Dr. Dragon Issue #8

DR. DRAGON HSMSE’S MATH, SCIENCE, ENGINEERING AND ARCHITECTURE MAGAZINE

WHAT’S INSIDE

THE BODY AND WATER FOOD WASTE COLOR BLINDNESS EXTREME ANIMALS COLLATZ CONJECTURE AND MUCH MORE!

winter

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Dear Readers, Welcome to our first ever print winter issue! The last semester has been incredible as I have seen the club members and board rally around the magazine as it grows. However, this could not have been possible without our advisor, Mr. Choi, who continually kept us on track, and our new advertising sponsors without whom we could not have gone to print. Together, we have created a magazine I am immensely proud of. We would also like to specially thank Alsion and Ken Aun for their contribution to Dr. Dragon. I hope you all enjoy the magazine as much as I do.

Carin Queener, president P.S. Look out for Dear Dr. Dragon next semester and send us your questions!

STAFF PRESIDENT

DESIGNER IN CHIEF

VICE-PRESIDENT

DESIGNER

TREASURER

EDITOR IN CHIEF

SECRETARY

EDITORS

CARIN QUEENER REBECCA MIKOFSKY JOY AUN

HAMIDA CHUMPA

FACULTY ADVISOR RONALD CHOI 1

FATEMA BEGUM JIM HUYNH

SHARON YOUNG

RICHELLE CHEN JEENERA HOSSAIN IRENE OK TAYLA GUENZBURGER

WRITERS

JONAH ROHLFING CARIN QUEENER REBECCA MIKOFSKY LISETTE PERES EANA BACCHIOCCHI ISAIAH JEREMIE ALEX SOLIVAN KUN YU AFSANA RAHMAN

SPECIAL THANKS HSMSE PTA

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Koenigsegg Regera

THE BODY AND FOOD WATER WASTE

JONAH ROHLFING

CARIN QUEENER

EANA BACCHIOCCHI

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EXTREME ANIMALS

COLOR BLINDNESS

ABC CONJECTURE

REBECCA MIKOFSKY

LISETTE PERES

12 13 MIT SCANDAL

ISAIAH JEREMIE

COMPUTER SIMULATIONS AFSANA RAHMAN

ALEX SOLIVAN

15 Collatz Conjecture KUN YU

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the Koenigsegg Regera The Koenigsegg Regera is different. When I first saw a picture of this car called the Koenigsegg Regera I was totally transfixed; I genuinely believe that it is a work of art. It IS handbuilt and the founder of the company looks over each individual car. This has been going on since the dawn of cars, and enthusiasts become so addicted and accustomed to the looks, the noise, and the act of driving,that they have become synonymous to pleasure for many people. These cars become more than just cars, and begin to embody art.

Like many family hatchbacks, it is a hybrid, which means it runs on both electricity and gasoline. This, now, is not as uncommon in high-end cars as you would think. There are many different high-performance cars that use electric and gasoline powered engines to get incredible amounts of power. The hybrid system on performance cars gives about 10 -15 times the power of a normal gasoline powered hatchback, so there is no doubt that hybrid systems work extremely well. This seems all fine and good, especially with the Regera’s claimed top speed of 280 miles an hour.

However, some devoted car fans are raising their eyebrows at this car, myself included, at simply the way it gets to that speed.

The act of changing gears has been an essential part of the intricacies of the performance car. The Regera however, does not have any gears. There have been attempts at this before, but none of them have been really successful. To solve this, Koenigsegg has been very smart, by controlling the amount of power that comes out of both the electrical and the gasoline powered engines. The car’s two engines are a 1136 horsepower gasoline powered engine, and a 700 horsepower electric motor. That adds up to 1,836

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horsepower. The thing is, not all the power is delivered at the same time. The engine is high wound, so it can make the car go very fast, but only if the car is already travelling at a very high speed. Think of it like this. If you are riding a bicycle stuck in a high gear, it is really hard to get the bike moving. But, once it starts moving, you can make it go much faster than you would be able to if you were in a lower gear. In cars, while you are trying to move from a slow speed or from a stop, the electric motor pushes the car when it is most difficult for the engine to move the car.

