Ramanujan And Mahalanobis

Number Theory Level 5

In 1914, Ramanujan's friend P. C. Mahalanobis gave him a problem he had read in the English magazine Strand. The problem was to determine the number \( x \) of a particular house on a street where the houses were numbered \( 1,2,3,\ldots,n \). The house with number \( x \) had the property that the sum of the house numbers to the left of it equaled the sum of the house numbers to the right of it. The problem specified that \( 50 < n < 500 \).

Ramanujan quickly dictated a continued fraction for Mahalanobis to write down. The numerators and denominators of the convergents to that continued fraction gave all solutions \( (n,x) \) to the problem (not just the particular one where \( 50 < n < 500 \)). Mahalanobis was astonished, and asked Ramanujan how he had found the solution.

Ramanujan responded: "...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."

This is not the most illuminating answer! If we cannot duplicate the genius of Ramanujan, let us at least find the solution to the original problem. What is \( x \)?

Bonus: Which continued fraction did Ramanujan give Mahalanobis?

This anecdote and problem is taken from The Man Who Knew Infinity, a biography of Ramanujan by Robert Kanigel.

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