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A simple example of the Siamese method. Starting with "1", boxes
are filled diagonally up and right (↗). 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.
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, as he was
returning from his 1687 embassy to the kingdom of Siam.^{[1]}^{[2]}^{[3]} The
Siamese method makes the creation of magic squares straightforward.
Publication
A description of the Siamese method in
Simon
de la Loubère's 1693
A new historical relation of the
kingdom of Siam.
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.^{[4]}
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:^{[5]}
"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.^{[6]}
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.^{[7]}
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
Order5
magic squares

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.^{[8]}
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.^{[9]}^{[10]}
See also
Notes and
references
 ^
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 [3]
 ^
A new historical relation, Tome II, p.228 [4]
 ^
A new historical relation of the kingdom of Siam p.228 [5]
 ^
A new historical relation of the kingdom of Siam p.228 [6]
 ^
A new historical relation of the kingdom of Siam p229[7]
 ^
A new historical relation of the kingdom of Siam p229[8]
 ^
The Zen of Magic Squares, Circles, and Stars by Clifford
A. Pickover,2002 p.37 [9]