 Siamese method

The Siamese method, or De la Loubère method, is a simple method to construct any size of nodd magic squares (i.e. number squares in which the sums of all rows, columns and diagonals are identical). The method was brought to France in 1688 by the French mathematician and diplomat Simon de la Loubère,^{[1]} as he was returning from his 1687 embassy to the kingdom of Siam.^{[2]}^{[3]}^{[4]} The Siamese method makes the creation of magic squares straightforward.
Contents
Publication
De la Loubère published his findings in his book A new historical relation of the kingdom of Siam (Du Royaume de Siam, 1693), under the chapter entitled The problem of the magical square according to the Indians.^{[5]} Although the method is generally qualified as "Siamese", which refers to de la Loubère's travel to the country of Siam, de la Loubère himself learnt it from a Frenchman named M.Vincent (a doctor, who had first travelled to Persia and then to Siam, and was returning to France with the de la Loubère embassy), who himself had learnt it in the city of Surat in India:^{[6]}
"Mr. Vincent, whom I have so often mentioned in my Relations, seeing me one day in the ship, during our return, studiously to range the Magical Squares after the method of Bachet, informed me that the Indians of Suratte ranged them with much more facility, and taught me their method for the unequal squares only, having, he said, forgot that of the equal"—Simon de la Loubère, A new historical relation of the kingdom of Siam.^{[7]}The method
The method was surprising in its effectiveness and simplicity:
"I hope that it will not be unacceptable that I give the rules and the demonstration of this method, which is surprising for its extreme facility to execute a thing, which has appeared difficult to our Mathematicians"—Simon de la Loubère, A new historical relation of the kingdom of Siam.^{[8]}First, an arithmetic progression has to be chosen (such as the simple progression 1,2,3,4,5,6,7,8,9 for a square with three rows and columns (the Lo Shu square)).
Then, starting from the central box of the first row with the number 1 (or the first number of any arithmetic progression), the fundamental movement for filling the boxes is diagonally up and right (↗), one step at a time. When a move would leave the square, it is wrapped around to the last row or first column, respectively.
If a filled box is encountered, one moves vertically down one box (↓) instead, then continuing as before.
Order3 magic squares
step 1 1 . . step 2 1 . 2 step 3 1 3 2 step 4 1 3 4 2 step 5 1 6 3 5 4 2 step 6 1 6 3 5 7 4 2 step 7 8 1 6 3 5 7 4 2 step 8 8 1 6 3 5 7 4 9 2 Order5 magic squares
Step 1 1 . . . . Step 2 1 . . . 3 . 2 Step 3 1 5 4 . 3 2 Step 4 1 8 5 7 4 6 . 3 2 Step 5 1 8 15 5 7 14 4 6 13 10 12 3 11 2 9 Step 6 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 Other sizes
Any nodd square ("oddorder square") can be thus built into a magic square. The Siamese method does not work however for neven squares ("evenorder squares", such as 2 rows/ 2 columns, 4 rows/ 4 columns etc...).
Order 3 8 1 6 3 5 7 4 9 2 Order 5 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 Order 9 47 58 69 80 1 12 23 34 45 57 68 79 9 11 22 33 44 46 67 78 8 10 21 32 43 54 56 77 7 18 20 31 42 53 55 66 6 17 19 30 41 52 63 65 76 16 27 29 40 51 62 64 75 5 26 28 39 50 61 72 74 4 15 36 38 49 60 71 73 3 14 25 37 48 59 70 81 2 13 24 35 Other values
Any sequence of numbers can be used, provided they form an arithmetic progression (i.e. the difference of any two successive members of the sequence is a constant). Also, any starting number is possible. For example the following sequence can be used to form an order 3 magic square according to the Siamese method (9 boxes): 5, 10, 15, 20, 25, 30, 35, 40, 45 (the magic sum gives 75, for all rows, columns and diagonals).
Order 3 40 5 30 15 25 35 20 45 10 Other starting points
It is possible not to start the arithmetic progression from the middle of the top row, but then only the row and column sums will be identical and result in a magic sum, whereas the diagonal sums will differ. The result will thus not be a true magic square:
Order 3 500 700 300 900 200 400 100 600 800 Rotations and reflexions
Numerous other magic squares can be deduced from the above by simple rotations and reflections.
