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**Macau stamps featuring magic squares**

Dedicated to Yuval on his birthday Pay attention: Stamps price convert the sequence in a magic square.

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**Macau's magic square stamps just made philately even more nerdy**

Postage goes meta as new Chinese stamps celebrate an ancient number pattern by themselves appearing in a pattern Old-age mutant number tortoise: Macau stamp displays the origin myth of the magic square. According to Chinese legend a turtle like the one above crept out of the Yellow River about 4000 years ago. It looks like it is riddled with spots, or bullet holes. But if you look carefully, the dots on its back represent the digits from 1 to 9 arranged in the following way: 492 357 816 If you add the numbers in each row together, they are all equal to 15. For example = 15, and so on. If you add the columns, they sum to 15 also. For example, = 15. And yes, you guessed it, the diagonals do too.

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4 pataca*: Dürer Magic squares can be bigger than three rows and columns. The best known of the 4 x 4 squares was immortalized by German artist Albrecht Dürer in his wood carving Melencolia 1. The magic square appears written in the background, behind a sulky angel. Each row, column and diagonal adds up to 34. In fact, many more combinations of four numbers add up to 34, such as the outer corners, some of the 2x2 subsquares, and many more. But the geekiest aspect of the Dürer square is that it includes the date of when he thought it up – 1514, which we see on the bottom row. * 1 dollar = 8 pataca

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9 pataca: de la Loubère There are many ways to create magic squares. One of the most famous methods is named after Simon de la Loubère, a seventeenth century French diplomat who spent time in what used to be called Siam, now Thailand. The method only works for squares that have an odd number of rows/columns. You start with a 1 halfway along a side, as the stamp shows, and then progress diagonally (NE) with the rule that if you leave the square on the top, you reappear on the bottom, and if you leave the square on the right you reappear on the left. Each free square you reach you must write down the next number up, and if a square is not free, you place the new number on the square below it.

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2 pataca – Sator Okay, this isn’t a magic square. But is a square of ancient mystical interest whose power comes from its playful arrangement of letters. It contains the Latin words SATOR (sower) AREPO (Arepo, probably a proper name) TENET (holds) OPERA (the works) ROTAS (rolling) Which can be read forwards, backwards, downwards and upwards. The meaning is unclear, but suggestions have been made like “The sower Arepo keeps the world rolling.” Several Sator squares have been found in excavations, including one in the ruins of Pompeii.

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full diagonals 52-17=228 45-16=292 ‘broken diagonal’. 11-22=260 16-17=260 3 pataca – Franklin Has there ever been an overachiever like Benjamin Franklin? Thinker, politician, scientist, Founding Father, musician, inventor, statesman, author…and magic square legend! One of his inventions was the ‘broken diagonal’. The 8x8 square in the stamp is not strictly a magic square since the full diagonals do not add up to 260. But the rows, columns and broken diagonals – colour-coded in the stamp – do. Columns 45-17=260 13-49=260 Rows 50-47=260 16-17=260 9-24=260

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5 pataca – Su Hui Again, not a magic square, but pretty amazing all the same. It is a ‘palindromic poem’ composed by the Chinese poet Su Hui around the fourth century AD. In the full version, the poem is a 29 x 29 square where each position has a single Chinese character. The poem can be read forwards, backwards, upwards and downwards. In fact, there are 2848 different ways to read it. The stamp contains the 15x15 central section of the full poem. Su Hui is said to have written the poem to her husband who had moved to live far from her, and then married another woman. When the husband read it, he returned to Su Hui.

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**Su Hui with her great palindrome, the Xuanji Tu.**

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** The original, of Su Hui's Xuanji Tu palindrome poem.**

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** A simplified Chinese reduction, from the palindrome poem by Xuanji Tu from Su Hui ..**

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**History of the poem and its retelling**

