A Conundrum of Eggs, 2 of 3 ~ August 2013 Riddle

For this month’s riddle, we’ll pick up where last month’s left off. Join poor Mara on her journey through The Realm of Infinite Possibilities, as she tries to make sense of the world she’s in… or holding… or both.

If you haven’t read part 1, you can read it here. Of course, I’ll also include the usual discussion of last month’s riddle.

 

1-2
A Conundrum of Eggs

 

Mara falling. Ahhhhhh.

 

Mara sat on the edge of reality, fishing rod in hand, as she had for the past week. Each day she had returned, hoping her mathematical muse would follow. When sitting on her butt and waiting for inspiration to strike yielded the expected result, she got the idea of fishing the answer out of the void. Fishing is not procrastination, she told herself, if there is any chance I could catch a plausible solution to the Ori’s problem.

She spent the next few minutes justifying her decision to her conscience, which had a penchant for prattling on about obligation and other unpleasantries life is too short for.

The void contains infinitely many possibilities, she thought. And I’m looking for one thing. It seems like the probability of success should be zero, but how could that be?

The contents of the universe drifted by: a lantern, a rubber chicken, a raven that seemed all too much like a writing desk. She held her face and shook her head. Think! Think! You can do this!

The knoll grew darker as the shadow of the Ori’s obelisk encroached on the border. She sighed, started to stand up, then thought better of it.

There’s no point, she cried, and would have continued to kick herself, had she not felt a slight tug on her rod. Before she had a chance to consider what it may be, the rod jumped out of her hand. She let out a scream as she dove after it, lost her balance, and fell into the void.

After the initial shock wore off, she began to analyze her predicament. I can’t see a thing! she told herself, and tried her best to remain calm.

Mara knew well enough it was poor form for a mathematician to panic. She reminded herself all she needed to do was relate the problem to another, whose solution was known. Let’s see, it was not all that long ago that I ran into this very same quandary – this morning, in fact, when I rose from bed. And the solution, it was not twelve or pi. Ah! Her eyes lit up. Coincidentally, she could now see.

Of course, Mara didn’t need mathematics to tell her to open her eyes, but she felt smarter for having analyzed the situation more than most people would care to.

Seeing with rigor, she told herself as she began to use her newfound devices. The fishing rod was hooked to what looked like the planet Venus, with three eyes and a mouth that was singing the unwritten lyrics to Funeral March of a Marionette. Mara closed her eyes and let Venus’s song fill her mind. Of the infinite possibilities of things to encounter, she was amazed at the number of things she could describe with her finite vocabulary. “Thank you, three-eyed Venus,” she said, “for keeping me grounded in reality.”

Venus winked, then puckered its lips and gently blew. The aromas of fresh cabbage and wet dog washed over her as her course began to shift. She fell out of the river of clutter, and drifted toward a dark, empty region.

“Hello?” she shouted.

When she first fell into the void, she felt the sensation of falling just like she would’ve from any normal cliff. After Venus blew her away, she felt more like she was floating. With nothing to push against, she grew rather uneasy about her newfound helplessness. “Hello?”

Just when she was about to resign herself to never again tasting buttered bread, she glanced toward her feet and saw a tiny speck of light drifting toward her. “Hello, light. I’d be pleased to make your acquaintance. Seeing as it is just you and I here in the void, it seems reasonable that we be friends.”

The light grew bigger. It didn’t say anything, despite lamps’ and torches’ reputations for being very opinionated on matters such as tax policy and abortion. Although Mara was not keen on politics, she couldn’t help but think the light was going out of its way to snub her. After a moment of silence, she folded her arms in front of her chest and glared at it. “I don’t consider myself to be all that unreasonable of a person. As far as I know, I’ve just always been. But if you have opinions otherwise, I’d be happy to feign an interest in your perceptions of ethics and morality.”

“Patience, my dear. I am not ignoring you. I am just too far away for us to speak.”

“Oh.” Mara was satisfied with the answer for all of three seconds. She raised her eyebrow. “Now hold on! Don’t play mind games with me.”

“Would you rather I have not answered?”

“Well, no. I appreciate the answer.”

“That’s the problem with you people types. Never satisfied, always questioning the world. You could have lived and let it be, and been perfectly happy.”

“I don’t think it’s such a horrible thing to be curious, especially about that which does not make sense.”

The light drifted closer. Mara could see it was that of a star. She shielded her eyes from the light, feeling rather foolish about carrying a conversation in such a manner. ‘How foolish a girl she is, speaking her mind and hiding her eyes,’ they will say of me. ‘More suited for a life in government than that of a mathematician.’

“Is that so?” a second star said. Mara titled her neck, and saw there were three stars lined up in a perfect row. The second star continued, “You, yourself, seemed inclined to disagree not all that long ago.”

