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Secondary Magazine - Issue 44: Focus on


Created on 28 September 2009 by ncetm_administrator
Updated on 19 September 2014 by ncetm_administrator

 

Secondary Magazine Issue 44magic square illustration
 

 

Focus on... Magic Squares

Wolfram Mathworld defines a magic square as a square array of numbers consisting of the distinct positive integers 1, 2, ..., n2 arranged such that the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number. While this definition only allows for the integers 1 to n, it is common for squares to be considered magic if they contain any set of numbers for which the condition the sum of the numbers in any horizontal, vertical, or main diagonal line is always the same number.
 

The Lo Shu magic square is the only 3×3 magic square featuring the numbers 1 to 9 and may date from as early as 3000BC. Legend has it that there was a terrible flood in China and the king tried to calm the river god with sacrifice but the sacrifice wasn’t accepted. One day a turtle was seen in the river with a pattern of dots on its shell which may have looked like this:
 
Lo Shu magic square

The local people took this as a sign that, to overcome the flood, they would have to increase their offerings to 15. They did this and the river god was appeased.
 

Although the Lo Shu is the only 3×3 magic square to feature the numbers 1 to n (all others are simply transformations of the Lo Shu square), there are many ways of creating a 4×4 square (808 unique solutions) and 275 305 224 ways of creating a 5×5 square. It is not known how many possible 6×6 magic squares can be created, but Pinn and Wieczerkowski use statistical analysis to generate an estimate of a staggering (1.7745 ± 0.0016) × 1019.
 
John Mason uses a magic square with the numbers removed to help develop mathematical thinking and reasoning in this activity.
  
If the grid below is a magic square then convince yourself that it’s true that
sum (orange square) – sum (yellow square) = 0
 
John Mason magic square
 
What other arrangements are there in a 3×3 magic square for which this is true?
How about in a 4×4 magic square?
 

The 4×4 magic square in Albrecht Dürer’s engraving, Melancholia, features the year that the engraving was completed (1514) in the middle two columns of the bottom row. 
 
 
In 1690 Simon de la Loubère, the French ambassador to the King of Siam, developed a method for creating a magic square with sides which are odd. The method starts by placing 1 in the top row of the middle column. The aim is to place the next integer one square up and to the right. If this isn’t possible because the number would ‘fall out’ of the top then it rejoins at the bottom; if the number would fall out of the right hand side of the square then it rejoins on the right hand side. If it’s not possible to place a number because the square one up and to the right is already occupied then the number is instead placed directly below the most recently placed integer. An example of this method can be seen on Wolfram Mathworld.
 
 
In 1770, Euler generated the first known magic square constructed entirely of square numbers, sending it to Joseph Lagrange.
 
Euler's magic square
 
Electronic engineer Lee Sallows created this magic square
 
Lee Swallows magic square

Counting the letters in each word gives
 
Lee Swallows second magic square

Which is a second magic square!
 
 
 
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