1

4TITLE SPONSOR FOR MATHFEST 2

5Message from the HOD, 6

Editorial, 7

ArticlesTurning the Game Over, 8

How to lose friends and alienate people, 10

The Weirstrass Function, 12

Math in Video Games, 14

The Mathematician : Paul Erdos, 16

The Grand Hilbert Hotel, 18

Square values of Mathematical Expressions, 21

Interview with Krutika Tawri, 24

FunPuzzles, 25

Comics, 26

Puzzles Solution, 27

Contents

msg from HOD

A Message from the HOD

As another exciting academic year is coming to an end, the second issue of Convergence is now in your pious hands. Before going further, I would like to acknowledge that this has been made possible by the hard work, greater spirit, and innovation of our students. They are unswerving in their determination to taking the knowledge of mathematics to new heights.

We in BITS, have adopted a unique educational process which makes students think at a progressively higher level, and carry their learning beyond the memorization domain. Towards the end of their gradu-ation, every student feels his or her critical and creative thinking skill in mathematics enhanced phenom-enally.

An annual magazine is a great way to share our experience with the readers. It should have a great educa-tive value. I am sure, Convergence explores and expresses the power of creative-revolutionary-innovative ideas that how our students approach and treat mathematics!

I hope you will enjoy reading our annual magazine and learning about all the wonderful things that our students and faculty are doing. I congratulate and thank all the students who have contributed their valuable creations for publication. I also appreciate the strenuous efforts made by the editorial board in shaping this edition a top class one.

I convey my best wishes to all and wish you a lively and informative reading.

Dr. Prasanna Kumar(Head of Department, Mathematics)BITS Pilani K K Birla Goa Campus

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Editorialmsg from HOD

The Editorial

If youre reading this, the editorial of a magazine called Convergence, it is a safe assumption that you must have had a taste of the beauty that Mathematics has to offer. You were, at one point in your life, amused with the orderliness, the rationality and the truthfulness which it brings to the table. You marveled at the existence of more than one ways to solve a problem, the analysis of a problem as if it were a real life situation, the visualisation of concepts in two or three dimensions and so on. The journey began with numbers and the various elementary operations that you could do on them, followed by equations, and trigonometry, and calculus and so on.

Some of us, daunted by the growing complexity, chose to drop out from this lovely journey. While some of us still use it in myriad ways as a foundation for various applied sciences. In doing so, we often forget that Eureka moment or that little eye-brow raising moment, which made us taste the beauty of Maths in the first place. Convergence, in its second edition, tries to relive that beauty and the love for the subject which we all once discovered as a kid. From applied Math concepts like Game theory to pure Math topics like square values of ancient mathematical expressions, from combinatorial Maths to infinite paradoxes and pathological functions, we have attempted to bring up that eye-brow raiser. Convergence also, in a way or two, strives to imbibe a liking towards the courses offered by the Department of Mathematics, BITS Pilani K. K. Birla Goa Campus. It also hopes to bring about that immense joy that one gets in giving ones head a nice scratch while going about solving mathematical puzzles, and at the very least tries to tickle your funny bone with some mathematically hilarious comic strips.

Regardless of whatever equation you have had with Mathematics, we hope you find your fodder, some-thing you can relate to and thoroughly enjoy reading in this edition of Convergence. We do promise you that each little write-up in this edition will leave you completely intrigued, enlightened and educated. And to be really honest, on the path to excellence which most of us have set to achieve, one just cannot ask for anything more. Read on!

Siddhartha GovilkarEditor-in-Chief

7

TURNING THE GAME OVER

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ARTICLE

BY RUTUJA SURVE

Chess, combat, zeroing in on a movie to watch, a game of cards, shopping at a mall, ordering food at a restaurant, negotiations- all these deliberately chosen examples have five features in common:

1. Conflicting entities, or players, such as chess opponents or negotiating parties.2. Making choices, such as ordering Chinese at the restaurant.3. Information, based on which you make your choices like considering movie reviews before you decide to watch one.4. Desired results, like hoping your bluff works whilst playing a game of poker.5. Results of choices, like victory or a loss in Chess games, the Chinese food you ordered not quite living up to the expectations, and so on.

Game theory is the branch of mathematics which deals with such conflicting situations, tackling them with particular perspectives and strategies. It was first presented to the public and the academic world in the monumental treatise by John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, published in 1943. The two principal areas of its application were war and economics.

Now, lets turn detective, applying game theory to the study of crime. The players, of course, are the criminals and the police. The choices that the criminals have to make are:

1) Potential places to rob (for this case: Bank A and Bank B) 2) The time of operation (in the broad daylight or at night)

As for the police, different patrolling schedules including 1)Location, and 2)Time serve as choices.

