Quadramagicology

Perhaps no other area of non-practical mathematics has been so popular for so long as magic squares. Mathematicians, artists and mystics have long been fascinated by the mesmerising patterns that they produce.

A traditional magic square is a square grid of numbers in which the values of each row, column, and diagonal add up to the same sum. A magic square's order is the number of cells in each row. 1) …… For a magic square of consecutive natural numbers starting with 1, let n equal the magic square's order and C equal the magic constant. Then, for an order-3 magic square starting with 1, C = 15.

Mathematicians trace an order-3 magic square back to ancient China, Babylonia, and Mayan culture. In India amulets with magic squares were worn as protective charms, while in western Europe Renaissance astrologers equated them with planets.

According to the legend the Loh Shu was the first magic square, turning up in the dots on the shell of a tortoise that crept out of the Yellow River in China about four millennia ago. 2) …… The Chinese gave it spiritual importance, believing that it encapsulated the harmonies of the universe. Feng Shui, the Chinese system of arranging objects, is in part based on the Lo Shu. 3) …… .

The first documented magic square in the West was a 4x4 array of integers one through to 16 which appeared in an engraving Melancholia created by Albrecht Dürer in 1514.

A magic square from the 12th-century temple at Khajuraho in India gives an idea of the extra possibilities offered by a 4-by-4 square. Its magic constant is 34, but there's a new twist. Place the table next to copies of itself, and it creates an infinite "magic carpet": any four adjacent entries along a straight line − horizontal, vertical or diagonal − sum to 34. 4) …… .

Squares with the magic-carpet property were called panmagic. Soon they were joined by antimagic squares − in which all the row, column and diagonal sums were different − and by nested and knight's-tour magic squares, in which each number from 1 to 64 is a chess knight's move apart from the next one.

Some great mathematicians studied magic squares – such as Leonhard Euler in the 18th century, and Édouard Lucas and Arthur Cayley in the 19th. The American statesman and scientist Benjamin Franklin liked to spend his spare time constructing innovative variations of magic squares. 5) ……. .

Today, magic squares are studied in relation to factor analysis, combinatorial mathematics, matrices, modular arithmetic, and geometry.

12. Work in pairs and see if you can remember the following words and phrases. Take turns to ask each other. Use the Mini-Dictionary (UNIT 8) if necessary.

    an artist	to trace back
    to be fascinated	protective charms
    mesmerizing patterns	to equate with planets
    a cell of a grid	millennium / millennia
    a row / a column	a tortoise
    consecutive natural numbers	to encapsulate the harmonies of the universe
    square's order	array of integers
    magic constant	an engraving
    adjacent entries	the magic-carpet property
    statesman	subsquare
    matrix / matrices	to abandon modesty
    to be inspired	a corner

13. Underline the stressed sound in each word as in the example. Choose any 4 words and use them in your own sentences.

millennium, spiritual, universe, array, integer, infinite, horizontal, diagonal, vertical, innovative, combinatorial, subsquare, likewise

14. Explain the meaning of words and phrases below. Choose 3 words you like and write sentences with the words chosen.

protective charms, mystics, a grid, a tortoise, Feng Shui, adjacent entries, to give a new twist, magic-carpet properties, row’s values, statesman.

15. Find words or phrases in the article that match the meanings (A-E).

1. technique of making prints from metal plates

2. ancient Chinese book of prophecy and wisdom

3. the period in European civilization that marked the transition from medieval to modern times

4. object or formula that credulity and superstition have endowed with the power of warding off harmful influences

5. edifice or sometimes merely an enclosed area dedicated to religious worship.

16. Match the words and collocations (1-8) with their definitions (a-h).

    1	antimagic square	a	a square in which the rows, columns and diagonals sum to different numbers
    2	panmagic square	b	measure of size - the length of the side of a magic square, which is the square root of the total number of cells in the square
    3	consecutive	c	the ordinary counting numbers
    4	square’s order	d	a systematic arrangement of objects, usually in rows and columns
    5	magic square	e	a magic square in which all the broken diagonals also sum to the magic sum
    6	magic constant	f	following one after another without interruption
    7	array	g	the total of the numbers in any row, column, and diagonal of the magic square
    8	natural number	h	square matrix often divided into cells, filled with numbers or letters in particular arrangements that were once thought to have special, magical properties

17. Arrange the following words in pairs of synonyms.

equate, fascinate, date back, infinite, condense, spellbind, appear, encapsulate, trace back, turn up, charm, mesmerize, endless, relate

18. Fill in the blanks with appropriate words from the list below.

reconstruct, shapes, participates, arcs, determine, reinvented, sequence, forms, represents, described, space

In Cubism, the subject of the artwork is transformed into a 1) ….. of planes, lines, and 2) …. Cubism has been 3) ….. as an intellectual style because the artists analyzed the 4) ….. of their subjects and 5) .…. them on the canvas. The viewer must 6) ….. the subject and 7) ….. of the work by comparing the different shapes and 8) ..… to 9) ..… what each one 10) ….. Through this process, the viewer 11) ..… with the artist in making the artwork make sense.