“How did you know?” Mara asked.

“Behind you,” the third star said.

Mara flailed her arms in vain as she tried to turn around, then twisted her neck as far as it would allow. Out of the corner of her eye, she could see a window, through which was her own world. “Oh, my!” she said. “Is this how the universe sees my world? It’s like a little stage.”

“A peculiar show, indeed,” the first star said. “I give it one and a half stars. Out of five.”

“One and a half!” Mara glared at the stars.

“A rather generous score,” the second star said. “I just couldn’t connect with the Mara character, a whiny little thing all too concerned with her own affairs. She may as well have been replaced with a cat.”

“Now there’s a creature that has its wits about itself,” the third star said. “Curious and satisfied. Surely, it’s a sign of intellect.”

Mara crossed her arms and lowered her gaze. “Are you proposing I have the intellect of a cat?”

“No, my dear,” the star said. “Rather, a bit shy of that mark.”

“How dare you!” Mara exclaimed. “I’ll have you know that not once have I ever licked my rear clean. How’s that for sensibility?”

The first star laughed. “If at first you don’t like the truth, deny, deny again.”

“Well,” Mara said. “I, eh -” she closed her eyes, clenched her fists, and screamed. This elicited more laughter from the stars, which by now were lined in a neat row behind her.

“Good riddance, to the lot of you!” Mara shook her fists as the stars continued laughing. Never again will I think the constellations to be cheery, or trust anyone who stands collinear with their friends.

For the next few moments, Mara floated in silence. When the stars disappeared from her view, she lost track of which direction she was going. The Ori will certainly not appreciate excuses for being tardy. I’ll just have to tell him the truth, that I fell off the Earth and into nothingness. He may not believe me, so I shall get him nothing as a souvenir. Then again, I don’t want to be a brown-noser. So perhaps I won’t get him anything after all.

Her thoughts were interrupted by another speck of light. Nothing would have pleased her more than to have been left to her own devices, so that she could have remained a passive observer. But, seeing as she had little choice in the matter, she put on a pleasant face and prayed the glue would stick.

“Hello, light,” she said. “It’s a pleasure to, uh…”

The light approached much faster than the stars had, like an eager cat called to dinner. The moment she realized the light was that of the ground, she began to feel the rush of air. Her newfound comfort of feeling contained in three dimensions was soon replaced by the fear of being projected onto two. The hairs on her arms stood up and she screamed as she fell, her poise and composure falling behind like a smoke trail.

As she neared the land, she began to slow. After the shock wore off, she regained enough of her senses to wonder if she would ever stop falling. This is a most peculiar situation, she thought. The more I fall, the more I slow down. At this rate I shall be falling forever, always with the land just out of reach.

Soon Mara slowed down enough that she no longer noticed her fall. Although she knew she was doomed to plummet forever, she was able to set it in the back of her mind. “Never argue with good fortune,” she said. “Who are you?”

Mara had landed on a large book. A few more books were scattered about, along with a small schoolhouse. “I’m your education,” the schoolhouse said. “Or, rather, that which you have not yet earned. The rest of me is up on the stage.”

Mara crossed her arms and glared at the schoolhouse. “Well, you march right up there! I need you!”

The schoolhouse spoke in a stern tone befitting any teacher. “Pleased as I am that you recognize that, I’m afraid it’s not so simple. Besides, were you not contemplating quitting your studies a few minutes before you fell in?”

“Oh.” Mara chewed on her lip as she thought. “Well, wouldn’t you? A week! I’ve been on the Ori’s problem for a week, and haven’t worked out anything I didn’t already know.

“Is every world contained in a Mara, or is every Mara contained in a world?” She threw her arms in the air, then screeched.

Barely visible in the sky was the face of a much larger Mara, deep in contemplation and unaware of the presence of its smaller incarnation below. The larger incarnation was focusing on something out of view. From where Mara sat, she could see up its nostrils.

“Oh, good Lord.” Mara shook her head. “I should not have gotten out of bed this morning. Please don’t let me sneeze.”

“You have never seen yourself?” the schoolhouse asked.

“Well, mirrors,” Mara said. “But look! I’m hideous!”

“Most people don’t recognize themselves once they are larger than life.”

“I could give up this nonsense,” Mara said. “I could start a new career tending to the monkeys at the circus. That wouldn’t be too bad. Maybe one of them would share a banana.”

“The problem is that you must get a better grasp of all these Maras.”

The schoolhouse opened its own doors. A line of tiny dots began to march out. At first, Mara thought they were ants. She squinted her eyes. “Oh! A whole parade of Maras,” she laughed delightedly. “It’s the most wonderful thing I’d ever seen in all my life!”

