скачать Министерство образования Российской Федерации Часть I Методические указания по английскому языку для студентов 1 курса математического факультета. 2004 От составителя Данные методические указания предназначены для студентов 1 курса математического факультета. Целью работы является развитие умений чтения литературы по специальности, а также оформления монологического высказывания по проблемам прочитанного. Учебные материалы данных методических указаний объединены в 3 блока:  треугольники;  простые числа;  теория игр. Тематика разработки охватывает не основные, базовые понятия математической науки, так как достаточная обеспеченность математического факультета учебными материалами снимает необходимость обращаться к ним. В методических указаниях затрагиваются мало освещенные в научной литературе на английском языке, доступной студентам, но довольнотаки актуальные проблемы современной высшей, а также прикладной математики. Перед текстами и после них даются упражнения, которые различны по содержанию, целевой направленности и форме выполнения. В том числе упражнения на отработку вокабуляра, включающего специальную и общенаучную лексику, упражнения на проверку понимания содержания текста, упражнения по некоторым аспектам грамматики. Блоки содержат творческие задания (кроссворды, головоломки), задачи, требующие логического осмысления, а также математические шутки. Тексты некоторых блоков сопровождаются иллюстративным материалом. Диаграммы, таблицы, схемы и чертежи призваны облегчить понимание тех или иных текстов, а также разнообразить комплекс послетекстовых упражнений. В приложение включены правила чтения и названия некоторых чисел и цифр на английском языке; словарь математических символов и ответы к задачам, встречающимся в блоках. Настоящая работа предназначена как для аудиторной работы под руководством преподавателя, так и для самостоятельной работы студентов. ^ Triangles Text 1 Before you read: Can you give a definition of a triangle? What features of a humble triangle can you remind yourself of from your geometry classes? Read the following text and find out whether you are right or not. ^ Everyone is familiar with the simplest mathematical objects, such as straight lines and circles and squares, and the counting numbers. To be a mathematician all you have to do is to learn to look at these objects with some insight and imagination, maybe do a few experiments too, and be able to draw reasonable conclusions. The result of these activities – which are also quite familiar to you from everyday life – is that you soon see the square as more than something with four equal sides and four right angles; a circle as more than just a plain circle; and the number 8 as much more than merely the next number after 7. We are going to start this mathematical adventure by looking at another very simple and common mathematical object, the humble triangle. What features does it have? The three sides could be any length at all – except that the two shorter sides together must be longer than the longest side, or the triangle would not close. You cannot make a triangle out of three strips of wood of length 3, 5 and 12 meters. The three angles cannot be chosen as freely as the three sides. In fact, when we know the size of two of them, the other one can be calculated, because their sum is constant and is equal to 180°. What other properties does the original triangle have? None, until we use our imagination and start asking some searching questions. As soon as we start to pose problems, and to solve them, we inevitably find ourselves discovering more of its many features. One natural question is: how big is this triangle? What is its area? The simplest and traditional way to find the area is to divide the triangle into two rightangled triangles, by drawing an altitude, as in Fig. 1.1, and then drawing a horizontal line which bisects the altitude at right angles. This dissects each rightangled triangle into a rectangle. The original triangle has been transformed into a rectangle of the same height, and half the width. This trick can be performed in three different ways, if all three angles of the triangle are less than a right angle, by starting with each of the three sides. This at once tells us something about the lengths of the altitudes and the sides: BC × AD = CA × BE = AB × CF. Fig. 1.1 Assignments: 1. Answer the following questions:
^ Insight, to draw reasonable conclusions, to pose problems, to bisect, to dissect, to be performed. 3. Give definitions of the following geometrical figures: A square, a circle, a triangle, a rectangle. ^