The payoffs are a successful raid and a safe escape for the criminals and a successful red-handed capture of the criminals for the police.

BY RUTUJA SURVE

9

This scenario can be neatly illustrated in the table that follows:

Let us assume that the criminals are captured and held as prisoners. During interrogation, they are asked to give details of their entire gang, including the ones who were not present during the robbery. The police end up bringing these suspects behind bars as well. The police divide these newly captured suspects into two groups and decide to interrogate them separately. Lets look at the famous Prisoners dilemma matrix, illustrating possible actions that the two suspect groups might take. Assume that both groups have two choices. They can either confess or they can deny their hand in the crime. Assume also a set of outcomes corresponding to each set of choices: for example, if both groups deny the crime, then each of them gets five years of jail, whereas if only one of them confesses, then that group gets 20 years of jail and the other one is set free. The features of the conflict situation might be represented in the matrix at the end of the page. This matrix includes the players (Group A suspects, Group B suspects), choices (confess or deny), outcomes or results (time to be spent in jail), and the dependence of the outcome on the actions of each player. Each player has the goal of getting acquitted.

The first number in each pair is the payoff to Group A (number of years in jail) , the second the payoff to Group B . If the numbers in the pair are equal and of opposite sign, it becomes a zero sum game.

On the right is an illustration of the MaxMin concept of game theory:

This is a strategy for B that ensures the same return irrespective of the strategy used by A. Let x be the average payoff to player B, p1 be the probability of making choice B1 and p2 be the probability of making choice B2. Thenx = 5p 1 1p2 = -4p1 + 3p2 and p1 + p2 = 1, since probability of making a choice is 1.

The mathematical techniques used in game theory aim towards maximization of the minimum payoff, where the minimum payoff is the least amount a player can receive from a strategy choice. The solution (set of expected outcomes) to the game when this strategy is adopted by each player is the equilibrium solution, so-called because neither player can gain by changing his strategy unless the other player also changes his strategy. Solutions for two-person, zero-sum games with finite numbers of choices are easily solvable. This can be extended to n-person games, infinite games, non-zero-sum games using mixed strategies.

When stuck up at the precipice of making indispensable choices in life, the fascinating concepts of game theory can surely help you in turning the game over to your side!

HOW TO LOSE FRIENDS & ALIENATE PEOPLE

ARTICLE

BY SIDDHARTHA GOVILKAR

Meet-ups come with a barrel of fun! Consider one such meet-up, with 6 people, where two people are friends if they know each other and strangers otherwise. Now, in our little meet-up of 6 people, each person isnt necessarily friends with every other person, or (goes without saying), a stranger to every other person . Suppose we draw lines joining every pair of people in the room and colour them blue if the two are friends, and colour them red if they are strangers. We might get a scenario similar to this:

An interesting question can be asked at this point - in such a meet-up, can we find at least three peo-ple who are either all friends (blue triangle) or all

strangers (red triangle)? In the scenario above, the answer is yes. There does exist a red triangle be-tween Trent, Mike and Mary thereby implying the three are strangers to each other. But what if the original graph had been different? Would we al-ways have been able to find an orderly set of three people? The answer to that too is yes. One way of proving it is to list all possible colourings, and check each one in turn. But in the case of 6 people, there are over 30,000 such possible colourings! This proof clearly is not the most efficient one and is quite mind-taxing. An easier method is to first choose any point in your graph, and note that five lines come out of it - one to each of the other five points:

These five lines are either coloured blue or red. So, there must exist at least three lines of the same colour (pigeon-hole principle). Without loss of gen-erality, assume three lines out of five to be red.

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BY SIDDHARTHA GOVILKAR

Consider, these three lines for now.

Among the four points in the diagram on the left, if any one of the remaining three lines is red, we have a red triangle. And if all of the remaining ones are blue, then we have a blue triangle. Hence, proved.

Thus, for a meet-up of 6 (or more) people, there must be at least one group of three friends or of three strangers. An obvious question now arises: what is the minimum number of people that we need to either find a group of three friends or three strangers? This number is what we call as Ramseys number, and for this case it is written as R(3,3). It is named after Frank Plumpton Ramsey, who was a great mathematician, philosopher and economist. We can show that R(3,3) has to be 6, and any value less than 6 wont enable us to have a red or a blue triangle. As for five points, there does exist the following case where we dont get a triangle of one

colour.

