By Ryan Bose-Roy
“Imagine that you are on a street with n houses marked 1 through n. Let’s say there is a house marked by a number X between House 1 and House n, such that the sum of the numbers marked on the houses to the left of X is equal to the sum of the numbers marked on the houses to the right. If n is between 50 and 500, what is n, and what is X?”
Often one of the defining features of student mathematicians is their delight to see others struggle with problems they have solved. Perhaps this was the case when mathematician P.C. Mahalanobis posed this newspaper problem to his colleague, Srinivasa Ramanujan, when they were roommates at Cambridge.
Ramanujan is said not only to have solved the problem in minutes, but to have also remarked that he knew how to find all the possible solutions if n was any number from 1 to infinity. Both Mahalanobis and I were astounded.
So, what was his solution? I’ll let Ramanujan explain:
“The minute I heard the problem, I knew that the answer was a continued fraction. ‘Which continued fraction?’, I asked myself. Then the answer came to my mind.”
We actually can find the answer by a continued fraction (For derivation, check out: https://www.math.auckland.ac.nz/~butcher/miniature/miniature2.pdf or http://www.campusgate.co.in/2016/10/applications-continued-fractions-irrational-numbers.html?m=1 ). And if you were wondering, the total number of houses is 288, and X = 204.
I think it’s pretty clear that if Ramanujan were assigned this problem on one of Ms. Kuberska’ s tests, he would fail due to Lack of Proper Proof. Yet his answer wasn’t just a lucky guess. At the same time, he didn’t know how to prove it. As a matter of fact, he had never heard of some of the most basic and fundamental theorems of algebra, and barely knew what a function or a complex variable was.
Throughout his short lifetime of 30 years, Ramanujan has proved himself to be one of the greatest Mathematicians of the 20th Century. Of more than the 1000 mathematical identities he derived from “pure intuition,” nearly all of them have been proven right in the last 40 years. These identities are not simply obvious musing. Rather, they are ideas that have baffled Mathematicians, and yet seem too brilliant to be false:
Much as how we initially skipped imaginary solutions to quadratics, Ramanujan often did not include concrete proofs in his work. His approach suggests a more experimental approach to mathematics:
“I have observed that P(e2πx is of such a nature that its value is very small when x lies between 0 and 3 ( its value is les than a few hundredths when x =3) and rapidly increases when x is greater than 3.”
At face value, it looks as if these concepts are familiar from college-level calculus. But they are not the same. If we look closely, each of his formulas and concepts seem to involve a much more complicated and different level of mathematics.
And so, we find ourselves discovering what British Mathematician G.H. Hardy discovered in 1913 when he first saw Ramanujan’s notebooks – these ridiculous yet correct formulas had to have come from some kind of genius.
Srinivasa Ramanujan was born in Erode, Madras (modern-day Tamil Nadu), a small town in south India under British Rule. He was sent to school, and was said to have an excellent memory – being able to recite digits of numbers like pi. He was given a scholarship for college in recognition of his mathematical prowess. It became clear, however, that mathematics was Ramanujan’s only skill – he failed his other classes, and ran away to Chennai. He soon became known as a math whiz, and began publishing papers in the Journal of the Indian Mathematical Study. Eventually, however, he ended up as an accounting clerk – basically a human calculator – at a big shipping hub. His boss noticed his particular talent for mathematics, and encouraged his pursuits.
Ramanujan’s big break occurred when he sent a letter to the famous Cambridge mathematician G.H. Hardy. Hardy was revered as somewhat of a stickler for proofs.