Then, when the car gets faster and faster, the engine produces more power, as the electric motor decreases in power. This is what makes it possible for the car to reach 280. Now, while it may feel like something is missing when you are denied the ability to shift, it may be worth it, as the lack of gears can help acceleration immensely. With gears, there are gaps in power while shifting, and with no gears there are no gaps in that powerband. Therefore, this hybrid system serves a double purpose; it increases the possible performance of the cars, but it also helps ease the car enthusiasts away from the gasoline powered cars. Not only is this true, but this car is a

gateway for mainstream performance cars to be hybrids too. The BMW i3 is smilar to the Koenigsegg. The i3, unlike the Koenigsegg, is fully electric. It is not built for performance, as it is built for sensible city driving. It is practical as well; it is a hatchback. However, it is hitting a market that no other cars have ever gone before. It is a tiny electric hatchback car, but it is very beautiful.

While that may be subjective, it really looks like nothing else on the road. It is extremely striking, and it pays tribute to “concept cars.” This is so important, because it has received great reviews even from devoted car enthusiasts. This, like the Koenigsegg helps gently move the lovers of cars to electric cars, as that is where the future is going.

The reason that we have gasoline cars still are for multiple reasons. First, the availability of gasoline is enormous. Yes, it won’t always be this way, but for right now, the convenience is much better. Also, it is much more familiar. It is comforting not to change your routine, especially for something as important as your car. Also, with gasoline in an average car, you can go 300-400 miles per tank. In an electric car, it is only about 150 miles until you run out of charge. When that happens, it takes a few minutes to refill a gas powered car, while it

takes many hours to recharge an electric car, from 1020 hours using an outlet. Also, when cars get as expensive as they do, they become more than just things that move you around, they are moving works of art. And the removal of gasoline engines, while very good for the environment, takes away some of the spirit of the car itself. So this is why the Regera is so vastly different; it helps the car stay artful, while helping the environment. So the Regera changes things for the car world, and definitely for the better.

-Jonah Rohlfing Astrén, Eddie. “Regera - Koenigsegg.” Koenigsegg.

Koenigsegg, 2015. Web. 31 Dec. 2015.

Duff, Mike. “2016 Koenigsegg Regera: 1500 Horsepower, Single-Speed Gearbox.” Car and Driver. 2015 Geneva Auto Convention, Mar. 2015. Web. 8 Feb. 2016.

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got water?

Go a few hours without water and you will find yourself pretty thirsty. A little longer, you will have a rapid heartbeat and extremely dry eyes. Within 3-5 days, most people die.

Clearly, water matters. But why? The brain, heart, lungs, skin, muscles, and kidneys all consist of large quantities of water, such that the human body is composed of a vast amount of water. This has been the center of a plethora of jokes and pickup lines, such as “If the human body is 60% water, why aren’t we constantly drowning? ... Luckily it’s just the bottom 60%, so that’s only a little over waist deep.” However, is this really true? While some humans are actually 60% water, the reality is much more complex.

The first factor that dictates how much water a person contains is his/her age. Adults are usually about 60% water, which is the basis for most popular statistic. However, when babies are born they can be up to 80-90% water, which is why their skin is so soft.

Conversely, the elderly can be as little as 50% water.

This reduction in water concentration is due to the increased fat and bone mass that comes with age.

Neither fat nor bones contain high amounts of water, resulting in a lower overall concentration.

This is also why women tend to be less watery than men; women tend to have more fat, whereas men tend to have more muscle, which contains a much higher percentage of water.

Going into more detail, each part of the human body contains an exact amount of water required for it to function.

•The vitreous body ofthe eye is 99% water.

•Blood is 90% water.

•The heart is 79% water.

•Muscles and internal organs are 75% water.

•The brain is 75% water.

•Bones are 30% water.

•Teeth enamel is only 1.8% water.

