Magic Squares Explained
In my novel Rescuing Ranu, I have the main character Nela bond with Ranu over Indian mathematician Ramanajun's "magic squares".
Nela asked, in Malyalam, "Can you read?" She flipped open a page, ran her finger down the margin. The girl patted the pad as if it were a pet, then shook her head and backed away.
"Don't be afraid," Nela smiled. "The words can't hurt you. Do you know your numbers?" The girl nodded. Of course she must—it was probably up to her to haggle over prices at the market. She must have learned to add, subtract, multiply, and divide, at a young age.
A game the young Ramanajun had played with his schoolmates--his "magic squares" might appeal to the girl, so Nela pulled out a blank sheet of paper, scored it with three columns, and wrote numbers in each square. The columns added up to the same number, in all directions. The girl laughed with delight. She wanted to try it, too. Nela pushed the paper and pencil toward her.
"What is your name?" she asked.
"I am Ranu," the girl said.
Here is a description of magic squares from the Asia-Pacific Mathematics Newsletter:
Note that in a magic square, the sum of the elements in the columns; the sum of the elements in the three rows; and the sum of the three elements along the diagonal and the skew-diagonal all add to give the number. Above, the 3 x 3 magic square is filled with the first nine natural numbers 1 to 9. The interested reader can try to form 3 x 3 magic squares for any number greater than 15 and realise that this is recreational mathematics.
One can form also 4 x 4 (Date) magic squares and higher dimensional magic squares. Only Chapter 1 of his first Notebook has a title: "Magic Squares", all other chapters in his Notebooks are untitled. Chapter 1 of his first Notebook has 3 pages devoted to this topic and Chapter 1 of his second Notebook, has 8 pages, with 43 entries. While his first Notebook has 16 chapters and 134 pages, his second Notebook has 21 chapters and 252 pages. So, experts consider the second Notebook a revised, lengthened version of the first Notebook. Turning the second Notebook around, Ramanujan wrote down some more entries in an unorganised manner (unlike in his well-organised first and second Notebooks) and this is considered as his third Notebook which has 33 pages, containing about 600 theorems and proofs are being provided in an ongoing project by Bruce Berndt and George Andrews.
Again, in retrospect, we may conjecture that magic squares is perhaps his first introduction to partitions of integers. For, can we not say that he is looking at the partitions of 15 in 3 parts and the problem is equivalent to solving a set of equations, which is a consequence of:
a + b + c = d + e + f = g + h + i = a + d + g = b + e + h = g + h + i = a + e + i = c + e + g = 15?
An admirer and friend of Ramanujan took him to see his Uncle, Dewan Bahadur Ramachandra Rao, who was the Collector of Nellore, stationed at Tirukkoilur. On the first 4 occasions, they were unable to meet the Collector, and it was only on the fifth occasion that they met him. Although Ramachandra Rao was initrally highly skeptical about the prowess of Ramanujan, when he saw the mathematical entries in Ramanujan's 2 thick Notebooks, he could neither make head or tail of the gamut of notes/entries he saw in them. Note that Ramachandra Rao considered himself very knowledgeable as compared to a school boy as he was a MA in mathematics.
Nela asked, in Malyalam, "Can you read?" She flipped open a page, ran her finger down the margin. The girl patted the pad as if it were a pet, then shook her head and backed away.
"Don't be afraid," Nela smiled. "The words can't hurt you. Do you know your numbers?" The girl nodded. Of course she must—it was probably up to her to haggle over prices at the market. She must have learned to add, subtract, multiply, and divide, at a young age.
A game the young Ramanajun had played with his schoolmates--his "magic squares" might appeal to the girl, so Nela pulled out a blank sheet of paper, scored it with three columns, and wrote numbers in each square. The columns added up to the same number, in all directions. The girl laughed with delight. She wanted to try it, too. Nela pushed the paper and pencil toward her.
"What is your name?" she asked.
"I am Ranu," the girl said.
Here is a description of magic squares from the Asia-Pacific Mathematics Newsletter:
Note that in a magic square, the sum of the elements in the columns; the sum of the elements in the three rows; and the sum of the three elements along the diagonal and the skew-diagonal all add to give the number. Above, the 3 x 3 magic square is filled with the first nine natural numbers 1 to 9. The interested reader can try to form 3 x 3 magic squares for any number greater than 15 and realise that this is recreational mathematics.
One can form also 4 x 4 (Date) magic squares and higher dimensional magic squares. Only Chapter 1 of his first Notebook has a title: "Magic Squares", all other chapters in his Notebooks are untitled. Chapter 1 of his first Notebook has 3 pages devoted to this topic and Chapter 1 of his second Notebook, has 8 pages, with 43 entries. While his first Notebook has 16 chapters and 134 pages, his second Notebook has 21 chapters and 252 pages. So, experts consider the second Notebook a revised, lengthened version of the first Notebook. Turning the second Notebook around, Ramanujan wrote down some more entries in an unorganised manner (unlike in his well-organised first and second Notebooks) and this is considered as his third Notebook which has 33 pages, containing about 600 theorems and proofs are being provided in an ongoing project by Bruce Berndt and George Andrews.
Again, in retrospect, we may conjecture that magic squares is perhaps his first introduction to partitions of integers. For, can we not say that he is looking at the partitions of 15 in 3 parts and the problem is equivalent to solving a set of equations, which is a consequence of:
a + b + c = d + e + f = g + h + i = a + d + g = b + e + h = g + h + i = a + e + i = c + e + g = 15?
An admirer and friend of Ramanujan took him to see his Uncle, Dewan Bahadur Ramachandra Rao, who was the Collector of Nellore, stationed at Tirukkoilur. On the first 4 occasions, they were unable to meet the Collector, and it was only on the fifth occasion that they met him. Although Ramachandra Rao was initrally highly skeptical about the prowess of Ramanujan, when he saw the mathematical entries in Ramanujan's 2 thick Notebooks, he could neither make head or tail of the gamut of notes/entries he saw in them. Note that Ramachandra Rao considered himself very knowledgeable as compared to a school boy as he was a MA in mathematics.
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