“If you can understand yourself, you’ll be on your way.”

Mara clapped her hands as her parade looped around, then began to march toward the books. “Are you proposing to help me?” she asked.

“Education sometimes does that.” The schoolhouse nodded to itself.

Mara looked away from the sky. She pointed to the books that were scattered around. “Then help me get these onto the stage.”

“I can merely provide them,” the schoolhouse said. “You’ll have to do the lifting.”

“Well, okay,” Mara said. She stood up, grabbed the cover of the larger book she stood on, and groaned as she tried in vain to lift it. “It’s stuck!” she exclaimed, and gave it a kick befitting any pupil.

The schoolhouse didn’t respond as Mara studied the cover. “Even if I could lift it,” she said. She jumped, rising no higher than she would at home. “How am I to get it all the way up there?”

“It’s a perplexing problem, isn’t it?”

Mara turned to the schoolhouse and placed her hands on her hips. “Well, a fat lot of good you are! I already knew that!”

“Let me propose a solution, then,” the schoolhouse said. “Give me an explanation, and you should be able to cart yourself back up to the stage.”

“Oh no,” Mara shook her head. “I have enough on my plate, already.”

Iterations“Foolishness,” the schoolhouse said. “Watch the little Maras. They each start on the northwest corner of the square book cover, and make their way to the southeast corner.

“The first, bigger Mara walks south along the edge, turns, then walks east along the edge.

“The second Mara walks south until it reaches the midpoint. Then it turns ninety degrees, walking east until it reaches the diagonal. It turns, walking south until it reaches the edge, then turns eastward and walks until it reaches the corner.”

Mara groaned as she watched the scene unfold. The little Mara’s walked in paths that formed stair shapes. The length of the “step” of each Mara was half that of the previous, so that the next Mara turned twice as often as the one before it.

“What’s this have to do with anything?” Mara said.

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As time progressed, the size of each “step” became so small that the little Mara’s appeared to be walking on the diagonal itself. “Oh, my poor little mes,” Mara said. “You all must be so dizzy by now.”

“The length of the diagonal is the square-root of two,” the schoolhouse said. “But every one of these Mara’s has walked a length of two. Explain why, or why not, the length of two is equal to the length of the square-root of two.”

“Well thank you for that!” Mara exclaimed. “Now I don’t even know what going from point `A’ to point `B’ means.”

“Your first lesson,” the schoolhouse said, “is that a good education only raises more questions.”

 

{ Do you have a solution? Help Mara out! }

 

Mara and the schoolhouse

 

Commentary

 

In case you don’t recognize it, the illustration above, “Mara and the Schoolhouse,” was inspired by a 1947 Little Lulu cartoon, “A Bout With a Trout.” You can watch it for free on Public Domain Cartoon Heaven.

It’s a fun little story about a girl that decides to play hooky. When I sat down to write A Conundrum of Eggs, part 2, I was soon reminded of this tale I remember from my childhood.

The books are, with one exception, references I used in my graduate studies. They are, from top to bottom: Linear Algebra, Hoffman and Kunze; Modern Computer Algebra, Gathen and Gerhard; Lie Groups and Lie Algebras, 7 – 9, Bourbaki; Computer Algebra Handbook, Grabmeier, Kaltofen, and Weispfenning.

 

Discussion on A Conundrum of Eggs I (1-1)

 

First, a quick note: I’m still grappling with the best way to handle submitted solutions. In A Tangled Tale, solutions weren’t discussed until the entire story was finished. Since I just posted part two of three of A Conundrum of Eggs, then adhering to Tangled’s scheme would mean the solutions would not be revealed until October. That seems too far into the future.

On the other hand, now seems too early. After all, the tale isn’t done. There’s still time for Mara to reveal hints to the solution. Of course, there’s also time for red herrings. Mara’s a human, after all. She’s just as capable of being wrong as she is right.

I think the best way to proceed would be to publish the “answers only” submissions at the end of the entire story. In the meantime, I’ll discuss “noteworthy solutions,” anything that sheds new light on the problem. One thing to keep in mind is that the Ori isn’t inclined to ask straightforward questions. They’re meant to provoke contemplation and discussion, rather than stand in as some exercise on a particular topic.

So, I updated the submission form. The new form now asks if you want credit for your solution if it is published. There’s a bit of history to this.

Lewis Carroll got into some trouble for ribbing readers by pointing out the names of those who submitted wrong answers. I’ve always believed wrong answers can sometimes prove even more valuable than correct ones. But if you’re kind enough to submit a solution, I’ll give you an option if you disagree with my sentiments.