5. Make up a plan of the text “What is a triangle?” and retell the text, according to your plan, and using the following words and expressions: It is well known, hence, consequently, therefore, so, it is obvious, it is evident, apparently, manifestly. Text 2 Before you read: What types of triangles do you know? Try to give a definition of each type. Read and translate the definitions of different types of triangles: ^ Equilateral triangle – a triangle with all three sides of the same length. Isosceles triangle – a triangle with two of the three sides of the same length. Rightangled triangle – a triangle with one angle equal to 90º. Acuteangled triangle – a triangle with all the angles less than 90º. Obtuseangled triangle – a triangle with one of the angles greater than 90º. Assignments: ^ Equ.lateral, isos.eles, acuteang.ed, eq.al, rig.tan.led, tri.ngle. 2. Match the words and their definitions:
^ 1. 2. 3. 4. 5. Text 3 Similarity criteria of triangles. Two triangles are similar, if: 1) all their sides are proportional; 2) all their corresponding angles are equal; 3) two sides of one triangle are proportional to two sides of another and the angles concluded between these sides are equal. Two rightangled triangles are similar, if 1) their legs are proportional; 2) a leg and a hypotenuse of one triangle are proportional to a leg and a hypotenuse of another; 3) two angles of one triangle are equal to two angles of another. Assignments: ^
^ It is known that…, we are quite familiar with…, every mathematician is sure of…, it should be pointed out…, in fact, thus. Text 4 Before you read: Consider Fig. 1.2 and try to prove Pythagorean Theorem, well familiar to you from your school years yourself. Pythagorean Theorem Fig. 1.2 In a rightangled triangle a square of the hypotenuse length is equal to a sum of squares of legs lengths. A proof of Pythagorean Theorem is clear from Fig.1.2. Consider a rightangled triangle ABC with legs a, b and a hypotenuse c. Build the square AKMB, using hypotenuse AB as its side. Then continue sides of the rightangled triangle ABC so, to receive the square CDEF, the side length of which is equal to a + b. Now it is clear, that an area of the square CDEF is equal to (a + b) ². On the other hand, this area is equal to a sum of areas of four rightangled triangles and a square AKMB, that is c² + 4 (ab / 2) = c² + 2 ab, hence, c² + 2 ab = (a + b) ², and finally, we have: c² = a² + b². Assignments: ^ It is clear that…, considering, using hypotenuse as a side, to receive the square, is equal to, on the one hand, on the other hand, hence, it is evident that… Text 5 Medians of a triangle David Wells in one of his books wrote: if the triangle were a real physical sheet, made of some uniform material, it would not only have an area, but also a centre of gravity; see Fig. 1.3. This is the point on which the triangular sheet would balance on a pinpoint. We understand that, if the triangle is suspended from a vertex, a vertical line through that vertex will also pass through the centre of gravity. If the triangular sheet were resting on the edge of a table, it would start to tip and fall over the edge, if its centre of gravity were over the edge. Fig. 1.3 You’ll find it natural to ask where the centre of gravity is. Actually, if the triangle is divided into numerous narrow parallel strips, each strip will balance about its midpoint, and all these midpoints appear to lie on the straight line joining the vertex to the midpoint of the opposite side (Fig. 1.4), called the median from that vertex. Fig. 1.4 The centre of gravity of all the strips together will lie somewhere on the same straight line. Bearing in mind that we found area of the triangle in three different ways by starting with each side as base in turn, it is natural to do the same for the centre of gravity of the triangle. Three constructions can be made, each with a line of midpoints. If the centre of gravity lies on each of these lines, then there must be one point where all three lines meet. To find it, we’ll join each vertex to the midpoint of the opposite side, and the three lines concur, at the centre of gravity (Fig. 1.5). We have a bonus in this case, also. Because we are confident that the three medians do indeed concur. We do not even need to draw a diagram to check this fact, whereas we only discovered that the altitudes concurred with the aid of a drawing. Fig. 1.5 Assignments: ^ a centre of gravity, to be suspended from, to bear in mind, to concur, to have a bonus, to be confident, with the aid of. 2. Answer the questions:
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4. Recall everything you know of Subjunctive Mood and insert suitable auxiliary verbs:
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^ It is proved, obviously, evidently, apparently, there is no doubt, beyond question, indisputably, unquestionably. Assignments for unit I: 1. Collect all the information about triangles, based on text “What is a triangle?”, according to your own plan. 2. Arrange the sentences, you have written out logically. 3. Unite all the sentences, using the following words and expressions: It is proved, obviously, evidently, apparently, there is no doubt, beyond question, indisputably, unquestionably, it is clear that…, considering, using hypotenuse as a side, to receive the square, is equal to, on the one hand, on the other hand, hence, it is evident that… It is known that…, we are quite familiar with…, every mathematician is sure of…, it should be pointed out…, in fact, thus, It is well known, consequently, therefore, so, it is obvious, it is evident, apparently, manifestly. 4. Do the same work (points 1, 2, 3) for each text: “Types of Triangles”, “Similarity criteria of triangles”, “Pythagorean Theorem”, “Medians of a triangle”. 5. Read the text, you have composed and make sure all the points of it are arranged logically. 6. Make changes if necessary. 7. Read the text one more time and think of the title. Try to solve Problem 1 Across the river Jake Hardy was standing on the river bank, looking across to the far side. ‘How wide do you recon it is?’ asked Harold. Jake adjusted the rim of his hat, and turned to look downstream. He paused and then walked with deliberate paces along the river bank, then turned and called out, ‘About thirty meters, give or take a few.’ How did he estimate the width of the river? Fig. 1.6 Smile About application of mathematics in linguistics A teacher of English was ill and a teacher of mathematics replaced him. He began to compose a table of irregular verbs: Then he said:  Okay, I mark this form as x . Then it’s possible to compose the proportion: ^ Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. The mathematicians of Pythagoras's school (500 BC to 300 BC) were interested in numbers for their mystical and numerological properties. They understood the idea of primality and were interested in perfect and amicable numbers. A perfect number is one whose proper divisors sum to the number itself. e.g. The number 6 has proper divisors 1, 2 and 3 and 1 + 2 + 3 = 6, 28 has divisors 1, 2, 4, 7 and 14 and 1 + 2 + 4 + 7 + 14 = 28. A pair of amicable numbers is a pair like 220 and 284 such that the proper divisors of one number sum to the other. In Book of the Elements, Euclid proves that there are infinitely many prime numbers. This is one of the first proofs known which uses the method of contradiction to establish a result. Euclid also gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way. Euclid also showed that if the number 2^{n}  1 is prime then the number 2^{n}^{1}(2^{n}  1) is a perfect number. The mathematician Euler (much later in 1747) was able to show that all even perfect numbers are of this form. It is not known to this day whether there are any odd perfect numbers. There is then a long gap in the history of prime numbers during what is usually called the Dark Ages. The next important developments were made by Fermat at the beginning of the 17^{th} Century. He proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares. He devised a new method of factorizing large numbers which he demonstrated by factorizing the number 2027651281 = 44021 46061. He proved what has come to be known as ^ (to distinguish it from his socalled Last Theorem). This states that if p is a prime then for any integer a we have a^{p} = a modulo p. This proves one half of what has been called the Chinese hypothesis which dates from about 2000 years earlier, that an integer n is prime if and only if the number 2^{n}  2 is divisible by n. The other half of this is false, since, for example, 2^{341}  2 is divisible by 341 even though 341 = 31 11 is composite. Fermat's Little Theorem is the basis for many other results in Number Theory and is the basis for methods of checking whether numbers are prime which are still in use on today's electronic computers. Number of the form 2^{n}  1 attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbers M_{n} because Mersenne studied them. Not all numbers of the form 2^{n}  1 with n prime are prime. For example 2^{11}  1 = 2047 = 23 89 is composite, though this was first noted as late as 1536. For many years numbers of this form provided the largest known primes. In 1952 the Mersenne numbers M_{521}, M_{607}, M_{1279}, M_{2203} and M_{2281} were proved to be prime by Robinson using an early computer and the electronic age had begun. By 2003 a total of 40 Mersenne primes have been found. The largest is M_{20996011} which has 6320430 decimal digits. There are still many open questions (some of them dating back hundreds of years) relating to prime numbers. Assignments: ^ Perfect number, amicable number, composite number, infinite, contradiction, integer, speculation, to devise. 2. Give the definition of the following notions in English: Perfect number, amicable number, composite number ^ 4. Give Russian equivalents to the following words and expressions: Primality, proper divisor, to establish a result, Dark Ages, to prove a speculation, to demonstrate by factorizing a number, to be divisible by, decimal digits. ^
What were the mathematicians of Pythagoras's school mainly interested in?
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^ The mathematicians of Pythagoras's school and their scientific work.
