Gödel, Escher, Bach An Eternal Golden Braid by Douglas R. Hofstadter

mediately jumped to fourth powers. After all, there are several other

reasonably natural generalizations of the equation

along different dimensions. For instance, there is the question about repre-

senting a number in three distinct ways as a sum of two cubes:

Or, one can use three different cubes:

Or one can even make a Grand Generalization in all dimensions at once:

There is a sense, however, in which Hardy's generalization is "the most

mathematician-like". Could this sense of mathematical esthetics ever be

programmed?

The other anecdote is taken from a biography of Ramanujan by his

countryman S. R. Ranganathan, where it is called "Ramanujan's Flash". It

is related by a Indian friend of Ramanujan's from his Cambridge days, Dr.

P. C. Mahalanobis.

On another occasion, I went to his room to have lunch with him. The First

World War had started some time earlier. I had in my hand a copy of the

monthly "Strand Magazine" which at that time used to publish a number of

puzzles to be solved by readers. Ramanujan was stirring something in a pan

over the fire for our lunch. I was sitting near the table, turning over the pages

of the Magazine. I got interested in a problem involving a relation between

two numbers. I have forgotten the details; but I remember the type of the

problem. Two British officers had been billeted in Paris in two different

houses in a long street; the door numbers of these houses were related in a

special way; the problem was to find out the two numbers. It was not at all

difficult. I got the solution in a few minutes by trial and error.

MAHALANOBIS (in a joking way): Now here is a problem for you.

RAMANUJAN: What problem, tell me. (He went on stirring the pan.)

I read out the question from the "Strand Magazine".

RAMANUJAN: Please take down the solution. (He dictated a continued

fraction.)

The first term was the solution which I had obtained. Each successive term

represented successive solutions for the same type of relation between two

numbers, as the number of houses in the street would increase indefinitely. I

was amazed.

MAHALANOBIS: Did you get the solution in a flash?

RAMANUJAN: Immediately I heard the problem, it was clear that the

solution was obviously a continued fraction; I then thought, "Which con-

tinued fraction?" and the answer came to my mind. It was just as simple as

this.4

Hardy, as Ramanujan's closest co-worker, was often asked after

Church, Turing, Tarski, and Others 565