With a bit more computation, we can also prove that R(4,4) is 18. But, can we be sure that there exists a definite value of R(a,b) for any given a and b? The answer is yes, and that existence result is called Ramseys theorem. However, with values like R(5,5) and greater numbers, we cannot be completely sure. Considering the research that has been done until now, we can at most predict that the value of R(5,5) lies somewhere between 43 and 49. One can ask why is it so difficult to get an accurate value. Well, the arguments involve finding upper bounds and lower bounds. We saw quite easily that R(3,3) could not be bigger than 6. But to show that it could not be as small as 5, we had to construct a graph with 5 points, as a counter-example. The problem is that we are looking for examples of

order, but the best counter-examples usually do have a lot of disorder. This makes it hard, and sometimes impossible to find a hard and fast rule that gives good counter-examples. Anything constructed by rule will probably have just too much order in it.

Another problem arises as the values of a and b in R(a,b) increase, because the upper bounds become too high. Examining all possible graphs to show that one has the right number of friends or strangers is a mammoth task. To show that R(5,5) is at most 49 we would have to look at 21176 possible colourings of a graph! This number is far, far bigger than the number of particles known in the Universe. We just might never know the answer to puzzle of this sort.

Joel Spencer, a combinatorialist who worked on Ramseys theory, once made a very popular quote, highlighting how explosive the values of R(a,b) can become.

Erds asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens!

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THE WEIRSTRASS FUNCTION

ARTICLE

BY DR. ALPESH DHORAJIA

Can you imagine a real valued function which is continuous but not differentiable? Well, if you re-member the very first classes that you took on continuity and differentiability of functions, you should remember being the one which typically comes to mind. is continuous everywhere on the real line but not differentiable at x=0. At x=0, the right hand derivative of this function is not equal to the left hand derivative, thereby making the func-tion not differentiable. The point x=0, is what we call a corner or a kink in mathematical terms. Now that we have recalled the definition of differ-entiability of a function, lets make things interest-ing a bit, shall we? Can you imagine a real valued function which is continuous everywhere but dif-ferentiable nowhere? Now, however ill-posed or counter-intuitive this may come across, a function like this does exist and is called as the Weirstrass Function, named after its discoverer Karl Weir-strass.

This pathological function, was the first published example (1872) which challenged the notion that every continuous function is differen-tiable except at a set of isolated points. In this arti-cle, I have made an attempt to prove the existence of such a function. Hold on tight, this is going to be quite a ride!

Theorem: There exists a real valued continuous function defined on the real line which is no-where differentiable.

Proof:To prove this result we will require the following two theorems;

Theorem 1: Suppose is a sequence of func-tions defined on an interval E, and suppose

(where )

Then, converges uniformly on E, if con-verges.Theorem 2: If is a sequence of continuous functions on E, and if converges to f(x) then f(x) is continuous on E.

Now,Define with and extend the definition of to real x by requiring that

.Then, for all , it can be easily seen that

, which implies is continuous on .

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BY DR. ALPESH DHORAJIA

Now define, ...(*)

Since , By Theorem 1, the series in (*) converges uniformly on .By Theorem 2, f(x) is continuous on .Now, we claim that for any , f(x) is not differentiable on x.

i.e. does not exist at every x.

Let, m be a positive integer and be a fixed real number.

Take ,

where the sign is chosen such that no integer lies between and .This can be done, since .

Define,

When, n>m, then is an even integer.In that case, as .

When 0nm, ..(since )Since , we conclude that

As . Thus, it follows that f is not differentiable at x.The graph of the Weirstrass function over the interval [-2,2] looks like this:

This function exhibits the property of self-similarity, i.e. every zoom is similar to the global plot.The non-differentiability of the function can easily be spotted graphically as the curves has corners or kinks at every point of R.

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MATH IN VIDEO GAMES

ARTICLE

BY ABHISHEK SHRIVASTAVA

I think the topic of this article should be enough for everyone (at least the game buffs) to get glued on to it. It was, at least for me. It is safe to say that videogames are an integral part of every teenagers life. Havent you, as-suming you have played some video game at some point in your life, been a little curious to know how a game is made? Let us start off by answering your curiosity. There are many as-pects to game creation. There is a staff of pro-ducers and publishers accompanied by a de-velopment team comprising of programmers, artists, graphic designers, level designers and testers. Our main focus in this article is on the mathematics used by the development team, primarily the programmers and designers.

Math is the foundation of game development. It is involved in everything - from having the ability to calculate the trajectory of an angry bird flying through the sky to ensuring that a character can jump and come back down to ground. Without the help of mathematics, games simply wouldnt work. To sum it up, it is the flour to the cake that game developers

attempt to bake. Without it, the cake wouldnt rise. To illustrate my point, we will glance through two types of games which are very popular namely, First Person Shooter (FPS) and Strategy Games.

