Cleverbot and other talking computers
One of the areas of research in artificial intelligence (AI) is to make computers talk like human beings. A theoretically perfect machine should be able to answer questions that you would mistake it for a human. Several such applications are available online. Cleverbot, Incogno, JabberWacky and Existor are only several AI applications that you can converse with on the internet. Converse with each of them and see if which machine is more advanced.
Math Applications
A question on how these machines work is a good topic for discussion in the K-12 classroom. Randomization (choosing random questions/answers from a database), probability distribution (choosing questions/answers that are more frequently mentioned), and combinatorics (combinations and permutations of questions/answers) are several of the mathematics behind the principle of artificial intelligence. A good activity would be to group students and ask them on how do they think talking computers are developed, and then ask them to list the steps or processes on how would they would go about it if they are the ones are to construct these computers.
Caution!
Talking to Cleverbot and similar applications is just like talking to a community on the internet. Since the application selects randomly from previously entered phrases, there might be phrases that are not suitable for children.
has coordinates 
and point 
has coordinates 
, then 
. It seems that there is nothing special about these points, but let us look at their position in the Cartesian coordinates.