^ Text 2 Before you read: Consider the following unsolved problems in the theory of prime numbers and give accurate translation. Some unsolved problems
Text 3 Before you read: Read the following records; try to memorize the numbers and the scientists, who have announced them. The Latest Prime Records The largest known prime (found by GIMPS [Great Internet Mersenne Prime Search] in November 2003) is the 40^{th} Mersenne prime: M_{20996011} which has 6320430 decimal digits. The largest known twin primes are 242206083 2^{38880} 1. They have 11713 digits and were announced by Indlekofer and Ja'rai in November, 1995. The largest known factorial prime (prime of the form n! 1) is 3610!  1. It is a number of 11277 digits and was announced by Caldwell in 1993. The largest known primorial prime (prime of the form n# 1 where n# is the product of all primes n) is 24029# + 1. It is a number of 10387 digits and was announced by Caldwell in 1993. Assignments: ^ Assignments for unit II: 1. Give English equivalents of the following words and expressions: совершенное число, десятичный знак, дружественные числа, простое число, факторизация, бесконечно много, арифметическая прогрессия, целое число, метод «от противного». 2. Using the above mentioned words and expressions (English translation) make up sentences of your own. 3. Remind yourself of what you were reading in this unit and answer the questions: a) Who is the founder of the study of prime numbers? b) Why was the certain period in the development of the study of prime numbers called the Dark Ages? c) What numbers do we call Mersenne numbers and why? d) It is possible to solve the following problem: are there infinitely many primes of the form n! + 1? e) What is the largest known prime? The largest known factorial prime? 4. Name: a) The scientists considering the study of prime numbers; b) the scientists who managed to pose unsolved problems in the theory of prime numbers; c) the scientists who announced the records of prime numbers. 5. Tell the group everything you know about these mathematicians, i.e. everything found out in this unit. Smile Einstein and telephone One woman asked Einstein to remember her telephone number: 361343. Einstein answered:  It’s very easy. 19 squared and 7 cubed. New about limits At a mathematics exam a professor asks a student to calculate the limit: The professor is surprised:  What is it? Why ? The student answers:  You explained at your lecture that and I have used this example. Unit III. Game Theory ^ Some Basics Game theory is a branch of applied mathematics fashioned to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests. In a typical game, decisionmaking “players,” who each have their own goals; try to outsmart one another by anticipating each other's decisions. (Encyclopedia Britannica) Game theory is a distinct and interdisciplinary approach to the study of human behavior. The disciplines most involved in game theory are mathematics, economics and the other social and behavioral sciences. Game theory (like computational theory and so many other contributions) was founded by the great mathematician John von Neumann. The first important book was The Theory of Games and Economic Behavior, which von Neumann wrote in collaboration with the great mathematical economist, Oscar Morgenstern. Certainly Morgenstern brought ideas from neoclassical economics into the partnership, but von Neumann, too, was well aware of them and had made other contributions to neoclassical economics. Assignments: ^
2. Give a brief summary of the article using the following words and expressions: a branch of applied mathematics, an interplay between parties, the study of human behavior, to be founded by, in collaboration with, to make contribution. ^ What is a game? Game: A competitive activity involving skill, chance, or endurance on the part of two or more persons who play according to a set of rules, usually for their own amusement or for that of spectators (The Random House Dictionary of the English Language, 1967). Конец формы A game is the set of rules that describe it. An instance of the game from beginning to end is known as a play of the game. And a pure strategy is an overall plan specifying moves to be taken in all eventualities that can arise in a play of the game. A game is said to have perfect information if, throughout its play, all the rules, possible choices, and past history of play by any player are known to all participants. Games like ticktacktoe, backgammon and chess are games with perfect information and such games are solved by pure strategies. But whereas one may be able to describe all such pure strategies for ticktacktoe, it is not possible to do so for chess, hence the latter's ageold intrigue. Games without perfect information, such as matching pennies, stonepaperscissors or poker offer the players a challenge because there is no pure strategy that ensures a win. Games such as headstails and stonepaperscissors are also called twoperson zerosum games. Zerosum means that any money Player 1 wins (or loses) is exactly the same amount of money that Player 2 loses (or wins). That is, no money is created or lost by playing the game. Most parlor games are manyperson zerosum games. Not all zerosum games are fair, although most twoperson zerosum parlor games are fair games. So why do people then play them? They are fun, everyone likes the competition, and, since the games are usually played for a short period of time, the average winnings could be different than 0. Assignments: ^ A competitive activity, a set of rules, an instance of the game, a play of the game, a pure strategy, eventuality, a game with (without) perfect information, a zerosum game, ticktacktoe, backgammon, chess, stonepaperscissors, matching pennies. ^ A game, a play of the game, a pure strategy, a game with perfect information, a game without perfect information, a zerosum game, a fair game. 3. Fill in the gaps with the words and expressions from the text.