First Person Shooter

It is a type of game where you run around 3D levels carrying a big gun shooting at ob-jects. Examples of this sort of game include Doom, Quake, Half Life etc. There are other games that look very similar, but arent first person shooters, for instance Zelda: Ocarina of Time or Mario 64. The most amazing part of an FPS are its incredible graphics. They look almost real, and they would not have been possible without the use of advanced math. To understand how these games work, you need to know a bit about geometry, vectors and transformations. Geometry is the study of shapes of various sort. The simplest shape is the point. Another simple shape is a straight line. A straight line is just the simplest shape joining two points together. A plane is a more

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BY ABHISHEK SHRIVASTAVA

complicated shape, it is a flat sheet, like a piece of paper or a wall. There are more complicated shapes, called solids, like a cube or a sphere. A vector is a mathematical way of representing a point in 3 dimension (space).A transformation moves a point (or an object) from one place to another. For instance, if I move to the right by 4 metres, this type of transformation is called a translation. Another type of transfor-mation is rotation. If you take hold of an object (a pen for instance), and twist your wrist, you have rotated that object. The basic idea of 3D graphics is to turn a mathematical description of a world into a picture of what that world would look like to someone inside the world. Of course, there is a lot more to it than just that: there is lighting, fog, animation, textures and hundreds of other things. Most of these use Math and Physics to a large extent.

Strategy Games

The Strategy games are divided into two main types, Real Time Strategy, and Turn Based Strategy. These games usually involve building and managing a city or civilization and also fighting wars by controlling troops. Examples of real time strategy games are Age of Empires, Command & Con-quer, Tiberian Sun. Examples of turn based strategy games are Civilization and Alpha Centauri.

Strategy games have much simpler graphics than FPS. So, they involve lesser usage of math in the graphics aspect. However, these games deal with a completely different aspect of Mathematics instead of geometry and vectors. When you click on a little soldier in a strategy game, and then click somewhere else, telling him that he should walk to the place where you have clicked, what happens inside the computer? How does the computer know how to make the soldier get from where he already is to where he is going? Remember, computers cant think for themselves (yet!) and so they need to be told exactly what to do. So you cant just say, look at the map and work out the best route to wherever you are going. A computer needs exact instructions for every step to be taken. This problem is called path finding. To explain how the computer works out the best route, you need to know what nodes, edges and graphs are.

The simplest example of nodes and graphs is a map of some cities, and the roads between them (or an underground map). Each city is a node, usually drawn as a circular blob. Each road is an edge, and connects two nodes (cities), these are usually drawn as straight lines. The whole collec-tion of nodes and edges (cities and roads) is called a graph. Sometimes there is a one way road, called a directed edge, and we draw an arrow on it to show which way you can travel along it. How does knowing about graphs help the computer guide troops around levels? It makes a graph where every interesting point is a node on the graph, and every way of walking from one node to another is an edge, and it then solves the corresponding path finding problem to guide the troops.

But it is not as easy as it sounds. For starters, what are the interesting points? You might think that every position on the entire level is interesting, but for most games this would lead to hundreds of thousands of interesting points, and finding the path would take years. Instead, the people making the game decide where the interesting points are. For instance, if there is a wide open expanse (a big field perhaps), you dont need a node at every point on the field, because the troops can walk in a straight line across the field. Basically, you only need nodes around obstacles. Once you have created a graph for a given map, the computer has to go through the following steps to guide the troops. Firstly, it has to work out what the nearest node that the troop can walk to in a straight line. This node is his starting node. Secondly, it has to work out the node which is nearest to his destination. This node is the destination node. Thirdly, it works out the shortest path connecting the starting node to the destination node. Now, all the troops have to do is walk to the starting node, then walk along all the nodes between the starting node and the destination node, along the connecting edges to their final destination. Generally, the computer assigns a cost to each path taken depending upon the difficulty associated with the path. Hence comes the problem of finding the path with the least cost.

So, we have seen two types of games which use different fields of math extensively. I hope these illustrations were interesting enough to prove my premise regarding the exemplary authenticity of Math in video games!

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PAUL ERDS

MATHEMATICIAN IN FOCUS

The stereotypical image of a prolific mathematician might be antisocial, mostly on his desk and young. Paul Erds was quite the exception. He was a freelance mathematician.

He had no home, no family, no possessions, no address. He went from math conference to math conference, from university to university, knocking on the doors of mathematicians throughout the world, declaring My brain is open and moving in. His colleagues, grateful for a few days collaboration with Erds - his mathematical breadth was as impressive as his depth - took him in.

Erds wrote around 1,525 mathematical articles in his lifetime, mostly with co-authors, more than any other mathematician in history. An astonishing legacy in a field where a lifetime product of 50 papers is considered extraordinary. He strongly believed in and practiced mathematics as a social activity.

Because of his prolific output, friends created the Erds number as a humorous tribute. The Erds number describes the collaborative distance between a person and mathematician Paul Erds, as measured by authorship of mathematical papers.

