Методические указания по английскому языку для студентов 1 курса математического факультета. 2004 (стр. 4 из 5)

4. It can be shown that heads-tails is a fair game ____ the same optimal mix of strategies that both players have.

5. Practise questions and answers. Ask your neighbour:

1. How many pure strategies there are for the game of coin tossing; for the game stone-paper-scissors?

2. What ways there are to control how to randomize?

3. What the optimal strategy is, given that the game is being played many times?

4. What fair games he/ she knows.

6. Give the main ideas of text “Strategies”. Retell the abovementioned text briefly using the main ideas as a plan of rendering.

Text 6

Before you read:

Have you ever had a chance to play any mathematical game? What king of game was it?

Read the text and outline puzzles and games mentioned in it. Try to solve some problems if possible.

Mathematical games and recreations

Mathematical 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 A 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 (2ⁿ⁺¹ - 1)/3 moves if n is odd and (2ⁿ⁺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 world-wide. 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:

1. Active vocabulary (turn the words into the active voice and memorize them):

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, world-wide, to realize.

3. Suggest antonyms to the following words and expressions.

Unsolved, deep problems, best known, to be famed for, insoluble, initial position.

4. Answer the questions:

1. What types of puzzles are the best-known types?

2. What was early Egyptian mathematics largely based on?

3. What is Fibonacci famed for?

4. What famous game did Cardan invent?

5. What problem was posed by Euler in 1779?

6. What was the most famous puzzle of Sam Loyd?

7. What does solving the Ernö Rubik’s cube show?

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 best-known types of puzzles are such games as matching pennies and stone-paper-scissors.

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.

6. Translate the sentences with the modal verbs, paying attention to their meaning in each specific context.

1. The Rhind papyrus shows that early Egyptian mathematics was to be largely based on puzzle type problems.

2. 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.

3. The problem asks how 15 school girls can walk in 5 rows of 3 each for 7 days so that no girl is to walk with any other girl in the same triplet more than once.

4. We can’t but ask the more general question about n school girls.

5. The 9 cubes forming one face can be rotated through 45

.

7. Match the problems with the descriptions.

Egyptian problem	This problem is about the rings, which are to be taken off the bar under certain conditions.
Fibonacci’s problem	The problem deals with 15 school girls, walking in rows for 7 days.
Cardan’s problem	This problem asks if it if possible to arrange six regiments consisting of six officers each of different ranks in a 6x6 square so that no rank or regiment will be repeated in any row or column.
Euler’s problem	This problem is about a pair of rabbits in a place surrounded on all sides by a wall.
Kirkman’s problem	We are to deal with cats, mice and grain while solving this problem.

8. Retell the text trying to point out as many mathematical games as possible.

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).

Try to solve

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

1. Saying 0 in English:

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.

2. Per cent

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!

4. Squares, cubes and roots

10² is ten squared

10³ 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:

1/2 - a half 1/4 - a quarter 3/4 - three quarters 3½ - three and a half

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 fifty-minute lesson not a fifty-minutes lesson

a sixteen-week semester, a fifteen-minute walk, a twenty-pound 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 01227-764000. 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 6-0 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 semi-segments (semi-intervals),