CSSE 220: Hardy’s Taxi Programming Problem
Carefully follow the turnin instructions at the bottom of this document. This is an individual assignment. As usual, you
may talk to others, but your code should be your own. Be sure to add comments that explain your approach. Javadoc
comments are not necessary.
This program will not take a lot of code, (probably less than 200 lines), but it will take a lot of thought, especially thought
about how to do it efficiently. In what order should you try various sums of cubes? What information should you store?
How do you avoid duplicates? How do you make sure you don't miss any solutions?
The following was adapted from http://cs.bilgi.edu.tr/pages/curiosity_corner/challenges/ramanujans_number.html
Write a Java application (Hardy.java) that reads a positive integer N from standard input (your program should not print a
prompt, or anything else except the solution in the form prescribed below) and then finds and prints (to standard output)
the Nth smallest positive integer that can be expressed as the sum of two cubes (of positive integers) in two or more
different ways, along with the evidence that it can be expressed that way.
Ramanujan and Hardy's taxi
G H Hardy, the famous British mathematician, brought Ramanujan, a poor man from India who was a natural
mathematical genius, to work with him at Cambridge University. They had a very important and creative
mathematical partnership. The British climate was bad for Ramanujan. He got tuberculosis. As he lay dying in
hospital, Hardy went to visit him. As he entered the room he said, "The number of my taxi was 1729. It seemed to
me rather a dull number." To which Ramanujan replied, "No, Hardy! No, Hardy! It is a very interesting number. It is
the smallest number expressible as the sum of two cubes in two different ways."
Ramanujan died at the age of 33.
Is the number in the story correct? Is 1729 the smallest number expressible as the sum of two cubes in two
different ways? It is fairly easy to show it is expressible as the sum of two cubes in two different ways, but is it the
smallest such number?
Obviously we can think of some other questions:
If 1729 is the smallest such number, what is the second smallest? the nth smallest? Can we write a program to find
this?
What do we mean by "different ways"?
What about a more general statement of the problem:
What is the kth smallest number that can be expressed as the sum of the nth powers of m numbers in r different
ways? Can we find a formula for it? (I doubt it) Can we write a program to do it? Are the results interesting?
The programming isn't as easy as it might at first seem. Some of these numbers are going to be rather big!
Chris Stephenson
Read the story of Ramanujan and Hardy in Curiosity Corner