Erdss Erds number is 0. Erdss coauthors have Erds number 1. People other than Erds who have written a joint paper with someone with Erds number 1 but not with Erds have Erds number 2, and so on. If there is no chain of co authorships connecting someone with Erds, then that persons Erds number is said to be infinite.

Born to high-school mathematics teachers in 1913, Erds had two sisters, ages three and five, who contracted scarlet fever and died the day he was born. His mother, fearing that he, too, might contract a fatal childhood disease, kept him home from school until the age of 10. With his father confined to a Russian prisoner-of-war camp for six years and his mother working long hours, Erds passed the time flipping through his parents mathematics books. I fell in love with numbers at a young age, Erds later recalled. They were my friends. I could depend on them to always be there and always behave in the same way.

Erds first did mathematics at the age of three, but for the last twenty-five years of his life, since the death of his mother when he was , he put in nineteen-hour days, keeping himself fortified with 10 to 20 milligrams of Benzedrine or Ritalin, strong espresso, and caffeine tablets. A mathematician,

BY DEVASHI GULATI

16

Erds was fond of saying, is a machine for turning coffee into theorems. When friends urged him to slow down, he always had the same response: Therell be plenty of time to rest in the grave.

To communicate with Erds you had to learn his language. When we met, said Martin Gardner, the mathematical essayist, his first question was `When did you arrive? I looked at my watch, but Graham whispered to me that it was Erdss way of asking, `When were you born? Erds often asked the same question another way: When did the misfortune of birth overtake you? His language had a special vocabulary--not just the SF and epsilon but also bosses (women), slaves (men), captured (married), liberated (divorced), recaptured (remarried), noise (music), poison (alcohol), preaching (giving a mathematics lecture), Sam (the United States), and Joe (the Soviet Union). When he said someone had died, Erds meant that the person had stopped doing mathematics. When he said someone had left, the person had died.

He believed that God, whom he affectionately called the S.F. or Supreme Fascist, had a transfinite book (transfinite being a mathematical concept for something larger than infinity) that contained the shortest, most beautiful proof for every conceivable mathematical problem. The highest compliment he could pay to a colleagues work was to say, Thats straight from The Book.

Erds never won the highest mathematical prize, the Fields Medal, nor did he coauthor a paper with anyone who did, a pattern that extends to other prizes. He did win the Wolf Prize, where his contribution is described as for his numerous contributions to number theory, combinatorics, probability, set theory and mathematical analysis, and for personally stimulating mathematicians the world over. In contrast, the works of the three winners after were recognized as outstanding, classic, and profound, and the three before as fundamental or seminal. He was elected to many of the worlds most prestigious scientific societies, including the Hungarian Academy of Science (1956), the U.S.National Academy of Sciences (1979), and the British Royal Society (1989). Defying the conventional wisdom that mathematics was a young mans game, Erds went on proving and conjecturing until the age of 83, succumbing to a heart attack only hours after disposing of a nettlesome problem in geometry at a conference in Warsaw in 1996.

Vegre nem butulok tovabb(Finally I am becoming stupider no more)

--the epitaph Paul Erds wrote for himself

BY DEVASHI GULATI

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THE GRAND HILBERT HOTEL

ARTICLE

BY JANVI PALAN

In a mythical land, not more than a few decades ago, there once stood a majestic hotel. The Hilbert Grand, as they now call it, was majestic not only in stature, but also in size. And what a tremen-dously large hotel it was! Room after room, floor after floor, all stacked up to infinity. No, really. The Hilbert Grand really contained infinitely many rooms.

Before we proceed further in the story, it is im-perative to be clear about what infinitely many rooms really constitutes. This story makes use of the concept of a countable infinity, because the rooms, though with a tendency to go towards the undefined, are still numbered. Room 1 is followed by room 2 is followed by room 3 and so on, but each room is still numbered. A countable infinity is when there is a One-One correspondence be-tween a set and the set of natural numbers. To be even simpler, in this case, it just means that there exists a relation between one room number and a natural number. As we have used natural numbers for naming rooms, the existence of such a corre-spondence is quite clear. Counting the room num-bers, or carrying food up the stairs to the infinite-th room might take forever, but if you do happen to know what room youre delivering to, then it can be a finite time operation. One coffee coming to

room 1312425632442346346 in a few moments, sir!

Lets get back to our grandiose setup. The Hilbert Grand was started in 1924 by famous German mathematician, David Hilbert, with a particular-ly odd liking for hotels and a fairly understanda-ble curiosity towards infinity. Under the watchful eyes of an exceptional night manager, business boomed and soon, theyd reached a stage when all the rooms of the hotel were completely booked.