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^ Hearts, spades, diamonds, clubs. 1) ♠ 2) ♣ 3) ♥ 4) ♦ 6. Answer the questions:
^ A game with (without) perfect information, a zerosum game, a twoperson (many person) game Finish the sentences suggesting your own ideas. Remember, that after ‘because of’ we use either a noun/pronoun or an –ing form.
Text 3 Read the rules of two games and translate them into Russian. How to play some games A Game of Coin Tossing: Two players take any coin at hand. Bigsize coins are more fun. One of them starts tossing the coin and continues for a long time. It will be convenient to assume that the tosses occur at equal intervals of time. If heads, Player 1 wins, say, one kopeck, if tails, Player 2 wins and his opponent pays. ^ : Two players show with their hands one of three figures: rock (that’s a fist), scissors (two fingers are shown), or paper (a player shows a palm); actually they are to choose randomly among these three options, with equal weights. Rock is stronger than scissors, scissors are stronger than paper and paper in stronger than rock. The one, who shows the stronger figure, wins. The fun of playing this game comes from trying to guess and exploit the other player's choices. Assignments: Pair work: think of some game you know very well, describe its rules to your neighbour, by analogy with the above mentioned games. Let your neighbour guess which game you were speaking about. ^ The Prisoner's Dilemma To be able to understand the strategies some games are played according to it is necessary first to get the idea of the prisoner’s dilemma. Two burglars, Bob and Al, are captured near the scene of a burglary and are given the "third degree" separately by the police. Each has to choose whether or not to confess and implicate the other. If neither man confesses, then both will serve one year on a charge of carrying a concealed weapon. If each confesses and implicates the other, both will go to prison for 10 years. However, if one burglar confesses and implicates the other, and the other burglar does not confess, the one who has collaborated with the police will go free, while the other burglar will go to prison for 20 years on the maximum charge. The strategies in this case are: confess or don't confess. The payoffs (penalties, actually) are the sentences served. We can express all this compactly in a "payoff table" of a kind that has become pretty standard in game theory. Here is the payoff table for the Prisoners' Dilemma game:
The table is read like this: Each prisoner chooses one of the two strategies. In effect, Al chooses a column and Bob chooses a row. The two numbers in each cell tell the outcomes for the two prisoners when the corresponding pair of strategies is chosen. The number to the left of the comma tells the payoff to the person who chooses the rows (Bob) while the number to the right of the column tells the payoff to the person who chooses the columns (Al). Thus (reading down the first column) if they both confess, each gets 10 years, but if Al confesses and Bob does not, Bob gets 20 and Al goes free. So: how to solve this game? What strategies are "rational" if both men want to minimize the time they spend in jail? Al might reason as follows: "Two things can happen: Bob can confess or Bob can keep quiet. Suppose Bob confesses. Then I get 20 years if I don't confess, 10 years if I do, so in that case it is best to confess. On the other hand, if Bob doesn't confess, and I don't either, I get a year; but in that case, if I confess I can go free. Either way, it is best if I confess. Therefore, I'll confess." But Bob can and presumably will reason in the same way – so that they both confess and go to prison for 10 years each. Yet, if they had acted "irrationally," and kept quiet, they each could have gotten off with one year each. What has happened here is that the two prisoners have fallen into something called a "dominant strategy equilibrium."
