The riddle
What Has Hands but Cannot Clap?
Which hands
On a clock face the hands are the pointers: a short one for the hour, a longer one for the minutes and, on many clocks, a thin one for the seconds. The clue "cannot clap" tells the listener to think of hands that are not attached to a person. A wristwatch with a dial has the same hands, so the riddle covers it too. A pair of hands that never clap sounds odd until the clock comes to mind.
When the hands meet
Two of those hands do come together, though not as often as most people guess. The hour and minute hands line up 11 times in 12 hours, not 12. After they cover each other at 12:00 they meet next at about 1:05, then at about 2:11, and so on, 65 minutes and 27 seconds apart. In a full day that makes 22 meetings, while the hour hand circles the dial twice and the minute hand 24 times.
A name that came from a bell
Merriam-Webster traces "clock" to a Middle Dutch word for a bell. The link makes sense, since many early mechanical clocks had no dial and announced the hour by striking a bell.
In a 1915 book
A close relative turns up in a 1915 collection by Edith B. Ordway, titled The Handbook of Conundrums: "What have hands but work not?" The answer given is "Clocks." Both versions use the same trick of a body word that belongs to an object, and in both the clue about what the hands cannot do is what points away from people.