Green Thumb - S6-E10
Character mistake: When Kim is trying to convince Lisbon to help the FBI on their case, she says Jane stopped the ten-year reign of a mass murderer. Red John was a serial killer, which, by the FBI's definition, is a criminal behavior distinctly different from mass murder; he may have had a large number of victims, but he killed them either one or a couple at a time, with a cooling-off period in between, not all at once.
Character mistake: When first discussing the projectile Murdoch has written on his board that 2 (sin45) =1. This is obviously incorrect to anyone who's studied even basic trigonometry. (00:05:54)
Character mistake: When Walter is saying his father was a scientific pioneer at the university of Berlin, Peter reacts by saying: "Yeah, I know. Who came here in 1933, which is long before the Nazi's ever took power." In fact it is EXACTLY when the Nazi's took power: 30 January 1933. (00:19:00)
Character mistake: Upon finding the Spanish flu virus missing, Hardison freaks out about it killing 50m people 100 years ago, saying it would be even more deadly today due to the higher population. There's no reason to think so - the death toll in 1918 was massively exacerbated by the aftermath of WWI. Not to mention today we have vaccines, plus most deaths then were actually caused by bacterial pneumonia, which today is easily treated with antibiotics.