I'm a big fan of concrete examples so here's one: Let's say you have two points on a line labeled 0 and 1. Let's also suppose there's a special property you're

I'm a big fan of concrete examples so here's one: Let's say you have two points on a line labeled 0 and 1. Let's also suppose there's a special property you're interested in at one of these locations, it minimizes some function, there's a pot of gold there, it's infested with lions, doesn't matter. Since there are only two points it's easy to just check both points and empirically find which one you want. We can extend this idea easily into 2 dimensions, now we have two axes each consisting of two points along perpendicular lines. Enumerated the points are [0,0], [0,1], [1,0], [1,1] and as you can see there are four of them. It should also be pretty easy to see that if we extend this further into 3 dimensions that you once again double the number of points in your space to 8. You can see this by starting to enumerate them in the same way: [0,0,0], [0,0,1], [0,1,0], [0,1,1], [1,0,0]...[1,1,1] From this it should follow that if you extend the problem into n-dimensions you end up with 2^n different points in your search space. If n is a small number, say 30 it's still possible to do an exhaustive search with a computer since 2^30 ~ 10^9 but soon you run out of resources regardless of how much money you're throwing at the problem. This is the heart of the curse of dimensionality, starting with stupidly easy problems in one dimension quickly gets you in trouble when you want to extend it. This is not to say that we can NEVER deal with high-dimensional problems, two wonderful areas where we are VERY good at solving problems are linear programming and quadratic programming, in the former problems with hundreds of thousands of dimensions are feasible and thousands of dimensions in the case of the latter.