So, if you are more bone than muscle, you are probably also a little less watery. Without water, your body’s entire function is disturbed. Your internal body temperature is disrupted, you can not metabolize

protein properly, your joints cannot be properly lubricated, and you cannot form saliva, among many other things. Water is essential for the body to sweat and release toxins. Without the flow of water there is no water to remove waste and toxins through your stool. Moreover water slows down the aging process and makes skin smooth.

This is why it is important to drink the proper amount of water every day. Because blood is 92% water, bones are 22% water, and muscles are 75% water, it is no surprise our immune system requires an ample amount of water.

However, it is also a myth that you need eight 8oz glasses daily; the amount of water you need varies depending on your total water composition. The fastest way to calculate the amount of water you need is to divide your bodyweight (in pounds) by 2 and drink at least that many ounces of water each day.

In the words of Da Vinci, “water is the driving force of all nature,” so make sure it is not neglected.

-Carin Queener 3, Drops Of Water. Water and Our Health (n.d.): n. pag.

United Nations Educational, Scientific and Cultural Oganization. Venice Office Regional Bureau for Science. Web. 30 Nov. 2015. <http://www.unesco.org/new/fileadmin/MULTIMEDIA/FIELD/Venice/ pdf/special_events/bozza_scheda_DOW03_1 .0.pdf>.

Reagan, - Jennifer M. “Half of Your Body Weight in Ounces of Water Per Day – Are You Drinking Enough?”

Bamboo Core. Jennifer Reagan & Bamboo Core, 2015.

Web. 30 Nov. 2015. <http://bamboocorefitness.com/ half-of-your-body-weight-in-ounces-of-water-per-dayare-you-drinking-enough/>.

Editorial Staff. “How Long Can You Live Without Water?” How Long Can You Live Without Water? Curiosity Aroused, 2015. Web. 30 Dec. 2015. <http://www.curiosityaroused.com/health/live-without-water/>.

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FOOD WASTE NOT Environmentally or Economically Sound

Think about every time you clear your plate. How much is left on it? A mouthful, a bite, perhaps a serving of a vegetable you do not like? While this may not seem like a lot, many people do not realize the toll a simple thing like food waste could have on significant environmental and economic issues. In reality, reducing your carbon footprint also comprises of simple actions such as cutting back on individual food waste.

Humans have existed on this planet for over 2,000 years, which prompts the environmentally sound question: Where does all our food waste go? Early settlers practiced the hunter and gatherer method, where human societies obtained their food from wild plants and animals, and then evolved into an agricultural lifestyle. Currently in 2015, our food is not as biodegradable as due to complex food technology.

About 60 million metric tons of food are wasted per year in the United States.

A simple banana peel takes around a month to biodegrade. Now, think of a banana peel per person per day just in the United States. That is about 320 million banana peels sitting in municipal landfills, for a month until they are completely decomposed within

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the earth. During that month of the banana peel sitting in a landfill, waiting to be decomposed, it emits methane, a potent greenhouse gas.

Greenhouse gases are any gaseous compounds in the atmosphere capable of absorbing infrared radiation, thus trapping and holding heat in the atmosphere. This results in the greenhouse effect, which then leads to climate change. Furthermore, food waste is not just hurting the environment once discarded, but also when it is produced and transported to consumers. Unless the produce is bought locally, the majority of your food has traveled thousands of miles from agricultural states to your grocery store and then to your own plate. Those thousands of miles driven to get there requires tons of fuel burned to refrigerate, process, and transport the food, thus contributes to air pollution.

The amount of food waste in United States is much greater than a single banana peel per person per day. About 60 million metric tons of food are wasted per year. The total cost of the land, water, and resources to supply the production of the wasted food forces the government to lose $162 billion.

This economic problem does not only exist in the United States; the total cost of food waste is globally estimated to be as high as $400 billion a year.

Reducing this waste from 20-50% could save $120$300 billion a year.

As world population increases, this amount and its cost will only grow. By 2030, consumer food waste is estimated to cost almost double, at $600 billion.

