Magic sQuares


The Chinese appear to have been the first to use magic squares - beautifully balanced rows of figures whose sums from left to right, from top to bottom, and diagonally are always equal. According to Chinese tradition, the mythical Emperor Yü first saw a magic square in a vision. This took place on the banks of the river Lo, after there had been great flooding in China. The people had offered sacrifices to appease the river Gods, but every time they did so, a turtle emerged from the river and walked around the offerings. In his vision the Emperor realised that the correct amount of offerings, fifteen, was shown in the square in the pattern on the turtle's back. That's why in its simplest form the magic square is called lo-shu. The square is shown below using arabic numerals. In Chinese, even numbers represent Yin - the feminine facet of life - and uneven numbers symbolise Yang - the masculine facet.













In mathematics a Magic Square is an array of distinct numbers so arranged in a square 
that the sums of each row, each column, and each main diagonal are equal. For example, 
the square array opposite is a magic square of order 3 (the order is the number of horizontal rows 
or vertical columns), the constant sum of which is 15. 
















In ancient times such configurations 
of numbers were thought to be good-luck charms or talismans. 
Later, mathematicians became interested in the magic square as a problem in mathematical analysis.



Below are several grids which you can fill in straight away.






































A magic square is called panmagic or pandiagonal if the sum of each broken diagonal 
is also equal to the magic-square constant. The magic square of order 3 shown above is 
not panmagic, but the magic square of order 4
can be panmagic because the numbers in each of the four rows, four columns, and eight diagonals 
all could add up to an equal total. Can you make this diabolical one? The constant sum should be 34. 













































A magic square is called bimagic or doubly magic if it remains a magic square when each 
element is replaced by its square; it is called trimagic or triply magic if it remains a 
magic square when each element is replaced by its square and its cube. 



Here you see the grid for a normal magic square of the order of five. 
The constant sum here is 65. You must fill in the numbers 1 to 25 in the square. 
The number 13 is given. 
Can you finish this one?






































This magic square cannot be made because the number four is not placed in a corner. 
However, by using fractions you can make a magic square of it with the constant sum of 15. 
As far as I know there are two solutions. 
Can you find one - or even both - of them?













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