Types of Numbers - Taxicab numbers

There was a mathematician named G. H. Hardy, who discovered and then later worked closely with a brilliant young mathematician named Srinivasa Ramanujan. One day in 1918 or 1919, Hardy went to visit Ramanujan in his private nursing home. Hardy said about Ramanujan:

"I remember once going to see him when he was lying ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me a rather dull one, and that I hoped it was not an unfavourable omen. 'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways'."

 This can be generalised to the following definition:

Ta(n) is the smallest number that can be expressed as the sum of 2 positive cubes in n distinct ways.

These numbers have been dubbed Taxicab numbers in honour of this anecdote. At the moment, only 6 taxicab numbers are known. The first few, with the cubes that make them up, are:

Ta(1) = 2 = 1^3 + 1^3

Ta(2) = 1,729 = 1^3 + 12^3
                      = 9^3 + 10^3

Ta(3) = 87,539,319 = 167^3 + 436^3
                                = 228^3 + 423^3
                                = 255^3 + 414^3

The remaining three known taxicab numbers are 6,963,472,309,248, 48,988,659,276,962,496 and 24,153,319,581,254,312,065,344.

I would love to know how they discover which number is the smallest one. Is is just a brute force algorithm that goes through every possible combination of cubes for each number? Or are there ways to narrow down the variables? At a guess, I would say it is mostly brute force, as there has been a lot of time between the discovery of each new number, the latest one only being discovered in 2003.

Although we only know the smallest number for six equal sums of two cubes, it has been proven by Hardy and Wright that the number of sums can be made arbitrarily large. In other words, the list of taxicab numbers goes on forever. There is not a biggest taxicab number.

Some people take taxicab numbers to be the list of numbers that are the sum of two cubes in two or more distinct ways. The first few are 1729, 4104, 13832, 20683, 32832, 39312, 40033, 46683, 64232, ...

I can see how you could find these numbers. You would have to limit the number you look up to, say to N. Then you take every cube number between 1 and N and add them to 1. Next add every cube number between 2^3 = 8 and N and add it to 8. Do the same up until N^3 + N^3, then sort all of the sums into order of the answers. Then you can go through the list and see which sums have the same answers. Obviously this would take very long to do by hand, but hey, that's what computers and algorithms are for!