There are simple ways to help alleviate this problem with steps any individual can take! Making the effort to be conscious of how much food you are throwing away after every meal and how biodegradable these foods is practical and essential. Another extra step is to compost. Composting involves throwing all your food waste together and allowing the materials to break down over a period of time. This is beneficial to the environment because it recycles organic resources while conserving landfill space as well as reducing the need for commercial soil conditioners and fertilizers for the soil.

Within New York City, there are tons of places to drop your compost off and then it goes straight back into the Earth.

- Eana Bacchiocchi

Nixon, Ron. “Food Waste Is Becoming Serious Economic and Environmental Issue, Report Says.” The New York Times. The New York Times, 25 Feb. 2015.

Web. 10 Dec. 2015.

Hatz, Diane. “End Waste Now- Facts.” Food Waste.

Natural Resources Defense Council, 2013. Web. 08 Feb. 2016. <http://www.endfoodwastenow.org/index.php/resources/facts>.

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Extremophiles Extremophiles are exactly what they sound like organisms, usually microorganisms, that “love” (phile means love in greek like pedophile) extreme temperatures or conditions. They can be found in places like the seafloor, deserts, underground, or hotsprings (like Yellowstone park). Many extremophiles are bacteria or archaea (prokaryotes), but they could be other types of organisms.

There are different characterizations of extremophiles based on what conditions they thrive or survive under •acidophilic - acidic environments (ph levels 1 to 5) •alkaliphilic - basic (ph above 9) •anaerobe - no oxygen •halophilic - salty environments •thermophilic - hot (140F to 176 F) •hyperthermophilic - extremely hot (176F or hotter) •psychrophilic - cold (32F to 68F, best at 60F) •piezophilic, barophilic - high hydrostatic pressure •radioresistant - resistant to high levels of radiation •oligotrophic - nutrient-low environments •endolithic - within a rock or mineral •metallotolerate - high levels of heavy metals •xerophilic - dry That is just a small subset of a growing list of extreme conditions in which certain organisms can tolerate.

Many extremophiles have more than one characteristic and are thus polyextremophiles. While these creatures may seem insignificant, they actually benefit human science. We can study and use their extremozymes (enzymes “that are functional under extreme conditions”) because industry and research can use them in conditions where normal enzymes are no longer effective. These extremozymes could benefit medicine and technology. Scientists can use the methods they have of surviving extreme environments

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to benefit humanity’s uses in extreme environments.

Also, extremophiles can help us understand how to survive under extremes and how extraterrestrial life survive.

Deinococcus radiodurans or “Conan the Bacterium” is extremely radioresistant (one of the most) as well as cold, dehydration, vacuum, and acid resistant.

These extremophiles are capable of transforming and repairing irradiated DNA, especially since they carry four copies of their genes. They are effective and consistent at DNA repair. This is a remarkable accomplishment and understanding this mechanism can help medicine and humans. It has also been used to detoxify especially in locations with radioactivity.

A well-known example is tardigrades or water bears, due to their resemblance. They are 0.05mm to 1.2mm in length and are invertebrates with 4 to 8 tiny claws on each other their 8 legs. They eat plant cells, animal cells, bacteria, and even other tardigrades. They tend to live in watery environments like deep sea or rainforests especially on lichen or moss. These little bears are extremophiles because they survive from around -300F to 300F, freezing and thawing, varying salt concentrations, lack of oxygen. lack of water, radiation, certain chemicals, vacuum (low pressure), and high pressure. They survive by shutting down their bodies so their metabolisms stop. This procedure is called cryptobiosis. Other organisms, especially other extremophiles also undergo cryptobiosis.

-Rebecca Mikofsky Niederberger, Thomas. “Extremophile | Biology.” Encyclopedia Britannica Online. Encyclopedia Britannica, 24 Nov. 2015. Web. 30 Dec. 2015.

color blindness Color Blindness affects approximately 32 million Americans and now there are glasses that can help treat this deficiency. EnChroma and Oxyamp are two brands that create these special glasses. The way they were discovered are ironically similar. Donald McPherson originally made laser safety eye wear for surgeons, but when one of his colorblind friends wore the glasses he was able to see a range of colors including reds and greens. Discovering that the glasses helped the colorblind, McPherson decided to shift his focus to creating eyewear for the colorblind and thus EnChroma began.

