The riddle
A Man Rode Into Town on Friday, Stayed Three Days and Rode Out on Friday. How Is That Possible?
Counting the days
Start on Friday and add three days: Saturday, Sunday, Monday. The man would ride out on Monday, not Friday. To leave on a Friday he would have had to stay 7 days, or any multiple of 7. Even if the arrival day counts as the first of the three days, the stay ends on Sunday. The numbers cannot be made to work, and that is the signal that the word Friday is not behaving like a day.
The name, not the day
Friday works as a name as well as a day. In both places it names the horse: the man rode into town on Friday and rode out on Friday. The calendar day is never given at all. Versions of the riddle change the number of days or the day of departure, and the solution stays the same.
A name with a history
A famous bearer of the name is a character in Daniel Defoe's 1719 novel Robinson Crusoe, whose hero names a man Friday because they first meet on that day, according to Wikipedia. The novel also has a sequel, The Farther Adventures of Robinson Crusoe. The expression "Man Friday" for a loyal, capable assistant comes from that character.
Making your own
The trick works with any day or month that can also be a name: a dog called Monday, a cat called April, a boat called Sunday. A group can write a new version in one minute by choosing the name first and building the impossible schedule around it.