# India

### The Mathematician Srinivasa Ramanujan

The famous Indian mathematician, Srinivasa Aiyangar Ramanujan, lived from 1887 to 1920. Early in his life he showed astounding prowess at mathematics, proposing new theorems of trigonometry by the age of twelve.  He published articles in Indian mathematical journals but received no formal university training.  He came to the attention of famous mathematicians of his day after he wrote to G.H.Hardy, then a leading English mathematician, proposing new theorems (but without giving any proofs).  Hardy recognized great mathematical insight in his work and arranged for Ramanujan to move to England in 1913 to study with him.

Ramanujan had a great love for numbers, much of his work focusing on prime numbers, summation formulas for constants such as Pi, and the theory of numbers. Ramanujan's contributions to mathematics come from his many notebooks which often contained unproven but important theorems.  Hardy's collaboration with him was very productive and they became good friends. Possibly because Ramanujan was a practicing Brahmin and therefore vegetarian, he was unable to maintain good health during the First World War (1914-1918) when fresh produce was hard to find.  In writing about this, Hardy related the following incident:  (Hardy Quotation )

"I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'"

Whether Ramanujan knew this in advance (demonstrating how he took extraordinary interest in individual numbers) or calculated it on the spot, it shows the mind of a great mathematician.

### Problem 1

1729 is the sum of two cubes in two different ways. What are those two different ways?

### Problem 2

What is the smallest number that is the sum of two squares in two different ways?

### Interesting Fact

In 1919 Ramanujan returned to India but was unable to regain his health, dying in 1920 after an incredibly short but productive life.  Some of his many notebooks recently resurfaced, and some of them have been shown to contain useful mathematics related to the number Pi. Mathematicians have long been interested in Pi because the means of calculating it have historically given rise to new branches of mathematics. Recent work by Canadian Mathematicians, among others, has produced more and more exact measures of Pi.  Go here to get your piece.

Famous Indian Mathematicians
http://turnbull.mcs.st-and.ac.uk/history/BirthplaceMaps/Countries/India.html

Towers of Hanoi
http://www.cut-the-knot.org/Curriculum/Combinatorics/TowerOfHanoi.shtml

Bicolor Towers of Hanoi
http://www.cut-the-knot.org/Curriculum/Combinatorics/BiColorTowerOfHanoi.shtml

### Surface area of a sphere and a cylinder

Ah yes, parallel lines on a sphere… they eventually all touch. However, when you view the map in an atlas, the lines are perfectly parallel. How does this happen? Well think about this firstly. Imagine the earth is a perfectly formed sphere. Now imagine it is completely surrounded by a map that fits snugly – like a cylinder, with a top and a bottom. The cylinder and sphere have the same height and diameter. Which has the greatest surface area?

Try solving the International Color Challenge!