The story of how O2Amp came to be is very similar.

Mark Changizi researched human vision and how it could see changes in skin color. He and his childhood friend, Tim Barber, began a research institute called 2AI Labs. Changizi designed glasses to enhance the vein by sharpening the ability to see the red- green or blue-yellow parts of the spectrum. When he tested it with people the feedback he received was unexpected. Some users said that it helped cure their color blindness. Both of these brands had a similar start but overall the results are incredible.

One such testimonial of these glasses is from David Pogue and in a New York Post article he talked about how he tried the Oxy-Iso ( which is made by 2AI Labs) but his color blindness was too severe to work.

When he tried the Enchroma glasses they actually allowed him to see much clearer. He said “I was floored…

All of a sudden, greens and reds looked richer.” These glasses could give hope to the millions of Americans who have it. There is a small catch. The glasses will not work indoors or on a computer screen which is fine because who wants to be inside when you can see the whole world in a new perspective.

You have heard about these glasses and how they work but how does color blindness work? Well in order to answer that question one must understand the eye and how it catches light. Some parts of the eye that you may know include the retina , iris, cornea, and etc.

A defective iris or cornea results in short sightedness or long sightedness. But the retina is what causes color blindness.

The retina has cones and rods. The rods help distinguish between black and white, which help you see in the dark. While the cones work in bright light, they help distinguish colors. The cones and rods work together to send signals to the optic nerve and brain, when light strikes the retina. This creates an image based on the colors you see. A normal eye has three types of cones: L, M, and S. Working together these cones will allow you to see the entire visible light spectrum. If one of these cones are malfunctioning or absent it results in colorblindness.

These glasses are revolutionary, especially for the millions of people who have to live their lives without the joy of color. This is one of the first steps for scientists, but the future is even more promising.

Hopefully scientists can use technology to help other types of people with eye problems and even one day the blind. This is just a beginning to a revolution between vision, technology, and science.

-Lisette Peres Yam, Kimberly. “New Glasses Transform The Way Colorblind People See The World.” The Huffington Post.

The HuffingtonPost.com, 13 Jan.Web. 31 Dec. 2015 Maldarelli, Claire. “How Does Colorblindness Work?” » Scienceline. Health, Blogs, 28 2014. Web. 31 . 2015

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sHinchi mochizuki &the ABC CONJECTURE

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a brilliant mathematician, Shinichi Mochizuki released a 500 page paper about a problem that has plagued mathematicians for over 30 years.

However even professionals cannot even understand his work. In fact, although the problem seems simple, even the question is difficult to understand, despite the fact that ironically, the problem revolves around the first three letters of the alphabet: A, B and C. This is the problem: Step One: Pick two numbers for the variables A and B. that do not share any factors beside 1.

Step Two: Add them together to get C. Step Three: Factorize A, B and C until there are only primes.

This is where things get interesting. If we gave A a value of 1,024 and B a value of 81, we would get a value of 1,105 for C. If we were to factor them out, we see that 1,024 is equivalent to 210 and 81 to 34, while 1,105 is equivalent to 5*13*17. The sentence can now be rewritten as 210*34=5*13*17.In this equation, the right side has less digits than the left side. This is normal.

Now, assign different values for A and B. If A is equal to 3 and B is equal to 125, C equals 128. When factored, A is 31, B is 53, and C is 27. Unlike the number of digits in the first equation, the number of digits in the second equation has more digits on the right side than it does on left side. This is incredibly rare.

For the next step, multiply the digits that make up the left and right sides. For the first equation, it would be 2*3*5*13*17, which equals 6,630. This value is larger than the value of C, which was 1,105. For the second equation, 2*3*5 is equal to 30, which is less than 128.