19. Fill in the chart below with the missing words.

Function	Sign	Process	Verb	Result
addition	+			the sum
	–		to subtract from
	×	times
	÷	divided by		the quotient (+ the remainder)

Now see how quickly you can find the answer to this rather long sum.

Four add two, divide by three, subtract one, multiply by eight, take away four, times three, plus two, minus four, halved, equals what?

20. Put the appropriate verb of measurement into the sentences, changing its form if necessary. Pay attention to the tenses.

A drop; gain; last; lose; measure; take; weigh

1. It was a long lecture: it … three and a half hours.

2. It was a big project: it … three weeks to finish.

3. That box looks heavy. How much does it…?

4. The time is 8.00. My watch says 7.55. It has … five minutes.

5. The time is 8.00. My watch says 8.05. It has … five minutes.

6. It was a cold evening, and temperature … 10 degrees in an hour.

7. The field mouse is quite a small animal: it … about 4 cms from nose to tail. B contain; cover; hold; register; seat; spend; total

1. The petrol tank is small: it only … 25 litres.

2. The bill for maintenance alone … $750 000 a year.

3. It is a big farm: it … more than 25000 acres.

4. This equipment measures sound: it can … up to 100000 decibels.

5. I … several hours every day just adding up figures.

6. This encyclopedia … over 25000 references.

7. It is a huge hall: it can … 2500 people.

21. Match the parts of the sentences from three columns to make complete sentences that make sense. Translate them into Ukrainian.

    1. In the pattern, called the Flower of life, found at the Temple of Osiris at Abydos, Egypt,	is to rearrange the seven forms to exactly reproduce a designated pattern	which have the property of no repeated values, in any of the 9 blocks of 3x3 cells.
    2. The tangram is an intellectual puzzle game	have abundant applications in tasks	in the shapes of triangles and quadrilaterals.
    3. Sudoku completed grids	,powerful good luck charms, are magic squares	which is either geometric in nature or patterned after a familiar object.
    4. Yantra magic squares	there were already prepared 30 Sudoku	forming a mesmerizing pattern of circles and lenses.
    5. Just after 2 weeks after Apple launched its online App Store in 2008, July,	are nothing more than Latin squares	such as efficiently assigning personnel to jobs or the statistical analysis of drug trials.
    6. Problems related to magic squares	the circles are placed with six-fold symmetry	for iPhone and iPod Touch users.
    7. The objective of the tangram puzzle	,originated in China, consisting of seven pieces,	built using your date of birth, and your Life Path number as the top row of the square.

22. Fill in the chart with the appropriate forms of the words.

Figure (noun)	Shape (adjective)	Figure (noun)	Shape (adjective)
cone		cube
oval		octagon
	rectangular		cylindrical
	prismatic		square
	spiral	circle
	spherical	quadrangle

23. Match the pictures with the words that describe shape.

spherical with an uneven surface; crescent shaped; pear-shaped; cylindrical with a convex top; round with a concave top; triangular, three-sided (acute triangle); prismatic; round with a serrated edge; spiral; pentagonal, regular hexagonal; semi-circular

1 Mail box		2 Goat cheese		3 Sunflower
4 Frugal soap		5. Tortilla chips		6. Puppy treats
6. Golf ball		7.Carabiner	8. Nautilus shell
9. Earring	10. Honey combs				11. Dumpling pastry

24. Match the adjectives on the left with a suitable noun on the right. Use each word only once. Translate the collocations into Ukrainian.

    1	hollow	a	staircase
    2	undulating	b	swimming pool
    3	pointed	c	countryside
    4	oval	d	chin
    5	spiral	e	roof
    6	sloping	f	tree

25. A drawing game. Try to draw each of the items below spending just a few seconds on each.

a dotted line; a crooked line; an octagon; a jagged edge; a rough sea; a steep heel; a sharp bend; a gentle curve; a smooth surface; rolling hills; wavy hair

26. Label the shapes with the suitable words. Then check your answers in the text below.

Two-dimensional, or 2D shapes have sides. Where two sides meet they make an angle. A circle only has one side and the distance all the way round this is called the circumference. When we measure a circle, the distance from one side to the other through the centre is the diameter and the distance from the side to the centre is called the radius. Three–dimensional, or 3D shapes are more complex because you can measure the height, width and depth. The surfaces on a 3D shape are called faces. Where two faces meet are the edges. Where two edges meet there is a corner. For example, a cube has twelve edges and eight corners.

27. A.What do you think life will be like in 100 years? Use perhaps, probably (not), certainly, I (don’t) think, I’m sure, I hope, I’d like to imagine.

Example: I think the world will still be powered mainly by fossil fuels.