One rainy night, a weary traveller walked into the (fully booked) Grand Hilbert, seeking a place to stay. He is informed that there is indeed room for the guest at the hotel. Perplexing, in a way, be-cause regardless of how there are infinitely many rooms at the hotel, theyre still all occupied. But at the Hilbert Grand, theres always room for every-one! How?

The guest cant be given the last room, since by definition (or in this case, un-definition), all the rooms are occupied. But since each room was occupied by a guest, the manager requested the guest in Room 1 to move to Room 2, the guest in Room 2 to move to Room 3, the guest in Room n to move to Room n + 1. Since the hotel had infinite-ly many rooms, there was no problem in moving,

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BY JANVI PALAN

there was always a room to move to. This left Room 1 vacant, and therefore, the guest was accommodated. Motto still stands. Yay.

The next night, a bus of 60 passengers arrived and they asked for one room for each passenger. The same thing happened. The manager requested the guest in Room 1 to move to Room 61, every guest in Room n to move to Room n + 60. Since the hotel had infinitely many rooms, there was no problem in moving, there was always a room to move to. This left 60 rooms vacant and therefore the hotel accommodated the 60 new guests. Motto still stands. Yay.

The next night, a bus infinitely long with an infinite number of passengers arrived. This looked like a big problem, but the manager shrugged it off. He requested the guest from Room 1 to move to Room 2, the guest from Room 2 to move to Room 4, the guest from Room 3 to move to Room 6 and all the guest in Room n to move to Room 2n. The guests didnt mind moving. This left all the rooms with odd numbers vacant, which still meant an infinite number of rooms. Motto still stands. Yay.

In each of the above cases, there is a one-one correspondence between the number of rooms freed and the number of passengers coming in. Take a look:

One night, the unthinkable happens. There appears to be, lined up outside the hotel, an infinitely long queue of infinitely large buses. The type of infinity that we deal with is still, mind you, countable. If this large number of passengers cannot be occupied, the hotel loses out on an infinite amount of money, and the manager, his job. Under pressure, he is reminded of Euclids second theorem, stating that there are an infinite number of prime numbers. So to accomplish this mammoth task of fitting in an infinite number of people into his large hotel, he moves his guests in such a way that the person residing in room n is moved to room number 2n. The hotel is assigned the prime number 2. Next, he assigns the prime number 3 to the first bus. Each passenger in Bus 1 is moved to room number 3n, where n is their seat number. This goes on for the next bus, with the next prime, and the bus after that, and then after that. Take a look above on the right.

Since each of the resulting room numbers has only one and the prime number base as its factors, there are no overlapping room numbers. All the passengers are given unique room numbers, and the night manager accomplishes his task. Of course, there are still rooms that are left vacant, such as Room 6, because it is not a power of any prime, but alls well that ends well.

This situation is called the Infinite Hotel paradox and was created to demonstrate the counter-intuitive properties of infinite sets. Thats a whole lot of trouble to go through just to have fun, what with building an enormous hotel that probably required an infinite amount of time to make simply because youd never be able to stop building it, because you wouldnt know when to!

The paradox of Hilberts Grand Hotel can be understood by using Cantors theory of transfinite numbers. Thus, while in an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms, over at Hilbert Grand Hotel, the quantity of odd-num-bered rooms is not smaller than total number of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is , or Aleph zero.

This sort of arrangement isnt always possible. If instead we used real numbers, then in that form of infin-

19

ity, these structured strate-gies would fail, because we have no organized way of including all numbers. We would have to have negative rooms (in the basement), rational number rooms (like room 8/9, which never real-ly seems just as big as the room to its left), and room Pi, where the guests expect free dessert!

Now I doubt anyone would want to work at such a giant mess, even for an infinitely large salary.

These cases constitute a par-adox not in the sense that they entail a logical contra-diction, but in the sense that

they demonstrate a counter-intuitive result that is provably true: the statements there is a guest to every room and no more guests can be accommodated are not equivalent when there are infinitely many rooms. What the infinite hotel paradox goes to show is, quite simply, how difficult it is for our minds to get a grasp over infinity.

Homer: This time tomorrow, youll be wearing high heels!Ned: Nope, you will.Homer: Fraid not.Ned: Fraid so!Homer: Fraid not.Ned: Fraid so!Homer: Fraid not infinity!Ned: Fraid so infinity plus one!Homer: Doh!

If it left even noted mathematical genius, Homer Simpson, stumped, what good are our simple minds?

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BY JITEN AMAR AHUJA

SQUARE VALUES OF MATH EXPRN.

ABSTRACT OF A LECTURE BY DR. MANJUL BHARGAVA IN TECHFEST AT IIT BOMBAY

This is an abstract from the lecture given by Dr.Manjul Bhargava in Techfest at IIT Bombay on 4th January, 2015. The title of the lecture was : Square values of mathematical expressions from ancient times.