This variation is incredibly fascinating.

Now, everything discussed so far can be summed up into one formula: rad(abc)k. which can be rewritten as (a*b*c)k. If the value of K is 1, there is

an infinite number of solutions in which rad(abc)1<c.

However, if the value of K is anything greater than one, there is a finite number of possibilities where rad(abc)k>1<c.Think about that for a second. There are more numbers than there are seconds in the life of the Universe and more combinations of numbers than there are grains of sand on every beach on the earth. Yet just because one value has changed, the solutions go from the number of stars in the universe to something far more tangible.

if true, it could revolutionize mathematics by solving one of its biggest mysteries.

We are now able to see that the questions behind the ABC Conjecture ultimately intrigued Mochizuki to invent a new branch of math which then allowed him to write a 500 page evidence behind the ABC Conjecture.

-Alex Solivan Castelvecchi, Davide. “The Biggest Mystery in Mathematics: Shinichi Mochizuki and the Impenetrable Proof.” Nature.com. Nature Publishing Group, 7 Oct. 2015. Web. 14 Jan. 2016.

Hartnett, Kevin. “Hope Rekindled for Perplexing Proof” Quanta Magazine. Simons Foundation, 21 Dec. 2015.

Web. 14 Jan. 2016.

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the mit lottery scandal With $600,000 dollars they were able to win a total of $8 million

Students at the prestigious Massachusetts Institute of Technology (MIT) are expected to use their intelligence and capabilities for good, for example: pioneering nuclear fission. However, this is not always the case.

In 2012 it was revealed that a group of students and a researcher at MIT participated in the scamming of the Massachusetts Lottery.

Typically the goal when playing the lottery is to hit the jackpot, but this group took a different approach and was able to profit millions without ever hitting the extremely rare six-digit prize. They realized that when the lottery is not won, the prize is added to the next drawing, creating a larger jackpot, which is capped at $2 million.

When the jackpot hits $2 million and still has not been won, it is “rolled down” to smaller prizes. However these prizes are still between 5 and 10 times larger than the average prizes. The students purchased $600,000 dollars worth of tickets, which almost guaranteed then a 15-20% return on investment because of the

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rollover prizes. The group was supported by investors who would receive some of the profits. With $600,000 dollars they were able to win a total of $8 million, causing them to quit their day jobs and live off of the scam. This is not the first time MIT has been involved in something like this. Approximately ten years prior a group of students were involved in a card counting scandal in Las Vegas casinos.

-Isaiah Jeremie Kavoussi, Bonnie. “Massachusetts Cash WinFall Lottery Was Monopolized By Gamblers As State Turned Blind Eye: Report.” The Huffington Post. TheHuffingtonPost.com, 08 Feb. 2012. Web. 18 Jan. 2016.

London, Jay. “A Calculated Approach to Winning the Lottery.” Slice of MIT from the MIT Alumni Association RSS. IN THE NEWS, RESEARCH, MANAGEMENT, 14 Aug. 2012. Web. 18 Jan. 2016.

Gorman, Ryan. “MIT Students Scammed Mass. Lottery for $8M .” NY Daily News. New York Daily News, 3 Aug. 2012. Web. 18 Jan. 2016.

are we real?

What if there were binary numbers hidden deep within us, proxy servers built in the skies, and a motherboard watching from above, causing us to live in a simulated program?This may be closer to reality than you think it is. Philosophy professor, Nick Bostrom, at Oxford University proposed this first hand, believing that what we observe as our universe may be a very complex and advanced simulation created from our distant prodigies. Bostrom perceives that we live in a reality based on the laws of math. When we discover any mathematical truth, we are simply discovering parts of the code that our “programmer” used.

This may seem unsettling, but Bostrom argues that we are likely living in a simulation. There would be a greater amount of simulated worlds than non-simulated ones, indicating that we may be present in a simulated reality. Bostrom’s theory, known as the Simulation Theory, has five components: (i) Other intelligent civilizations exist. (ii) Their technologies grow exponentially. (iii) They do not all go extinct. (iv) There is no universal ban or barrier for running simulations. (v) Consciousness can be simulated. All of these five premises must be true, with no exceptions, in order for the universe to be simulated and created.

