Hanuman’s tail – continued fractions and Ramanujan

V S S Sastry

P. C. Mahalanobis, who was a student at Cambridge and who later became an eminent statistician and founder of the Indian Statistical Institute, Kolkata, was a friend of Srinivasa Ramanujan. One day, while visiting Ramanujan, he read about a problem in Strand Magazine. Ramanujan was in the kitchen, cooking. Mahalanobis tried to solve the problem and thought he would ask his good friend about it. So he turned to Ramanujan and said, “Here’s a problem for you.” “What problem? Tell me,” said Ramanujan, still stirring the vegetables. Mahalanobis read out the problem.

A certain street has between 50 and 500 houses in a row, numbered 1, 2, 3, 4, … consecutively. There is a certain house on the street such that the sum of all the house numbers to the left side of it is equal to the sum of all the house numbers to its right. Find the number of this house.

Ramanujan: Did you find the house number?
Mahalanobis: Yes, I did. If the street had 15 houses, then the required house number is 6, since the sum of all the house numbers to the left side of it is 1 + 2 + 3 + 4 + 5 = 15 which is equal to the sum of all the house numbers to its right 7 + 8. But unfortunately, this doesn’t provide the required answer since it is given that the street has between 50 and 500 houses. I got stuck here.

Ramanujan: That means there are many solutions to the stated problem isn’t it?
Mahalanobis: Yes, I think so.

Ramanujan immediately said: Take down my solution which is in the form of continued fraction.
Mahalanobis: What? A continued fraction?

Ramanujan: Yes, take it down.

The author is a math communicator. He has experimented with all types of communication tools available to take maths to people. He creates math cartoons, builds math models, does glove puppetry for math tales, teaches math activities through origami, designs popup books, and writes scripts for math videos. He can be reached at [email protected].

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