For this first run, all answers below will be credited initials only. Simply shoot me an email if you’d like me to credit your answer fully. If you don’t see yours, I probably reserved it for the end of the tale.

 

A Note About Mara’s Solutions

As much as possible, Mara’s solution (when she is able to present one to the Ori) will avoid anything above basic Calculus. There’s a good reason for this.

In case it’s not clear from the Gutenberg reference, Mara’s story takes place in an “alternate” 1443 Italy. A bonus cookie goes to anyone who gives an answer she could give. Keep in mind that 1443 predates just about anything taught in a standard undergraduate mathematics curriculum – including most of Calculus.

Mathematical rigor is always appreciated here. But Mara’s own solution may prove disappointing to those who demand it – which is a good reason why I’m inclined to publish reader solutions.

 

My Notes – A Partial Solution

Mara’s original interpretation of the problem, before the Ori added the complexity of infinitely many Mara’s going up as well, is a standard exercise in infinite series. The topic of infinite series is a Calculus II course staple, but you don’t need Calculus at all to get started on the solution. Basic arithmetic is all that is required.

Let’s figure out the total “volume” of Mara’s, first. There’s Mara, of course. So chalk up “1” onto the tally.

Mara is holding a world of size ten percent of herself. And there’s another Mara of size ten percent of the world. This means the “next Mara” is of size one percent of the first. One percent of 1 is .01, so add that to the tally. So far we’re up to 1.01 Mara’s.

So far, so good. The second Mara down is of size one percent of the first little Mara. Hence, she is of size one percent of one percent of the “original Mara.” One percent of one percent of 1 is .0001. Adding that to the tally, we are up to 1.0101 Mara’s.

Hopefully you see the pattern. Adding the next Mara down will bring us to 1.010101, and with each new Mara, we tack on another “01” to the list of decimal digits.

We can keep doing this forever.

$latex \begin{array}{ccccc} & 1. \\ + & 0. & 01 \\ + & 0. & 00 & 01 \\ + & 0. & 00 & 00 & 01 \ldots \\ \hline \\ & 1. & 01 & 01 & 01 \ldots \end{array} &bg=e6eaea&s=0$

So, what’s the final tally? You may have recognized we have repeating digits. We can convert this decimal representation into the fraction $latex \frac{100}{99}&bg=e6eaea&s=0$.

The formula for the volume of an ellipsoid is well known. If the egg-shaped Mara is the center of the ellipsoid, her height is 27.5 inches from center, width is 7 inches from center, and depth is 3.5 inches from center.

 

Mara - egg shaped

 

Divide 55, 14, and 7 in half, then multiply the product of all three numbers by $latex \frac{4}{3}\pi&bg=e6eaea&s=0$ to get Mara’s volume of approximately 2822.197 cubic inches. Multiply by $latex \frac{100}{99}&bg=e6eaea&s=0$ to get a volume of approximately 2850.704 cubic inches.

If you knew the volume of the room, you’d be one step away from answering the question. Well, the original question. But the Ori, he’s a tricky one.

The fact that the series of Mara’s continue upward as well adds a whole new dimension to the problem.

The size of the room is now completely irrelevant. There is no room. Every world is being held by some Mara, and some Mara is contained in a world – studying all the flaws of her bigger incarnation.

 

Selected Reader Submitted Solutions

Reminder, “answers only” solutions will be left to the very end.

 

1. Submitted by: T.

I believe the solution is 10/99 ~= 10.1%

Proof (hoping my math is correct):

On a given world of size 1:
– Mara’s size is 1/10
– The world’s size in her hands is 1/100
– Mara’s size in the smaller world is 1/1000
etc.

The size of all the Maras is then 1/10 + 1/1000 + … = sum (i = 1 to infinity) 10^(-2i+1) = (1/10) * sum (i = 1 to infinity) 100^(-i) = (1/10)*(1/(1-1/100)) = 10/99 ~= 10.1%
Very nice story and drawing by the way 🙂

 

Commentary. Thanks for the compliment! As for the proof, here’s the next thing to think about: Can we assume the size of the world is 1, when the worlds “go upward” as well?

 

2. Submitted by: S. F.

The answer appears to be 90.909090909…% (repeating) assuming that Mara is the beginning of the universe.  Since this is a geometric series, I am not sure how to find out where the beginning besides using context clues, such as the last line that Mara can crush the world.   If the world is the beginning of the geometric series, then Mara would be 9.09090909…% (repeating) of this universe.   The geometric series would be [sum from 0 to infinity of 100(.1)^x*(-1)^x].

 

Commentary. I think this answer reveals the real issue (aside from the series) in this problem. It’s the problem Mara herself was complaining about when she landed by the schoolhouse.

There are two ways to write a geometric series. One way (the above solution) produces 90.9090…%. There’s another way, which produces an answer of 1.0101…%.