Mathematics can be divided into two broad branches- Applied Mathematics and Pure Mathematics. Applied mathematics, the one which most of us like to study and understand, deals with mathematical methods that find use in science, engineering, business, computer science, and industry. Pure mathematics deals with abstract entities with respect to their intrinsic nature, without being concerned with how they manifest in the real world.

This article deals with one of the topics in pure mathematics whose fundamental question is highly recreational in nature and which has been instrumental in shaping the very structure of pure mathematics. Pure Mathematics is something that exists for its own sake. Much of it doesnt have any immediate application in the real world. Most of it has been developed only by some generic questions. Questions simple enough to understand but quite difficult to crack. One of them is -

When is it that a mathematical expression

admitting real values will result in a perfect square? And if it has solutions, how often do we get them? ------------------------------ (1)

Or more generally,

When is it that a mathematical expression admitting real values will result in a perfect power? -------------------------------- (2)

The first known attempt at solving question (1) goes back to 2500 BC in Egypt. The megalithic monuments contain right-angled triangles with side-lengths of integral values. Another example is that of the Plimpton 322 Tablet dating back to 1800 BC. The Mesopotamian tablet lists some huge numbers which satisfy the property . The magnitude of these numbers suggests that they must have known a method to find what we call as Pythagorean triplets.

BY JITEN AMAR AHUJA

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Some other early well-known examples of the attempts made at (1) are as follows:

1) Baudhayana-Pythagoras Theorem: When is ?

2) Pells Equation*: How to find all the square values of the form ?

3) Fermats Last Theorem: For what non-zero integral values of a,b,c and integral values of n does the re-lation hold? Note that finding rational a, b and c for the above expression can also be considered as an alternate question.

4) Ramanujan-Nagell Equation: Ramanujans question was show that is a square only when n=3, 4, 5, 7, 15. This question was open for 35 years until it was solved by Nagell using methods previously un-known. As is with most of the questions posed by Ramanujan, the above is the only equation with 5 solu-tions for the kind with a, b and n integers such that is a perfect square. Rest have at most 2 solutions in n for a=2 and fixed b, unless b=7.

We shall now move on to the topic of hyperelliptic and elliptic curves. The simplest kind of a square equation is that of a hyperelliptic curve.

If be a hyperelliptic curve and (x,y) is a solution; then (x,-y) is also a solution.Now, has no rational solution whereas has infinitely many rational solutions.

The question arises - How many rational solutions can a hyperelliptic equation have?

It has been proved that hyperelliptic curves have- 1. Infinitely many or no solution if n=1,2. 2. Finite or infinitely many if n=3,4. 3. Finite if n 5

An algorithm is known for how to find the solutions in the case of n=1, 2 but not for n 3. In particular, if n=3, 4 one cannot say whether there will be finite or infinite solutions.

The question remains-If random equations are taken, what is the probability that there will be finitely many or infinitely many solu-tions?

Elliptic curves are a special case of hyperelliptic curves with n=3. Elliptic curves have been used to find the solution of Fermats Last Theorem.

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The Plimpton 322 Tablet The numbers on the Tablet

The speciality of elliptic curves is that they naturally form a group under addition**. The group law is constructed geometrically but has got nothing to do with ellipses or conic sections at all. In simpler geometrical terms, what this group law means is that if you take a reflection of the point, the point that is the intersection of the line joining two points on the curve and the curve, will also lie on the curve. This figure might help you understand what this exactly means.

Let E: .

Do the elliptic curves in E when ordered in a particular manner tend to have finitely many or infinitely many rational solutions?

The answers for above questions have been provided by Dr.Manjul Bhargava in association with others.

Theorem: (Dr.Manjul Bhargava, Charles Skinner, 2013)When all elliptic curves E: are ordered in a particular manner, there is a great-er than 20% probability that there will be infinitely many rational solutions.Dr.Manjul Bhargava conjectures that above two theorems holds for n = 4 too.

Theorem: (Dr.Manjul Bhargava, Gross, Poonen-Stoll + pre-print)For n 6, most (>50%) hyperelliptic curves have no rational points as their solutions. It can be said that as n increases, number of solutions decreases exponentially.

For n=10, 99% equations have no solutions.

As far as equations with degree 5 are concerned, there havent been any significant results proved as of yet. With the accomplishment of the above task, we will inch closer to solving the Birch and Swinnerton-Dyer Conjecture which is one of the unsolved Millennium problems.