Assignments: ^ To confess, to implicate, to collaborate, payoff, penalty, interaction, to assume. 2. Find synonyms among the following words: To cooperate, the most influential, penalty, to commit, to guide, to assume, prison, to lead, dominant, to accomplish, jail, to suppose, punishment, to interact ^
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Text 5 Strategies For the game of coin tossing: there are two pure strategies: play heads or tails. For stonepaperscissors there are three pure strategies: play stone or paper or scissors. In both instances one cannot just continually play a pure strategy like heads or stone because the opponent will soon catch on and play the associated winning strategy. What to do? There are some ways to control how to randomize. For example, for stonepaperscissors one can toss a sixsided die and decide to select stone half the time (the numbers 1, 2 or 3 are tossed), select paper one third of the time (the numbers 4 or 5 are tossed) or select scissors one sixth of the time (the number 6 is tossed). Doing so would tend to hide one’s choice from the opponent. For twoperson zerosum games, the 20th century’s most famous mathematician, John von Neumann, proved that all such games have optimal strategies for both players, with an associated expected value of the game. Here the optimal strategy, given that the game is being played many times, is a specialized random mix of the individual pure strategies. The value of the game, denoted by v, is the value that a player, say Player 1, is guaranteed to at least win if he sticks to the designated optimal mix of strategies no matter what mix of strategies Player 2 uses. Similarly, Player 2 is guaranteed not to lose more than v if he sticks to the designated optimal mix of strategies no matter what mix of strategies Player 1 uses. If v is a positive amount, then Player 1 can expect to win that amount, averaged out over many plays, and Player 2 can expect to lose that amount. The opposite is the case if v is a negative amount. Such a game is said to be fair if v = 0. That is, both players can expect to win 0 over a long run of plays. The mathematical description of a zerosum twoperson game is not difficult to construct, and determining the optimal strategies and the value of the game is computationally straightforward. It can be shown that headstails is a fair game and that both players have the same optimal mix of strategies that randomizes the selection of heads or tails 50 percent of the time for each. Stonepaperscissors is also a fair game and both players have optimal strategies that employ each choice one third of the time. Assignments: ^ Associated winning strategy, to randomize, to toss, designated optimal mix of strategies, computationally. 2. Arrange the following words according to the parts of speech they belong to: Instance, individual, guarantee, strategy, expect, amount, determine, selection, optimal, randomize, description, value, specialized, averaged, computationally. ^ Чистая стратегия, игра с ненулевой суммой, безразлично (неважно), подбрасывать игральную кость, честная игра, «каменьножницыбумага». ^
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6. Give the main ideas of text “Strategies”. Retell the abovementioned text briefly using the main ideas as a plan of rendering. Text 6 ^ Read the text and outline puzzles and games mentioned in it. Try to solve some problems if possible.Mathematical games and recreationsMathematical puzzles vary from the simple to deep problems which are still unsolved. The whole history of mathematics is interwoven with mathematical games which have led to the study of many areas of mathematics. Number games, geometrical puzzles, network problems and combinatorial problems are among the best known types of puzzles. The Rhind papyrus shows that early Egyptian mathematics was largely based on puzzle type problems. For example the papyrus, written in around 1850 BC, contains a rather familiar type of puzzle: Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hectares of wheat. What is the total of all of these? Fibonacci is famed for his invention of the sequence 1, 1, 2, 3, 5, 8, 13, ... where each number is the sum of the previous two. In fact a wealth of mathematics has arisen from this sequence and today there are lots of problems related to the sequence. Here is the famous Rabbit Problem. A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive? Cardan (1501 – 1576) invented a game consisting of a number of rings on a bar. Fig. 3.1 It appears in the 1550 edition of his book De Subtililate . The rings were arranged so that only the ring ^ at one end could be taken on and off without problems. To take any other off the ring next to it towards A had to be on the bar and all others towards A had to be off the bar. To take all the rings off requires (2^{n}^{+1}  