Here’s a followup question to anyone who has reached a conclusion similar to the one given above: Is every world contained in a Mara, or is every Mara contained in a world? Or, does it matter?

A Conundrum of Eggs ~ July 2013 Riddle

This month’s riddle is presented in the form of a short story. I am revising the sequel to A Foundation in Wisdom, currently titled An Ember in the Wind. This is not an excerpt, but rather, a side tale that branches off the main work.

This little tale is an experiment of sorts. Afterward, I’ll explain a little bit about it, and what direction I’m thinking of taking with the monthly riddles.

Also, I’ll reveal the solution to last month’s riddle.

1-1
A Conundrum of Eggs

fishing hole

Mara’s fishing hole at the edge of reality.

 

Mara sat on the edge of reality, dangling her feet over the border and watching them fade out and return to existence. “Hello, feet,” she said as she swung her leg back toward her. She thought for a moment about what grand adventures they may have had without her, but ultimately decided she couldn’t have missed out on much if they stayed connected to her knees.

The grassy knoll Mara sat on separated the city of Locana from Fordham Forest. It stood in the shadow of a tall, obelisk structure. Part of the knoll jutted into the woods, forming a little cliff that overlooked a dip. When she first peered over the edge, she saw nothing but a dark abyss. This must be the edge of the world, she thought. Nothing exists beyond this point. It is a blank canvas on which anything can be created.

Mara closed her eyes and listened to the gentle trickling below. The world is rushing out, like a waterfall.

Earlier in the day, Mara came up with the idea of fishing from the void. She spent the previous morning imagining what wonderful treasures she could pull. Oh, how wonderful it would be to fish out a book, she thought. A real one. A real book with illuminations, and not one made by whatever Gutenberg is calling that machine.

Mara stared at the void. Or a friend. Yes, I could pull out a friend… someone to share the knoll with me so that neither of us got too lonely. I could tell him all about what it is like to exist, and he could tell me all about what it is like to not exist.

After sitting for a while, she felt a tug on the line. Her heart raced as she stood up and began reeling in her catch. She pulled it out of the shadow, and grimaced. She had retrieved another catfish.

“Is that all you have for me?” she spoke into the void. “Out of the entire realm of possibility, the nexus of man’s wildest dreams, fish?”

She tossed the fish back into the void. The fabric of the universe splashed, and she thought how peculiarly similar it sounded to water.

Mara then felt a tickling sensation on her shoulder. She saw a bee out of the corner of her eye, screamed, and ducked. Her fishing rod slipped out of her hand and tumbled into the void with a sploosh.

Mara sighed and looked away from the void. By now, the shadow of the obelisk had almost reached the edge of the world, which meant it was about time to meet with her tutor.

She sat on the edge, dangled her feet over the void, and listened to the voices of the university students in the distance. How nice it would be to be a real student here, she thought, and not receive curious glances. At least I have the obelisk.

The obelisk structure was a forgotten tower, part of the University of Locana. It seemed peculiar to Mara that an entire wing would have been omitted from the minds of the Locanans. But, indeed, the obelisk was not at all cared for. Vines grew around and up the structure, and even into the structure – pushing through the brittle mortar that held the stones together.

But Mara knew it was not abandoned. It was the home of her tutor, known as the Ori. Nobody was allowed to meet with the Ori. At least, that was what she gathered from the fact that nobody at the university was allowed to speak about him.

On the day of her first appointment, Mara wandered the entire grounds looking for him. Nobody seemed to know where he lived, or even who he was. They must have taken a vow of silence about the matter, she thought.

But Mara was persistent, and soon ruled out all locales except the dilapidated obelisk. When she pushed away the vines and revealed a door, she knew this must be it. And, indeed, there was the Ori. He already knew who she was, and that she was sent by Aphrael, the priestess, to learn about the nature of reality.

For the past couple of months, since the start of the planting season, Mara had met with the Ori once a week. Each week he had given her a new problem, and discussed his disappointment in her failure to solve the previous.

Mara wasn’t inclined to give up. For the tenth time she pounded on the heavy door. Little bits of mortar rained down onto her hair as she pushed the door open and stepped inside with cautious confidence.

“Have you solved your problem?” the Ori asked, as he always greeted her.

She returned her usual greeting in reply, “No.” Her confidence fluttered out the window and joined the birds flying above.

“I see,” the Ori said, diverting his gaze for a moment.

“Do not be disappointed, Ori,” Mara exclaimed. “I tried. I really tried.”

The Ori nodded. “But if you wish to hold the world, open it up and see inside, you need to learn more than how to fish at the end of it.”

