I- Catcer Consol-web Moneter

peach and Gaudí magic squares on the infinite

In the previous post The magic of numbers , we refer to some interesting considerations about the numbers and their relationship to magic and superstition. Always related to this subject, talk about today of magic squares. The first known, in ancient China in the third century BC, is:

is so called, magic square, they were used to build the numbers 1 to 9 and by adding the elements found in its columns, rows or diagonals will always get the same result 15, which is called the magic sum .

Now consider the square of order 4 which is obtained by placing the numbers 1 to 16 as follows:

Se puede comprobar, para verificar que es un cuadrado mágico, que la suma de los elemento de las filas, columnas o las dos diagonales es el mismo valor de 34, su suma mágica. Este cuadrado fue incorporado por el pintor Alberto Durero en su ob ra M elancolía , en la parte superior derecha de la obra y cumple algunas propiedades adicionales, como por ejemplo, incluye la fecha de la pintura en la parte inferior al centro, año 1514, la suma de las cuatro celdas centrales (10,11,6,7), las cuatro esquinas (16,13,4,1) y otras combinaciones simétricas, also added 34, among many other properties, which give a more than magic in this square.

I say there, other than those given above, several squares of order 3 and many others of order 4, and of course the higher we make the order or size of this square, the greater the difficulty of sorting the numbers so as to meet the sum of the elements of each row, column and diagonal is the same. Interestingly, for those of order 4, De Bessy established in 1693 that there are 880 different magic squares. Later it has been shown that there are magic squares of order 275305224 5. For larger orders we only have estimates.

The great American inventor Benjamin Franklin found the magic square of order 8 given by

In each row, column and diagonal add 260, he was passionate about this type of problems that occupied his spare time while doing clerical work in Philadelphia, a curious fact, and evidence of his genius, it is said that in one afternoon, managed to solve the square order 16.

Leonard Euler, the most prolific and great Swiss mathematician found a magic square of order 8, where each row and each column adds 260, but in addition, each row and column of each of the four subquadrate of order 4 obtained, if we start this square into four , numbered 130 and, further, that in this "magic board" of order 8 describes the path of the horse's movement around the board without being twice in any of them known as the " problem of motion horse " .

variations of magic squares, the one that caught my attention is that of order 4 which is incorporated The Passion façade the expiatory church of La Sagrada Familia in Barcelona , monumental and amazing work of Antonio Gaudí , see the kiss of Judas:

The Cryptogram Gaudí, as they call it, is extended:

In this case the magic constant of the square is 33, the age of Jesus Christ for this, two numbers (12 and 16 ) are reduced by two units (10 and 14) with two replicates displayed in this way is achieved the magic constant lower by 1 unit, 34 to 33.

This magical sum is always 33, in 310 different combinations in groups of four, just to name a few: 1,14,11,7 (subquadrate top left) or 8,10,13,2 (subquadrate bottom left) or 7,6,10,10 (subquadrate interior) or 14,11,6,2 (a cross) or 14,7,9,3 (cross) or 14, 6,10,3 (shingles) or 14,14,2,3 11,8,9,5 or , including yourself can try using geometric shapes or paths.

Thus, we can use in our classrooms the magic of these squares to motivate our students and, in turn, demonstrate the close relationship that exists between mathematics and art.

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