1)/3 moves if n is odd and (2^{n}^{+1}  2)/3 moves if n is even. The Thirty Six Officers Problem, posed by Euler in 1779, asks if it is possible to arrange 6 regiments consisting of 6 officers each of different ranks in a 6 6 square so that no rank or regiment will be repeated in any row or column. The problem is insoluble but it has led to important work in combinatorics. Another famous problem was Kirkman's School Girl Problem. The problem, posed in 1850, asks how 15 school girls can walk in 5 rows of 3 each for 7 days so that no girl walks with any other girl in the same triplet more than once. In fact, provided n is divisible by 3, we can ask the more general question about n school girls walking for (n  1)/2 days so that no girl walks with any other girl in the same triplet more than once. Solutions for n = 9, 15, 27 were given in 1850 and much work was done on the problem thereafter. It is important in the modern theory of combinatorics. Around this time Sam Loyd's most famous game was the 15 puzzle. Fig.3.2 It illustrates important properties of permutations. The most famous of recent puzzles is the of Rubik's cube invented by the Hungarian Ernö Rubik. It's fame is incredible. Invented in 1974, patented in 1975 it was put on the market in Hungary in 1977. However it did not really begin as a craze until 1981. By 1982 10 million cubes had been sold in Hungary, more than the population of the country. It is estimated that 100 million were sold worldwide. It is really a group theory puzzle, although not many people realize this. The cube consists of 3 3 3 smaller cubes which, in the initial configuration, are coloured so that the 6 faces of the large cube are coloured in 6 distinct colours. The 9 cubes forming one face can be rotated through 45. There are 43,252,003,274,489,856,000 different arrangements of the small cubes, only one of these arrangements being the initial position. Solving the cube shows the importance of conjugates and commutators in a group. Assignments: ^ To be interwoven, to be famed, to be related, to be based on, to be arranged, to be posed, to be patented, to be estimated, to be rotated. 2. Suggest synonyms to the following words. Simple, unsolved, familiar, to suppose, to arrange, to be divisible, a craze, worldwide, to realize. ^ Unsolved, deep problems, best known, to be famed for, insoluble, initial position. 4. Answer the questions:
5. Contradict the following statements. Begin your answer with: “You are mistaken…, that’s not true…, I can’t agree with it…, I doubt the statement…, It’s just the other way round. Quite the reverse…” 1. Mathematical puzzles are the simplest problems which are mathematicians solve every day. 2. The bestknown types of puzzles are such games as matching pennies and stonepaperscissors. 3. Fibonacci is famed for his invention of the sequence 1, 2, 3, 4, 5… 4. Cardan’s problem deals with seven cats and seven mice. 5. The Thirty Six Officers Problem, posed by Riemann, asks if it is possible to arrange 6 regiments consisting of six officers in a 6x6 square. 6. The most famous of recent puzzles is the one of Rubik’s square. ^
The 9 cubes forming one face can be rotated through 45. ^
^ Assignments for unit III: 1. Sum up everything you now know about games and game theory and answer the following questions: a) What is game theory? Who has found it? b) What is a game? What types of games do you know? Give examples. Describe the rules of a game you know perfectly well. c) What is the essence of the Prisoner’s Dilemma? d) What is a pure strategy? Describe a pure strategy for the game of coin tossing. e) What are the most famous mathematical games and puzzles? 2) Speak on the topic “Game Theory” according to the plan: a) The definition of game theory, its connection with other sciences. b) A game (Its definition, types, rules of some games). c) The Prisoner’s Dilemma as a strategy. d) Mathematical games and puzzles (give examples). ^ Problem 2 Three men in the hotel Three men go to a hotel. They ask the clerk, «How much is a room?" and the clerk tells them it is $30. They each pay the clerk $10 and go to the room. The clerk knew that the room rate was really only $25 and started to feel guilty about overcharging the men so he gave the bellboy a $5 bill and told him to return the money to the men. The bellboy knew that $5 didn't divide evenly among the three men, so he kept the $5 bill and returned one dollar to each of the men, keeping the extra $2 for himself. So each of the three men paid $9 for the room ($27 total) and the bellboy kept $2. Where is the other dollar? Supplement Saying numbers ^ We say oh after a decimal point 5.03 five point oh three in telephone numbers 67 01 38 six seven oh one three eight in bus numbers No. 701 get the seven oh one in hotel room numbers Room I’m in room two oh six in years 1905 nineteen oh five We say nought before the decimal point 0.02 nought point oh two We say zero for the number 0 the number zero for temperature 5˚C five degrees below zero We say nil in football scores 5  0 Argentina won five nil We say love in tennis 15 – 0 The score is fifteen love Say the following: 1) The exact figure is 0.002. 