A star gauge was a spherical instrument used to calculate and predict the motion of planets and stars. Early sources focused on the circular poem composing the outer border of the grid, consisting of 112 characters. Later sources described the whole grid of 840 characters (not counting the central character xin, meaning "heart", which lends meaning to the whole but is not part of any of the smaller poems). The text of the poem was circulated continuously in medieval China and was never lost, but during the Song Dynasty it became scarce. The 112 character version was included in early sources. The earliest surviving excerpts of the entire grid version date from a 10th-century text by Li Fang. By the Tang period, the following story about the poem was current: Dou Tao of Qinzhou was exiled to the desert, away from his wife Lady Su. Upon departure from Su, Dou swore that he would not marry another person. However, as soon as he arrived in the desert region, he married someone. Lady Su composed a circular poem, wove it into a piece of brocade, and sent it to him. Another source, naming the poem as Xuanji Tu (Picture of the Turning Sphere), claims that the grid as a whole was a palindromic poem comprehensible only to Dou (which would explain why none of the Tang sources reprinted it), and that when he read it, he left his desert wife and returned to Su Hui. Some 13th century copies were attributed to famous women of the Song Dynasty, but falsely so. The poem was also mentioned in the story Flowers in the Mirror.s.* * Flowers in the Mirror is set in the reign of the Empress Wu Zetian (684–705) in the Tang dynasty. She took the throne from her own son, Emperor Zhongzong of Tang. Empress Wu lets the power she is given go to her head, and she demands that all of the flowers on the earth be in bloom by the next morning. The flower-spirits fear her and follow her orders, but they are then punished by the gods for doing so. Their punishment is to live on earth. Once their penance is complete, they will be allowed to go back to heaven again

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A geometric magic square, often abbreviated to geomagic square, is a generalization of magic squares invented by Lee Sallows in A traditional magic square is a square array of numbers (almost always positive integers) whose sum taken in any row, any column, or in either diagonal is the same target number. A geomagic square, on the other hand, is a square array of geometrical shapes in which those appearing in each row, column, or diagonal can be fitted together to create an identical shape called the target shape. As with numerical types, it is required that the entries in a geomagic square be distinct. Similarly, the eight trivial variants of any square resulting from its rotation and/or reflection, are all counted as the same square. By the dimension of a geomagic square is meant the dimension of the pieces it uses. Hitherto interest has focused mainly on 2D squares using planar pieces, but pieces of any dimension are permitted. Lee Sallows, a British recreational mathematician living in the Netherlands, has invented a whole new type of magic square. In a ‘geomagic square’ the shapes in each row, column and diagonal can be reassembled into the same master shape. The stamp shows a 3x3 geomagic square in the middle, and around the outside are how the constituent parts fit into the master shape, which is a 4x4 square with one unit taken out. Trying to unlock the secrets of magic squares has been the ruling passion of my life over the past 50 years. It has been a very fullfilling experience, and given the choice, I would happily do it all over again. In particular, nothing has exceeded the sheer excitement and intellectual pleasure I took in the exploration of a fascinating new world revealed through the emergence of two-dimensional or geomagic squares. That I was lucky enough to be the first to go prospecting in that land was a privilege I probably didn't deserve, but certainly appreciated and enjoyed in full measure.

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Inder taneja ixohoxi 88 Inder Taneja, Professor of the Department of Mathematics of University of Santa Catarina, Brazil, from 1978 to He has published more than 100 research papers in internationally renowned journals. IXOHOXI Magic Squares are a special series that not only show common properties like other Magic Squares, as well as being Pandiagonals, but also include alternative properties such as Symmetries, Rotations and Reflections.

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The word IXOHOXI is itself a Palindrome and Symmetric (Reflection), in relation to its center “H”. As the 10 digits (0 to 9) use the number style of a 7 Segments LED Display, in which only 5 digits (0, 1, 2, 5 and 8) remain the same after a 180 Degrees Rotation. It should be noted that the 4 digits (0, 1, 2 and 5) used to construct the Magic Square of Order 4, are precisely the same digits that constituted the year 2015, year of its publication as a stamp. Taking into consideration the 5 digits and their Symmetric Properties, Inder Taneja created the IXOHOXI Universal 88 Magic Square, reproduced in this stamp, has the following properties: The Magic Square still remains a Magic Square: • After a Rotation of 180 Degrees; • After changing the order of the digits in the Cell numbers, i.e. 82 to 28; • If it is seen in a mirror, or reflected in water or seen from the back of the sheet; • The Magic Sum S of the Magic Square of Order 4 is equal to 88, number that also enjoys Symmetrical properties.