Variations
A slightly more complicated variation of this method exists in which the first number is placed in the box just above the center box. The fundamental movement for filling the boxes remains up and right (↗), one step at a time. However, if a filled box is encountered, one moves vertically up two boxes instead, then continuing as before.
Order 5 23 6 19 2 15 10 18 1 14 22 17 5 13 21 9 4 12 25 8 16 11 24 7 20 3 Numerous variants can be obtained by simple rotations and reflections. The next square is equivalent to the above (a simple reflexion): the first number is placed in the box just below the center box. The fundamental movement for filling the boxes then becomes diagonally down and right (↘), one step at a time. If a filled box is encountered, one moves vertically down two boxes instead, then continuing as before.^{[9]}
Order 5 11 24 7 20 3 4 12 25 8 16 17 5 13 21 9 10 18 1 14 22 23 6 19 2 15 These variations, although not quite as simple as the basic Siamese method, are equivalent to the methods developed by earlier European scholars, Johann Faulhaber (1580–1635) and Claude Gaspard Bachet de Méziriac (1581–1638), and allowed to create magic squares similar to theirs.^{[10]}^{[11]}
See also
Notes and references
 ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 54. ISBN 9781848000001. footnote 8
 ^ Mathematical Circles Squared" By Phillip E. Johnson, Howard Whitley Eves, p.22
 ^ CRC Concise Encyclopedia of Mathematics By Eric W. Weisstein, Page 1839 [1]
 ^ The Zen of Magic Squares, Circles, and Stars By Clifford A. Pickover Page 38 [2]
 ^ A new historical relation of the kingdom of Siam p.228
 ^ A new historical relation, Tome II, p.228
 ^ A new historical relation of the kingdom of Siam p.228
 ^ A new historical relation of the kingdom of Siam p.228
 ^ A new historical relation of the kingdom of Siam p229
 ^ A new historical relation of the kingdom of Siam p229
 ^ The Zen of Magic Squares, Circles, and Stars by Clifford A. Pickover,2002 p.37 [3]
Categories: Magic squares
 Search algorithms
Wikimedia Foundation. 2010.
Look at other dictionaries:
Strachey method for magic squares — The Strachey method for magic squares is an algorithm for generating magic squares of singly even order 4 n +2.Example of magic square of order 6 constructed with the Strachey method: example 35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28… … Wikipedia
Conway's LUX method for magic squares — is an algorithm by John Horton Conway for creating magic squares of order 4n+2, where n is a natural number. Method Start by creating a (2n+1) by (2n+1) square array consisting of n+1 rows of Ls, 1 row of Us, and n 1 rows of Xs, and then exchange … Wikipedia
FrancoThai relations — France Thailand relations cover a period from the 17th century until modern times. Relations started in earnest during the reign of Louis XIV with numerous reciprocal embassies, and a major attempt by France to Christianize Siam (modern Thailand) … Wikipedia
Simon de la Loubère — A page from Simon de La Loubère : Du Royaume de Siam. Illustration from the English edition (1693). Simon de la Loubère (21 April 1642 – 26 March 1729) was a French diplomat, writer, mathematician and poet. Contents … Wikipedia
Magic square — In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant.[1] A normal magic… … Wikipedia
List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… … Wikipedia
Spawn (biology) — The spawn (eggs) of a clownfish. The black spots are the eyes developing. Spawn refers to the eggs and sperm released or deposited, usually into water, by aquatic animals. As a verb, spawn refers to the process of releasing the eggs and sperm,… … Wikipedia
Southeast Asian arts — Literary, performing, and visual arts of Myanmar (Burma), Thailand, Laos, Cambodia, Vietnam, Malaysia, Singapore, and the Philippines. The classical literatures of Southeast Asia can be divided into three major regions: the Sanskrit region of… … Universalium
Marine steam engine — Period cut away diagram of a triple expansion steam engine installation, circa 1918 A marine steam engine is a reciprocating steam engine that is used to power a ship or boat. Steam turbines and diesel engines largely replaced reciprocating steam … Wikipedia
Mongkut — Not to be confused with Vajiravudh (reigning title Phra Mongkut Klao Chaoyuhua). Rama IV redirects here. For the fourth book in the Rama series, see Rama Revealed. Mongkut King Rama IV … Wikipedia