2) Can you get back to me on 01244 24907? I’ll be there all morning. 3) Can you put that on my bill? I’m in room 804. 4) Do we have to hold the conference in Reykjavik? It’s 30 degrees below 0! 5) What’s the score? 2 – 0 to Juventus. ^ The stress is on the cent of per cent: ten perCENT We say: 0.5% a half of one per cent Say the following: 1) What’s 30% of 260? 2) 0.75% won’t make any difference. 3. The number 1,999 is said one thousand nine hundred and ninety nine The year 1999 is said nineteen ninety nine The year 2000 is said the year two thousand The year 2001 is said two thousand and one The year 2015 is said two thousand and fifteen or twenty fifteen 1,000,000 is said a million or ten to the power six 1,000,000,000 is said a billion or ten to the power nine Say the following: 1) It’s got 1001 different uses. 2) Profits will have doubled by the year 2000. 3) You are one in 1,000,000! 4) No, that’s 2,000,000,000 not 2,000,000! ^ 10^{2} is ten squared 10^{3} is ten cubed √10 is the square root of ten. 5. We usually give telephone and fax numbers as individual digits: 01273 736344 oh one two seven three, seven three six three four four 344 can also be said as three double four 44 26 77 double four two six double seven 777 can be said as seven double seven or seven seven seven 6. Notice the way of speaking about exchange rates: How many francs are there to the dollar? How many francs per dollar did you get? The current rate is 205 pesetas to the pound. 7. Fractions Fractions are mostly like ordinal numbers (fifth, sixth, twenty third etc.) 1/3  a third 1/5  a fifth 1/6  a sixth Notice, however, the following: ^ 8. Calculating 10 + 4 = 14 ten plus four is fourteen ten and four equals fourteen 10 – 4 = 6 ten minus four is six ten take away four equals six 10 Х 4 = 40 ten times four is (equals) forty ten multiplied by four is forty 10 ÷ 4 = 2½ ten divided by four is two and a half 9. When a number is used before a noun – like am adjective – it is always singular. We say: a fiftyminute lesson not a fiftyminutes lesson a sixteenweek semester, a fifteenminute walk, a twentypound reduction, a one and a half litre bottle. Check yourself How many of the following can you say aloud in under 1 minute? 1) 234, 567 2) 1,234, 567, 890 3) 1.234 4) 0.00234% 5) 19,999 6) In 1999 7) I think the phone number is 01227764000. 8) He was born in 1905 and died in 1987. 9) 30 Х 25 = 750 10) 30 ÷ 25 = 1.20 11) Let’s meet in 2023. 12) I can give you 367,086,566 apples. 13) The score is 60 to Zenit. 14) I’ll rent room 407. 15) My salary is $ 200 a month. 16) If he was born in 1964 and decided to start working at this problem in 1998, then 34 years had passed before he began doing it. 17) Did you say 0.225 or 0.229? 18) It’s white Lamborghini Diabolo, registration number MI 234662, and it looks as if it’s doing 225 kilometers an hour! 19) Have you got a pen? Their fax number is 00 33 567 32 49. 20) 2/5 21) 2¾ Mathematical symbols dictionary – identically equal, – approximately equal, ~ – approximately, 0.(12345) – the repeating decimal with the period 12345, N – the set of natural numbers, Z – the set of whole numbers (integers), R – the set of real numbers, Ø – an empty set, – an infinity sign, – an element x belongs to a set X, – a union of sets X and Y, – an intersection of sets X and Y, { u_{n} } – a sequence with a general term u_{n }, [ a, b ] – a numerical segment, – numerical semisegments (semiintervals), ( a, b ) – a numerical interval, – a scalar product of vectors , – a vector product of vectors , ==> – it follows, <=> – equivalent, – perpendicular, – parallel, – a triangle ABC, – a function and its derivative ^ Problem 1: Across the river In Fig. 4.1, the man has adjusted the brim of his hat so that his eyes just see past the brim of the hat to the distant point X on the other bank of the river. With the ground, this forms a triangle whose shape is fixed by angle at which the brim of his hat is tipped. By turning his head, and looking past the brim of his hat to a point on the ground which is on his side of the river, he will be identifying a point he can reach on foot, which is the required distance away – and all he has to do is pace out the distance. Fig. 4.1 Problem 2: Three men in the hotel The three men paid only $25 for the room, not $27; the bellboy returned $3 to them and kept the other $2 for himself. Therefore: $25 + $3 + $2 = $30 Problem 3: Practice questions
Content Стр. От составителя 3 Unit I. Triangles 4 Unit II. Prime Numbers 12 Unit III. Game Theory 17 Supplement 29 Bibliography 34
