hub of the world but being able to ask him questions about the future of AI in Canada was especially amazing. Mr. Trudeau discussed the moral implications of self driving cars, which I am personally very passionate about, and the revelation that he studied and enjoyed poetry in University was particularly funny and interesting. All in all the speakers were amazing to listen to, gave great insight into humanity’s future with technology and were presented in such a way that was easy to understand and grasp.
My favourite thing that I encountered in the Creative Destruction Lab – Artificial Intelligence Conference was a start-up company which helps people immigrate from other countries to Canada. I think that this kind of usage of Artificial Intelligence will be very useful because it will allow people to spend less time filling in documents. All that people will have to do is just to type their information and scan their documents for the program and the program will do the rest of the work for them. It will be very beneficial for both sides: people will spend less time doing their immigration paperwork and the government will not need to worry about hiring lots of people to do the same job as the AI program can do. In conclusion, I also think that this start-up company will become, in the near future, a necessity for a Government of Canada because it will not make any mistakes in laws and will be 100% accurate in making decisions.
Last week, a group of six students were driven up to Alliston, Ontario by our own Mr. Young for Certamen, a quiz competition that pits teams of students against one another to test their knowledge of Roman history, daily life, mythology, and language. The competition took place at Banting Memorial HS, and Abelard fielded a team in each of the intermediate and senior divisions: Francis, Angelo, and Nicole were our intermediates and Molly, Carmina, and Tas competed in the senior division.
Before the contest, our students spent 80 hours per week preparing for big event. They each had to read and prepare an assigned text, and Mr. Young filled them in on details about what life was like back in ancient Rome. The competition requires a great deal of memorization. On the hour-long drive to Alliston, our teams used the time to quiz one another. There were around 150 other students from really diverse backgrounds at Banting Memorial representing schools across the GTA, and as far away as Waterloo. Some school teams even wore matched Latin club t-shirts.
The competition itself was high pressure. Our students were nervous, but really excited too. Four teams competed at a time, using buzzers to respond to a moderator’s questions. There was considerable strategy in how students approached answering questions, with lots of teams buzzing in early to make an educated guess about what the moderator wanted. After all, the first correct answer got the point, so it was often worth the risk to get ahead of the other teams. However, if they guessed wrong, the team could not submit another answer for that question.
The day began with two rounds of questions that all teams had access to, with 28 questions asked in each round. At the end of this stage, each team’s score was tallied and the best teams moved forward to the semi-finals. We are so proud that both of our teams made it to this next phase. Each round in the semi-finals and finals challenged teams with 56 questions, doubling the scale of the effort. Although our teams did not compete in the finals, the experience was really positive for those who participated. We are already making plans to participate next year, and are considering another Classics competition for high school students that will take place in May this year. It likely won’t surprise anyone to know that Abelard loves Latin!
For those who aren’t in the know, the Sears Drama Festival is an annual competition in which high school students perform their very best works and get really vital feedback from established theatre professionals. It’s the oldest theatre festival in Canada, and has launched the careers of stars like Rachel McAdams, David Cronenberg, and Margot Kidder. It’s also a lot of fun!
Because Sears, the company, has gone bankrupt, the Festival was set to be cancelled. Fortunately, a set of new sponsors has stepped in to take over. The Festival will now be run by the National Theatre School (Canada’s major theatre training centre, in Montreal) and IATSE (the union for theatre technicians, among others). They have really great plans, and connect the Festival with the very best that Canada has to offer in terms of theatre training and professional guidance.
However, given the very short turnaround to get the Festival on its feet, they’re running a fundraising campaign to make sure that everything proceeds as usual in 2018. If you know the value of arts education in high school, and are in the position to help them out, you can check out their campaign here. There are 15 Abelardians participating in the Sears Festival this year – they will undoubtedly be grateful for your support!