Ramanujan’s Continued Fraction!

The popular English magazine Strand had long carried a page, entitled “Perplexities,” devoted to intriguing puzzles, numbered and charmingly titled, like “The Fly and the Honey,” or “The Tessellated Tiles,” the answers being furnished the following month. Each Christmas, though, “Perplexities” expanded, the author fitting his puzzles into a short story.
Now, in December 1914, “Puzzles at a Village Inn” took readers to the imaginary town of Little Wurzelfold, where the main topic of interest was what had just happened in Louvain.

In late August, pursuing an explicit policy of brutalization against civilian populations, German troops began burning the medieval Belgian city of Louvain, on the road between Liege and Brussels. House by house and street by street they set Louvain to the torch, destroying its great library, with its quarter million books and medieval manuscripts, and killing many civilians. The burning of Louvain horrified the world, galvanized public opinion against Germany, and united France, Russia, and England more irrevocably yet. “The March of the Hun,” English newspapers declared. “Treason to Civilization.” It was an early turning point of the war, doing much to set its tone. Louvain came to symbolize the breakdown of civilization. And now it reached even the “Perplexities” page of Strand.

One Sunday morning soon after the December issue appeared, P. C. Mahalanobis sat with it at a table in Ramanujan’s rooms in Whewell’s Court. Mahalanobis was the King’s College student, just then preparing for the natural sciences Tripos, who had found Ramanujan shivering by the fireplace and schooled him in the nuances of the English blanket. Now, with Ramanujan in the little back room stirring vegetables over the gas fire, Mahalanobis grew intrigued by the problem and figured he’d try it out on his friend.

“Now here’s a problem for you,” he yelled into the next room

“What problem? Tell me,” said Ramanujan, still stirring. And Mahalanobis read it to him.

“I was talking the other day,” said William Rogers to the other villagers gathered around the inn fire, “to a gentleman about the place called Louvain, what the Germans have burnt down. He said he knowed it well — used to visit a Belgian friend there. He said the house of his friend was in a long street, numbered on this side one, two, three, and so on, and that all the numbers on one side of him added up exactly the same as all the numbers on the other side of him. Funny thing that! He said he knew there was more than fifty houses on that side of the street, but not so many as five hundred, I made mention of the matter to our parson, and he took a pencil and worked out the number of the house where the Belgian lived, I don’t know how he done it.”

Perhaps the reader may like to discover the number of that house.

Through trial and error, Mahalanobis (who would go on to found the Indian Statistical Institute and become a Fellow of the Royal Society) had figured it out in a few minutes. Ramanujan figured it out, too, but with a twist, “Please take down the solution,” he said — and proceeded to dictate a continued fraction, a fraction whose denominator consists of a number plus a fraction, that fraction’s denominator can consisting of a number plus a fraction, ad infinitum. This wasn’t just the solution to the problem, it was the solution to the whole class of problems implicit in the puzzle. As stated, the problem had but one solution — house no. 204 in a street of 288 houses; 1+ 2 + … + 203 = 205 + 206 + … + 288. But without the 50-to-500 house constraint, there were other solutions. For example, on an eight-house street, no. 6 would be the answer: 1+2+3+4+5 on its left equaled 7+8 on its right. Ramanujan’s continued fraction comprised within a single expression all the correct answers.

Mahalanobis was astounded. How, he asked Ramanujan, had he done it?

“Immediately I heard the problem it was clear that the solution should obviously be a continued fraction; I then thought, Which continued fraction? And the answer came to my mind.”

From The Man Who Knew Infinity, by Robert Kanigel

I will follow up this post, with a description of how Ramanujan may have solved it, using Pell’s equation.

Follow up:

Prof John Butcher, in his Mathematical Miniatures, has an excellent account of the solution, using Pell’s Equation, entitled “On Ramanujan, continued fractions and an interesting street number”. Check it here.
I am sure Prof Butcher’s page on Mathematical Miniatures and Apologies would also be of great interest!


2 thoughts on “Ramanujan’s Continued Fraction!

  1. Hi Karthik,

    I was looking for details on this anecdote. Thanks for providing the same. Your blog looks interesting. I would be happy to know more about you.

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