(3) Decoding the Quadratum Mirabile!
Quadratum Mirabile means Wonderful Squares! And all the Magic Squares we are going to discuss are indeed Quadratum Mirabile!
Let's start by tackling the 3x3 'Quadratum Mirabile' puzzle first! The challenge is simple: Fill a 3x3 square with natural numbers (they can be consecutive or random) such that the sum of the numbers in all three rows, all three columns, and both main diagonals are equal.
To develop a reliable mathematical method for completing any 3x3 magic square, we'll use a little bit of algebra.
First of all we will introduce Variables: Let's represent the numbers in the grid using variables like a, x and y. By establishing relationships between these variables, we can construct the entire square.
Start with the Diagonal: A helpful rule of thumb for any order of magic square will be to always begin by filling a diagonal. This is often the easiest and most powerful way to set up the rest of the puzzle!
So let us start with the middle most term. Let us assume the value as equal to 'a' as shown in the part (i) of the above image. We want to keep the sum of all cells on each diagonals same. Hence we assume the value of the first cell as 'a-x' and the value of the last cell as 'a+x' as shown in the part (i) of the image. Now the sum of numbers on the first diagonal is equal to 3a.
Similarly, we can take the numbers on third cell and 7th cell as 'a-y' and 'a+y'. Now the sum of numbers on the second diagonal is also equal to 3a. See part (ii) of the above image.
Now we are left with cell numbers 2,4,6 and 8. Their values can be found by simple addition and subtraction such that sum of all Rows/Columns is also equal to 3a. Thus we get the final image of 3x3 square as below:
So this is the generalisation of 3X3 magic square or our own Quadratum Mirabile. We can take various values for a, x and y to make any magic square.
Suppose we take a=5, x=3 and y=1, then we will get the following basic square:
By rotation and reflection we can get seven more squares from the above square. ( Total eight squares.)
Don't be fooled by appearances! Beneath their seeming differences, they are all merely echoes of the original: revealed through a simple twist (rotation), a mirror-image flip or a water image flip (reflection).
The easier part ( 3x3 Magic Squares) is over! Can you now try to develop your own theory to create 4x4 Magic Squares!?
Hint!? Start with the Diagonals and choose the variables carefully. Finally put the values of variables to get the desired magic square.
In the upcoming blog post we will understand the mythic origins behind the very first known pattern—the 3x3 Lo Shu Square. Let us delve a bit into the fascinating history of Magic Squares that spanned continents and centuries before trying 4x4 or higher order magic squares.
Pankaj Khanna
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मेरे कुछ अन्य ब्लॉग:
हिन्दी में:
तवा संगीत : ग्रामोफोन का संगीत और कुछ किस्सागोई।
रेल संगीत: रेल और रेल पर बने हिंदी गानों के बारे में।
साइकल संगीत: साइकल पर आधारित हिंदी गाने।
कुछ भी: विभिन्न विषयों पर लेख।
तवा भाजी: वन्य भाजियों को बनाने की विधियां!
मालवा का ठिलवा बैंड: पिंचिस का आर्केस्टा!
ईक्षक इंदौरी: इंदौर के पर्यटक स्थल। (लेखन जारी है।)
अंग्रेजी में:
Love Thy Numbers : गणित में रुचि रखने वालों के लिए।
Epeolatry: अंग्रेजी भाषा में रुचि रखने वालों के लिए।
CAT-a-LOG: CAT-IIM कोचिंग।छात्र और पालक सभी पढ़ें।
Corruption in Oil Companies: HPCL के बारे में जहां 1984 से 2007 तक काम किया।
