What is the fewest number of seismographic stations that must record the arrival time of p and s waves in order for the epicenter of an earthquake to be located?

3

Explanation:

The process is called triangulation and involves at least 3 stations.

The principle is that from each station, you get a distance location from that station of the event. Then from each station you make a circle with the distance measured at that station.

All three circles done form the monitoring station will meet at a point somewhere between those three stations, and you'll get your location.

That's the triangulation process.  You do need 3 points for a precise reference, if you have only 2, you are usually left with 2 possibilities.