A team of researchers from the University of Bonn in Germany claim they have evidence that might prove this theory. In the paper, “Constraints on the Universe as a Numerical Simulation,” they indicate that the current simulations of the universe are too weak and small to naturally fit physical laws, thus setting limits to such laws. The problem with these simulations is that the laws of physics, which appear continuous, have to be placed onto a discrete three dimensional lattice which advances as time progresses.By being in a simulation, the computer would have to set limits within many aspects of the universe, such as the energy the particles can have within a program. These limits would be felt by those living in the simulation.

Therefore, it might be possible for these limits to exist and we might have found one of them.

Does the Simulation Theory express the existence of a god? Bostrom says that it could be a version of

either theism or atheism. It could also reveal a weaker form of the creation hypothesis because the simulators would have some of the characteristics we associate with God, in the sense that they have created our world.

We do not have all the answers for the occurrences and events leading to the creation of the universe, but this theory may hold the key for figuring out one part of our existence, why we are the way we are, and how we even exist in the first place.

Whether or not you choose to believe the idea that our reality is a simulation, this frontier may open doors for understanding more of what is surrounding us and aid future endeavors of exploring the universe. We can even warp the universe and create simulated realities ourselves with this theory, becoming part of a larger reality.

-Afsana Rahman “Are We Living in a Computer Simulation?” PBS. PBS. Web. 25 Dec. 2015.

Bostrom, Nick. “Simulation Argument.” Web. 25 Dec.15 Frenkel, Edward. “Is the Universe a Simulation?” TheNew York Times. The New York Times, 15 Feb. 2014.

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collatz conjecture Mathematics is an art, but also a language used frequently by almost everyone from office clerks to physicists and engineers, and people who study it deeply often discover and sometimes solve intriguing problems. Such problems can take decades, but some are imbedded into the foundation of mathematics that their significance is often forgotten, like the Pythagorean Theorem. However, not all problems have solutions or proofs.

One such problem is the Collatz conjecture, or 3n + 1 problem, which is quite possibly the simplest impossible problem in mathematics. The problem got its name from Lothar Collatz, a German mathematician, who first proposed the problem in 1937. The problem is really simple. It goes as follows: Given n is a positive integer and the following iterative sequence is for n: n → n/2 (n is even) n → 3n+1 (n is odd) For example: If we use the starting number 13, the following sequence is generated: 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2→1→4→2→1 The sequence repeats itself in 4-2-1 cycle forever and ever. The conjecture states that the sequence will eventually reach the number 1, regardless of which positive integer is chosen initially.

It may seem as if the problem can be solved by simply showing that all the numbers reach 1, however there is an infinite amount of positive integers,and only numbers up to 19 * 258 have been checked to reach 1. Over the past few decades, numerous mathematicians have worked on this problem, however it has evad-

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ed all attempts to prove it. Upon further careful analysis of the problem, the numbers follow somewhat of a pattern. If the current number in the sequence is odd, then the next term will be even, because multiplying it by 3 and adding one will always result in an even number. On the other hand, if the current number in the sequence is even, then the next term can be either odd or even, such as 6 and 8, which would produce 3 and 4 as the next term respectively.

Thus an interesting question appears. Can the numbers be halved infinitely? No, they cannot, because the number will reach 1, as the conjecture states.

Mathematicians propose 2 additional scenarios for numbers that will not reach 1. One such theory is that the sequence will approach another cycle.

Advanced math theory shows that if this were true, the sequence would have to have more than 10 billion terms.

Another theory is that the numbers will get arbitrarily large and increase towards infinity. However, no formal proofs have been developed for either cases.

Upon further analysis of the problem, the digits of the numbers seem to behave in a pattern as well. In the figure above, it shows the progression of the ones digit place for numbers in the Collatz sequence. For example, for the digit 4, there are two possibilities, either the tens digit is also even or it’s odd. The two different tens digits will produce two different paths in the figure above.