“Oh, I do want to hold the world,” Mara said. “See it all. That’s why I work. Hard. But, you see… please don’t be disappointed.”

“I think that if you want to understand the world, we would be best starting with yourself.”

“But I know my own self!” Mara exclaimed.

The Ori drew out a long hum as he stroked his beard. He had a penchant for prolonging his response just long enough to make Mara nervous. As soon as he saw her eyes look away, he replied, “Do you?”

“Oh, yes. See, I saw my image in the mirror, and immediately didn’t confuse me with my reflection.”

The Ori continued to hum, and after a moment Mara felt compelled to explain further. “Most people look in the mirror and see themselves. I look in the mirror and see another Mara who may look like me, but is much less creative and far more annoying in her way of always copying my actions.”

“That would be irritating, yes,” the Ori said. “However, I feel that you do not know yourself as well as you say you do.”

Mara shook her head. “Oh no, no, no. I know myself very well. Much more than my place in the world, it seems.”

“Well, then let us find out,” the Ori said. “One can hardly determine their place in the world if they don’t know how much of it they are occupying.”

“I don’t see how myself matters at all.”

“It matters all too much,” the Ori said. “If you wish to create a true model of the world, it must include yourself. Should it not?”

“Well, I don’t think I take up too much,” Mara said. “I demand very little, if anything at all – hardly any food or drink, and I never throw perfectly good things to waste.”

The Ori stroked his beard some more, and Mara grew very worried. He smiled. “How big are you? Do you know?”

“Yes,” Mara said. “I am 55 inches tall. And, I gather, 14 inches wide and 7 inches deep.”

“Then let us assume an ellipsoid Mara.”

Mara gasped. She crossed her arms in front of her chest and lowered her gaze. “Are you suggesting I have the build of an egg?”

“My dear,” the Ori said. “Just for ease in the calculation. It is only in the spirit of efficient mathematics.”

“Fine, so be it,” Mara said, “I see mathematics and flattery do not make good bedfellows.”

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“And you say you wish to be holding a little world. Suppose it’s ten percent the size of you, and on this world is another Mara of size ten percent of it.”

Mara nodded, humming away as she pretended to be in contemplation and not annoyance.

The Ori continued, “And, of course, the little Mara is holding another little world, with similar proportions, and it’s own little Mara, and so on and so forth.”

“Oh! Infinite me’s,” Mara laughed with elation. “Each one of them a darling and not at all egg-shaped.”

“Yes,” the Ori said, and smiled gently with only a hint of deviousness. “So, tell me, darling, what percent of this universe is Mara?”

“Oh, this should not do wonders for my ego, for I know the number will not be that much bigger than my presence in this room, so small all the same.” Mara stared at the ceiling while she thought. “But it shall be easy to compute. All I need to know are the dimensions of the room, of course.”

“Not so much,” the Ori said. “Because a true model of the world includes you, there’s no reason to believe that you’re not on another iteration of the model, with a bigger Mara watching you on a similarly proportioned sphere. And an even bigger Mara watching her, and so on and so forth upward as well.”

“Oh, I do not think I like this anymore,” Mara said as she bit her lip. “A bigger me would certainly have a bigger ego, and I shall fear I’d crush me like a bug out of the frustration of not knowing the answer.”

The Ori nodded solemnly, then smiled. “Then you best be off working on the solution. I much prefer for myself to be in three dimensions and not reduced to two, should you crush the world.”

 

{ Beat Mara to the solution! }

 

 

A Snippet of the History of Mathematics and Fiction

 

As mentioned at the start, this story was an experiment. It was mostly inspired by Lewis Carroll’s A Tangled Tale. Most people know Carroll for Alice in Wonderland – so A Tangled Tale seems to take the backseat in discussions of his work. The tale was a serial work, published in a magazine between 1880 and 1885. Each installment, called a knot, presented some number of mathematical problems.

Because A Tangled Tale was published in installments, Carroll would also discuss reader answers to the questions posed in the previous story.

Carroll was also known for the ribbing of readers who submitted incorrect answers, which was not always appreciated.

A Tangled Tale wasn’t the only such work attempting to combine mathematics and humor. A couple more examples would be Sideways Arithmetic from Wayside School, a spin-off of the Wayside series by Louis Sachar; and Dude, Can You Count? by Christian Constanda.

The latter two examples were published as books. A Tangled Tale was unique in that it included the dialogue between Carroll and the original readers. The entire text (the tale, and discussions) is available on Project Gutenberg.

As for the monthly riddles, here is a great opportunity to experiment a bit with the format of A Tangled Tale.

Each “riddle” presents a problem or two. At the end, the reader is invited to submit a solution before Mara does.