* To read more, refer to the chapter XXVIII Indeterminate Equations of the Second Degree in Higher Algebra by Hall and Knight

** If a, b and c belong to a set G, they are said to form a group under addition if all the elements of G satisfy following 4 properties: 1. 2. 3. . such that , 4. , such that

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Interview WITH KRUTIKA TAWRI

We had the pleasure of interacting with Krutika Tawri, a fellow Mathematics student of the 2011 batch who is currently doing her final year thesis at the National University of Singapore. During her journey here at BITS Goa, she also dropped her second degree in (the lucrative) Computer Science, keeping her best interests in mind. Convergence tries to find out how she went on about landing a thesis at a great university like NUS, her experience of studying mathematics here, and what really gets her ticking.

Q. You dropped your engineering dual of Computer Science. Could you take us through the pros and cons which you had to consider while taking that decision?Krutika Tawri: Well, I sort of knew even before I came to BITS that I wanted to study pure mathematics. So the de-cision that I took made more sense because it would give me a lot more time to invest in it. Having CS would have definitely given me more options but I think it would have been too time consuming.

Q. Could you tell us your exact thesis topic and how you went on about securing a thesis at NUS?Krutika Tawri: Here, Im working on isoperimetric inequality in two and three dimensions and its application in understanding partial differential equations(PDEs). The scope of my thesis is basically understanding the proofs of the inequality and later trying to prove it on my own and then apply it to solving the heat equation. What I did to get this thesis is email a few professors working in the field of PDEs. A couple of them agreed to mentor me. I also had a good project in the same field, which is what I think worked for me. Also as far as I know, ones CGPA does not matter when you are applying for a thesis or looking for summer internships in pure Math. I dont think I even had to mention mine in my resume. So the only indicator of your capability is research and experience.

Q. Which Math courses, apart from Partial Differential Equations, do you recommend one should do well in to go about in your field?Krutika Tawri: Complex and Real analysis, Functional Analysis, Ordinary Differential Equations, Partial Differential Equations and Topology.

Q. Could you give us a brief outline regarding the most ethical way of approaching professors of such big universities regarding thesis?Krutika Tawri: Well, mail only the ones who are working in the field you are interested to work in. Be polite. Try not bombarding them with multiple emails, and be precise.

Q. Did you do a summer internship after your third year? Does that help in getting a thesis?Krutika Tawri: Yes, I did pursue a summer internship at the Institute of Mathematical Sciences (IMSc), Chennai. It probably must have helped me a lot. I worked with two professors there. Initially, I started working under Prof. An-ilesh Mohari by studying Cantors construction of real numbers using axiom of choice and expanded it to Von-Neu-manns construction of Hilbert spaces. This helped me understand the Fourier series better which I applied into studying the heat equation. Then I went on to work under Prof. Kesavan Srinivasan, under whom I studied about the Fourier transform and Schwartz spaces from the book Topics in Functional Analysis and Applications.

Q. You have seen the Mathematics research here at BITS Goa and also at NUS. What are the things that you think we can improve upon?Krutika Tawri: I havent been involved in NUS as a student as such, so i cant make a fair comment. I feel that maybe teachers can be a bit more welcoming and encouraging to accepting students for projects, because those help a lot.

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Puzzles

1. A man owned a large, square, fenced-in field in which there were sixteen oak trees, as depicted in the illustration. He wished, for some eccentric reason, to put up five straight fences, so that every tree should be in a separate enclosure. How did he do it? Just take your pencil and draw five straight strokes across the field, so that every tree shall be fenced off from all the others. See figure on the left.

2. You have a piece of cheese in the shape of a cube. How can you cut it in two pieces with one straight cut of the knife so that the two new surfaces produced by the cut shall each be a perfect hexagon? Of course, if you cut in the direction of the dotted line the surfaces would be squares, now produce hexagons.

3. Find the four digit number x that satisfies these two properties: i. The digits of x add up to a number y where x equals y times the num-ber you get when you reverse the digits of y. ii. Reverse the digits of x and find the prime factors of the number you get. Then take the sum of the squares of these prime factors and halve it. Removing the digit 0 from the new number yields back x.

4. The Nurikabe puzzle: Grid cells must be filled in so that all the black cells form one contiguous region, not counting squares touch-ing at a corner to be adjacent, but it is not allowed to have a two by two square of black cells. Finally, each con-nected region of unfilled cells must contain exactly one number, which tells how many unfilled cells there are.On the left is an example puzzle and its only solution.

Solve this one now :

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Comics

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Puzzles Solutions

1. On the left.

2. Mark the mid-points in BC, CH, HE, EF, FG, and GB. Then insert the knife at the top and follow the direction indicated by the dotted plane. Then the two surfaces will each be a perfect hexagon, and the piece on the right will, in perspective, resemble Figure 2.

3. 1729

4. Figure On the left.

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