When the tens digit is an even number, it will be marked by a red arrow, and a blue arrow for an odd tens digit.

For instance, the next term after 34 will be 17, and so only the ones digit 7 is marked, whereas the next term after 44 will be 22, and so the ones digit 2 is marked.

However, odd numbers only have one possibility, which is marked by a green arrow and represent the next term after doing 3n+1 to it. All even numbers create two additional branches in this tree of digits. The tree stops when the digit has already been written somewhere else in the tree. The numbers seem to be stuck in an infinite loop, since all the numbers can eventually go back to 1. However, the same problem arises from analyzing the parity of the numbers: can the numbers be halved infinitely?

Other approaches to the problem include looking at the numbers in modulo 2, where numbers can either be 1 or 0, or looking at other versions of the problem to try to make conclusions based on them. One interesting result of the Collatz conjecture, if allowing all real numbers and then plotting them on a graph, is the formation of fascinating and aesthetically pleasing visuals such as the Collatz fractal.

Another version of the problem allows all the integers, both the positives and negatives. This version of the problem produces 3 additional cycles other than the 4-2-1 cycle and the meaningless 0 cycle. Most of the numbers will reach -1 and then the sequence will become a -2→-1 cycle. However, some numbers will end up in either a -5 or -17 loop.

-5, -14, -7, -20. -10, -5 -17, -50, -25, -74, -37, -110, -55, -164, -82, -41, -122, -61, -182, -91, -272, -136, -68, -34, -17 Pure mathematics is beautiful and complex, and often does not have solutions to its problems. Even something as simple as adding and dividing numbers can present a problem unsolved for decades. However, that should not deter anyone from trying these problems and coming up with his or her own conclusions. What other conclusions can be made? What other patterns are there? Is there a pattern at the length of the sequence? Math does not have all of the answers yet, but one day it might if we keep trying.

-Kun Yu Cook, John D. “Collatz 3n + 1 Conjecture Possibly Solved.” John D Cook Singular Value Consulting. N.p., 1 June 2011. Web. 27 Jan. 2016.

Davies, Jason. “Collatz Graph: All Numbers Lead to One.” Jason Davies. N.p., 2012. Web. 27 Jan. 2016.

Weisstein, Eric W. “Collatz Problem.” Wolfram MathWorld. Wolfram, n.d. Web. 27 Jan. 2016.

16

crossword

Across: 4) _____ metric tons of food are wasted per year in the US. 6) _____ might have solved the ABC Conjecture.

7) _____ MIT students won thousands of dollars by scamming the _____.

8) Red and _____ are the hardest colors for the colorblind to distinguish between.

17

Down: 1) _____ can live in environments too hostile for humans.

2) The _____ company manufactures a car without gears 3) _____ are composed of the highest percentage of water of all age groups of humans.

5) Our universe could be a _____. *Solutions are on the Dr. Dragon website.

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ISSUE #8

About Dr. Dragon Dr. Dragon is our school’s student produced magazine that focuses on math, science, and engineering. The mission of this magazine is to give HSMSE students the opportunity to take the school's core subjects and explore subtopics that particularly interest them. Students on the magazine staff research and write about subjects of their choice. They are also involved with the production of the magazine, and learn about everything from design to fundraising and budgeting.

If you are an HSMSE student and want to contribute your thoughts, please talk to our officers or our faculty advisor, Mr. Choi.

Contact information: Dr. Dragon email: hsmsedrdragon@gmail.com Mr. Choi: RChoi@hsmse.org Also, you can read our previous magazines, and check the answers to crossword puzzles and Sudoku puzzles by visiting our website: sites.Google.com/site/hsmsedrdragon/

Copyright © 2015 by Dr. Dragon All rights reserved. Published by Dr. Dragon No part of this publication may be reproduced or transmitted in any form by any means without prior written permission by the publisher.

Crystal Bonds, Principal