Speaking of solutions…

 

Solution to the June 2013 Riddle

 

If you haven’t worked out the solution to the June 2013 riddle for yourself, this is your last chance!

Seriously.

Once you read the answer, you can never solve it on your own… which, I may add, is an immensely satisfying feeling.

Of course, you could just argue with me over my solution, or how I worded the question. Depending on your personality, this may be just as fun.

 

Solution

The ages of the three children are 6, 6, 1.

 

Explanation

Textbook questions tend to give the student exactly what information they need to solve the problem… which I believe is poor preparation for the real world. But, I digress. There is just enough information in the narrative to solve the puzzle, and not a drop more.

The trick is rather devious. And it lies with the seemingly useless information that the mathematician gives his friend. The information itself isn’t useful (immediately), but the fact that it was needed does shed some light on the puzzle.

The product of the ages is 36. If you write all the ways you can multiply three numbers (positive integers, for those of you inclined to nitpick) to arrive at 36, you’ll find that most of the triples add up to a unique sum.

For example, 3, 3, 4 multiply to 36. These numbers also add up to 10. In fact, it is the only set of three positive integers that add to 10. If these were the ages of the children, the friend wouldn’t have needed the extra piece of information.

And this is the rub. There are only two sets of three positive integers that share the same sum: 6, 6, 1 and 9, 2, 2. The solution has to be one of these sets, because otherwise the friend would have been able to solve the problem without additional help. (Remember, he can see the number of windows).

Once you figure that out, the extra information itself is useful. 9, 2, 2 can’t be right, because there isn’t a “youngest child.” There are two “youngest children.” This means the correct answer is 6, 6, 1.

 

Winners List

 

No correct solutions were submitted – but thanks to everyone who submitted an answer.

Artwork Endeavor Five Year Anniversary

Drawing has been an on-and-off hobby of mine since, well, I can remember. When we were little, my brother and I used to spread huge pieces of moving paper on the living room coffee table and create mural-like drawings.

Thinking back, I’m sure some of them were inspired by Mark Kistler’s Secret City drawings. We were sometimes shown these videos in school.

 

[youtube=http://www.youtube.com/watch?v=4tK70tHKhME&w=400]

 

 

I always liked to doodle every so often, but for the most part, I considered my brother to be the family artist.

Five years ago my wife and I moved away from Chapel Hill, NC. She had just finished her degree, and since I was still attending NCSU, we moved closer to Raleigh.

With all of our stuff in boxes, all I had to entertain myself were a couple old boxes of colored pencils (the date stamped on the box goes back to the Clinton administration).

I was working on this project called History of the Wiener Dog, which would become An Orthogonal Universe. I had just become stuck on what would become A Foundation in Wisdom, when I had an epiphany. Why don’t I try drawing out the scenes I’m writing?

With that, I immediately got started procrastinating. Below is the very first drawing I completed in summer 2008.

 

Helicopters

R: You look real goofy driving that old thing.
L: Yeah? Well, that’s the most hideous fedora!

 

The above drawing has nothing to do with the plot of An Orthogonal Universe, but it did have one significance. I would soon start making a real effort to improve as an artist.

I took my old Clinton-era colored pencils and colored Ry, the Squirrel.

 

Ry the Squirrel

Poorly drawn Ry the Squirrel!

 

There was a novelty in seeing some of the characters I spent the past year writing about “come to life” in a visual form. Maybe that’s why I kept going, even after I broke through the writer’s block.

 

Start of Artwork Endeavor

 

I consider the image below to be the first drawing that came after a point I said to myself, “hey, I think I’d like to give improving a shot.”

This is a scene from An Ember in the Wind, the sequel to A Foundation in Wisdom.

 

Poorly drawn Mara

Mara and the Puppets! Mara has always had a Wolfpack Red tunic. The colors of Locana, the city she lives in, are Carolina Blue.

 

Eh, that wasn’t very good. The “puppets picture” is an important one, though. Every so often I redo it. The problem with learning any skill, especially if you’re self-teaching, is that progress seems to come so slowly. Improvement is measured in months and years, which is problematic if you get discouraged early on.

Below is the last colored pencil drawing I ever did. It’s John Bartlebee and Sheridan, the protagonists of the An Orthogonal Universe series.

 

LeSabre '88

The stars of An Orthogonal Universe: John Bartlebee and Sheridan! John drives an ’88 Buick LeSabre. John operates his school of the classical sort out of his classic car.

 

Sometime later I realized I could scan my line-art drawings and color them on the computer. I sat down with my copy of The GIMP, and griatch-art’s tutorials, and gave it a go.

Hey, it’s the puppets picture again!

 

Mara 2

Slightly better Mara and the Puppets. This is the very last drawing I did with a mouse.