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**McClintock / Ollerenshaw – Most Perfect**

A Most-Perfect Magic Square is a Pandiagonal Magic Square of Doubly Even Order – with additional two proprieties: • The Cells of any square of Order 2, (2×2 Cells) extracted from it, including Wrap-Around, sum up to the same constant value, 2(1+n2); • Along the Main or Broken Diagonals, any two numbers separated by n/2 Cells, are a Complementary Pair, i.e. sum 1+n2. In the case of the Most-Perfect Magic Square of Order 8 reproduced in the stamp, the mentioned properties show the following results: z 2(1+n2) = 2(1+82) = 130 E.g.: ( ) = ( ) = 130 z (1+n2) = (1+82) = 65 E.g.: (1+64)=(34+31) = (25+40) = 65 All the Pandiagonal Squares of Order 4 are Most-Perfect. However, when n>4 the proportion Pandiagnal to Most-Perfect decreases as n increases. It is not possible to establish the history of Most-Perfect Magic Squares without to mention Kathleen Timpson Ollerenshaw. In 1982, with Hermann Bondi, she developed a mathematical analytical construction that could verify the number 880 for the essentially different Magic Squares of Order 4. After this achievement she began to study Pandiagonal Magic Squares based on works published by Emory McClintock in After several years, in 1986, Kathleen Ollerenshaw published a paper where, making use of Symmetries, she proved that there are essential different Most-Perfect Magic Squares of Order 8. Step by step, finally she could discover how to construct and how to count the total number of Most-Perfect Magic Squares of all with an Order Multiple of 4. Together with David Brée, who helps her organize her research notes and proof-reading, they finally published the book “Most-Perfect Pandiagonal Magic Squares: Their Construction and Enumeration” in 1998.

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**Stamp (3/2): David Collison – Patchwork**

David M. Collison ( ) was born in United Kingdom and lived in Anaheim, California. He was a fruitful creator of Magic Squares and Cubes. He specialized in Generalized Shapes from which he created the Patchwork Magic Squares. A Patchwork Magic Square is an Inlaid Magic Square – one Magic Square that contains within it other Magic Squares or Odd Magic Shapes. The most common Shape is Magic Rectangle, but Diamond, Cross, Elbow and L Shapes can also be found. These Shapes are Magic if the Sum in each Direction is proportional to the number of Cells. The Patchwork Magic Square of Order 14 reproduced in this stamp has the following proprieties:

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**David Collison – Patchwork**

• Contain: Four Order 4 Magic Squares, 4×4, in the Quadrants; One Magic Cross, 6×6, in the Centre; Four Magic Tees, 6×4, on the Centre Sides; Four Magic Elbows, 4×4, in the Corners. • All the Shapes sum to a Constant which is directly proportional to the number of Cells in a Row, Column or Diagonal: S2=197; S4=394; S6=591; S14=1379.

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The stamps for the 8, 1 and 6 pataca stamps are due to be released next year, and when they are you will be able to buy a set of them that form their own lo shu magic square:

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**First Day Cover: Yang Hui Magic Circles**

The XIII Century was probably one of the most important periods in the History of Chinese Mathematics, with the publication of Shu Shu Jiu Zhang ‘1247, by Qin Jiu Shao ‘and Ce Yuan Hai Jing ‘by Li Ye ,followed 15 years later, by the works of Yang Hui . Yang Hui ( ), a Chinese mathematician born in Qiantang, Zhejiang Province during the late Song Dynasty ( ). His best known work was Yang Hui Suanfa), Yang Hui’s Methods of Computation, which was composed of 7 volumes and published in 1378. The topics covered by Yang Hui include Multiplication, Division, Root-extraction, Quadratic and System Equations, Series, Computations of Areas of Polygons as well as Magic Squares, Magic Circles, the Binomial Theorem and, the best known work, his contribution to the Yang Hui’s Triangle, which was later rediscovered by Blaise Pascal, 1653. The central picture of the cover of the first day presents the Yang Hui Magic Circles. These Nine Circles are composed by 72 Numbers, from 1 to 72, with each individual Circle having 8 Numbers. The Neighboring Numbers make Four Additional Circles, each with 8 Numbers, the total Sum of the 72 Numbers is 2628 and the Sum of the 8 Numbers in each Circle is 292.

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Horse jumping method

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**Math presentations on our website:**

מתמטיקאים מגטינגן – מצגת שנייה פפירוס רינדf מתמטיקה מצרית עתיקה תרגולי האתר ללימוד מתמטיקה האם ידעתם מקור המילים אלגוריתם ואלגברה מתמטיקאים יהודים באוניבאסיטת גטינגן – חכמים ספרדים מתמטיקאים ואסטרונומים

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**Clarita-Efraim pps: Sources: www.clarita-efraim.com**

Clarita-Efraim pps: October 2018

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