First, give the problem some thought, and see how you do. If you are lost after 5 minutes or so, or don't know how to get started, then read on.
The difficulty with this problem is dealing with the fact that you have potentially 1,000,000,000 dimensions to keep track of, and that seems daunting. However, let's think of the structure of a simple solution:

You'd probably like to keep track of the current point with a vector of integers, one for each dimension. However, with 1,000,000,000 dimensions, that's too big. Instead, you can keep track of the current point with a map whose keys and vals are both integers. Any entry in the map will have its key be the dimension and the val be the value of the point in that dimension. Any dimension not in the map will have a value of zero. Thus, the origin can be represented with an empty map. It can also be represented by any number of entries whose vals are zero.

Start by putting such a map into the RouteIntersection class, and write the code that prints out the points on each iteration. When you print a point, just print the non-zero values in the form "[dimension,value]". For example, Example 1 will go through the following points:

""              -- I.e. point (0,0)
"[1,1]"         -- I.e. point (1,0)
"[1,1][2,1]"    -- I.e. point (1,1)
"[2,1]"         -- I.e. point (0,1)
Example 2 will go through:
""                -- I.e. point (0,0,0)
"[1,1]"           -- I.e. point (1,0,0)
"[1,1][2,1]"      -- I.e. point (1,1,0)
"[1,1][2,1][3,1]" -- I.e. point (1,1,1)
"[2,1][3,1]"      -- I.e. point (0,1,1)
"[3,1]"           -- I.e. point (0,0,1)
Now, instead of printing those points to standard output, use a stringstream to turn them into strings. Each string represents a unique point, so you can keep a set of strings to represent the previous points, and you can use find() to look up the current point as a string.

So, now your program should have the following structure:

It is ok to have points in the map whose values are zero -- simply ignore them. It is easier to program it that way. For example, in Example 1 above, when you reach point (0,1), your map will have two values in it: [0,0] and [1,1]. However, when you print the point (or turn it into a string), you will ignore the [0,0] value. You may be tempted to remove it from the map, but it is much easier to simply keep it there with a value of zero.

When I first did this one, in July, 2010, it took me 6 minutes for 230 points.

My second time, in September, 2010, took less time for 240 points!

When I did it the third time, for class prep in March, 2012, it took me 12 minutes, as I made a mistake generating the string and had to submit twice. Only 160 points or so...

My 2012 solution is in RouteIntersection.cpp.