 

2009 – 2010

 

My wife got me a Wacom Bamboo pen for Christmas 2009. Once I got the hang of it, it made a huge difference.

But, starting in 2009, my free time dwindled considerably. In Fall 2008 I finished the last of the core coursework in my degree program, and began working with my research advisor.

Qualifier exams are notorious in just about any Ph.D. program. Despite their infamy, the workload only goes up once they’re over and done with. At least at NC State, though, the pressure goes down.

The qualifiers make up the last “gateway”, after which, few people leave the program. Still, I didn’t have much time for anything other than quick doodles. Here’s an illustration of Marcus, the protagonist of A Foundation in Wisdom.

 

Marcus

An old illustration of what is now chapter 12 from A Foundation in Wisdom. Marcus is contemplating the highway.

 

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2011 – today

 

I graduated in May 2010. After my first semester as a full-time professor, I began to have more free time.

Everything except my degree program was put on hold until I graduated. Afterward, I came up with the idea of an illuminated hypertext novel. I started creating a lot of full-color illustrations for A Foundation in Wisdom, one for each chapter. This version had 30 chapters.

Of the set, the image below is one of my favorites.

 

Eru

Chapter 17 of A Foundation in Wisdom. Marcus climbs the Mount of Mislor to meet with Eru, the wise man.
Random trivia: “Mislor” was my mom’s AOL username back in the mid 90’s. I think my brother came up with it.

 

If you’ve read the book, some of these scenes may look familiar. The one below won’t, unless you read the 2011 version.

 

Peoria

Peoria from A Foundation in Wisdom. This was an illustration of a scene in which Marcus and Peoria went inside a leaf. The scene was deleted from the final manuscript.

 

Vasigari

Vasigari, the Priestess from A Foundation in Wisdom. Poor Marcus – do legs bend like that? I still like that forest.

 

Like I said earlier, learning a skill can be frustratingly slow, unless you’re the patient type.

I like to think I’m of the “patient type,” at least, most of the time I am. But, sometimes, it’s easy to look at the work of people who have been at it much longer than you – and forget just that – that they’ve been at it longer.

That’s why I sometimes redo the “puppets picture.” It was the first illustration. This is the last version, and already it’s over a year old.

 

Mara 3

Mara and the Puppets – the latest version! Recognize the machinery?
Fun fact: The desk and lamp is a reference to the attic in Alone in the Dark.

 

It was completed in March 2012 (although it’s dated 1 April). At this point, I was still planning on releasing An Ember in the Wind as an “illuminated hypertext novel.” So I needed to redo this illustration for the new site, anyway.

This was also the last illustration I created with the intent of releasing An Ember in the Wind as an illuminated hypertext novel.

For various reasons, I pulled the project. A couple months later, I re-declared A Foundation in Wisdom as a novel, and began working with editor Kisa Whipkey to polish it.

While I was waiting for the manuscript to come back, I took requests from random people on deviantART.

 

 

superheroes

Someone on DeviantART asked me to draw her and her friend as superheroes. That’s a futuristic San Francisco in the background.

 

And, just because I felt bad for John Bartlebee and Sheridan, I redid their scene.

 

LeSabre 2

John Bartlebee and Sheridan make their return after 4.333 years. That’s a much nicer looking LeSabre!

 

Now we get to this year! Although I’m not doing illustrations for an “illuminated hypertext novel” anymore, I still enjoy drawing. Up above I mentioned griatch-art’s tutorials. Below is a scene depicting one of his characters.

 

Biltmore

griatch-art@deviantART asked people to draw his characters. That’s Ebb the Dragon, about to make a snack of the Biltmore estate.

 

We’re coming to the end! After nearly five years, I figured it was about time to try drawing myself. I tried using a mirror as a reference, but I kept getting myself backward.

Thinking about what happens next in a story can be hard work. Despite all those monitors, I still primarily use 15-cent Walmart notebooks.

 

self

Incomplete self portrait. Shown: me, hard at work revising A Foundation in Wisdom, while in actuality, putting the real task off.

 

And now, the latest illustration – completed five years after I drew the very first doodle in my notebook.

Even though I’m not working on an “illuminated hypertext novel,” I haven’t lost sight of why I started drawing these scenes in the first place. Drawing the scene out is a great way to break through writer’s block. And since I’m working on An Ember in the Wind, well, here’s Mara again.

 

Forest

Five years later, I finally drew a forest I’m happy with.
Mara still has her Wolfpack Red tunic. Go State!

 

After five years of self-teaching, I’m finally pleased with how I’m doing. As for the illustrations, I may have found a use for them.

I don’t think my experiments with the “hybrid novel”, or “illuminated hypertext” are done. But utility aside, sometimes it’s just fun to draw.