Can you be right, for good reasons, and still not know something?

Yes, and a three-page paper from 1963 made that famous. For a long time, many philosophers were happy with a neat definition: you know something if you believe it, it's true, and you have good reasons for it. Knowledge equals justified true belief.

Then the American philosopher Edmund Gettier offered counterexamples. Here's the spirit of them, in a version Bertrand Russell had already described. Alice glances at a clock that says two o'clock and concludes it's two. It is two. But unknown to her, the clock stopped exactly twelve hours ago. Alice believes it, it's true, and trusting a clock is a good reason. Yet she was only right by luck. Does she really know the time? Most people say no.

The backstory is part of the charm. According to philosopher Duncan Pritchard, Gettier wrote the paper to get tenure, never published anything else on the subject, and even declined to attend a conference celebrating its 50th anniversary. Meanwhile, his three pages "fundamentally altered the character of contemporary epistemology", as two philosophers later put it.

The idea wasn't entirely new: around 770 AD, the Indian philosopher Dharmottara described a traveller who sees a mirage of water and, by luck, finds real water hidden under a rock at that very spot. Since Gettier, philosophers have tried to add a fourth ingredient to knowledge that rules out lucky hits, and none has won universal agreement.