B. Write 3 sentences to say what you are doing this evening/tomorrow/next week.

Example: I’m doing my exams next week.

28. Put the verb into the correct future form to complete the sentences.

1. I'm hungry − Oh, I ____ (make) you a sandwich.

2. The futurist Ray Kurzweil predicts that we ______ (have) computers so powerful that they _______ (surpass) human intelligence.

3. He _____ (publish) the paper on the results of his research next month.

4. The 21st century ______ (see) technological change on an astonishing scale.

5. The flight ____ (leave) at 8 p.m.

6. By the end of this century we _____ (see) fundamental advances in our understanding of the underlying structure of matter and of the universe.

7. Look at this building! It _____ (collapse) any minute.

8. I think he ______ (be) very successful.

9. I hope in future we ______ (not / interface) with computers via keyboards.

10. We ______ (fly) to Austin next week for a meeting with the advisory board.

11. I promise you: I _______ (finish) my homework on time next week.

12. This time next week he ______ (test) new application.

29. Choose the best variant to complete the sentence.

1. Perhaps I become/will become a physicist when I get/will get a degree.

2. If you go/will go to China next year you need/will need a visa.

3. The meeting starts/will start when everybody has/will have arrived.

4. I am/will be surprised if she arrives/will arrive before seven o’clock.

5. If rubber is/will be cooled to -200^o C, it will become/becomes brittle.

6. If there is/will be a collision, the airbag inflates/will inflate.

30. Plan what you are going to do next time you have a few days free. Have you got any plans for the next year or so? Are you going to make any changes in your life? Think of something that you are never going to do again in your life.

31. Translate the following sentences into English.

1. Ігрове поле судоку складається з квадрата, розміром 9×9, розділеного на менші квадрати зі стороною 3 клітини. Мета гри — заповнити вільні клітини цифрами від 1 до 9 так, щоб в кожному рядку, в кожному стовпці і в кожному малому квадраті 3×3 кожна цифра зустрічалася б тільки один раз.

2. Магічні квадрати називають ще планетними таблицями. Легенда розповідає, що найстародавнішу з таких таблиць один східний мудрець уперше побачив на спині священної черепахи.

3. Площа трикутника великого розміру у два рази більша за площу середнього трикутника.

4. Тани, деталі танграма, одержують під час розрізання квадрата вздовж прямих на п’ять рівнобедрених прямокутних трикутників різних розмірів, одного квадрата і одного паралелограма.

5. На замовлення авіакомпанії генетики вивели сорт кукурудзи з "квадратними" зернами.

6. Моделі орігамі можуть бути як плоскими, так і тривимірними.

7. Використовуючи потужний інструментарій алгебри і геометрії, математики зараз можуть точно передбачити скільки існує магічних квадратів певного типу.

LISTENING

32. You are going to listen to the part of a radio program devoted to puzzle craze. Before you listen, check if you know what the following words mean: preoccupation; template; to coin; to superimpose; regiment; upstate; fad.

33. Now, listen and do the tasks that follow.

A. Look at these two types of squares. Which one is an example of a “Latin Square” and which is a “Graeco-Latin Square”?

1 2 3 3 1 2 2 3 1	1a 2c 3b 2b 3a 1c 3c 1b 2a
A	B

B. Decide if the statements below are true or false.

1. The Latin square is used to help create codes for transmitting information.

2. Leonard Euler was the first to create a Latin square.

3. Sudoku is a particular type of Latin square.

4. The “15 Puzzle” was a physical model of a magic square.

5. The “36 officers problem” was solved in 1908.

6. Typical Sudoku puzzle consists of 64 cells in the grid.

7. The “15 Puzzle” was the basis of the Rubik’s Cube.

1	2	3	4	5	6	7

34. Listen again and complete the descriptions.

1. A Latin square is a square in which …..

2. A Graeco-Latin square is a square in which ….

3. Sudoku is a puzzle to complete …

4. The "15 Puzzle is the …

SPEAKING

35. Explain the difference between

• rhombus and parallelogram?

• rhombus and kite?

• acute triangle and obtuse triangle?

• equilateral triangle and isosceles triangle?

• quadrilateral and rectangle?

• an angle and the vertex?

36. Discuss the following questions in small groups or pairs.

1. Which shapes in surrounding things (cars, phones etc.) do you like more: circular and curvy or angular? What’s your favourite shape? Why?

2. Do you like modern art? Can you describe any modern works of art that you either really liked or strongly disliked when you saw them?

3. Do you like round doors and windows like in hobbits’ dwellings? Would you like to live in a house like those in the photos in ex.38.

37. Prepare a short presentation for your classmates on one of the topics below. You may use information from this unit, Further Reading section or do some research on the Internet.

    Euler’s Latin and Graeco-Latin squares Magic figures History and types of magic squares

    Geometry in art and nature Geomagic squares Numerology Geometry in nature

38. Describe the following buildings in as many details as possible. Think about their shape, size, material.