A Zen koan primarily associated with Hakuin Ekaku zenji, the Japanese Tokugawa era revitalizer of Rinzai Zen. The phrase "the sound of one hand" does not involve clapping or anything of that nature. In its correct form, it is simply, "What is the sound of one hand?" This means, what does this hand, this eye, this foot, this breath, tell you about who and what you are? The phrase did not originate with Hakuin but was popularized by him and used as one of the miscellaneous koan practised before the Gateless Gate, the Blue Cliff Records series and so on.

An early (Song dynasty) occurence of the phrase is in Case 18 of the Blue Cliff Records, as follows:

Case 18: Huizhong's Seamless Stupa

The Koan:

Emperor Suzong asked National Teacher Huizhong, "After you die, what shall I do to honour you?"

The National Teacher answered, "Build this old monk a seamless stupa."

The emperor said, "Master, please tell me what style to build it in?"

The National Teacher was silent for a long time and then he said, "Do you understand?"

The emperor said, "I don't understand."

The National Teacher said, "I have a disciple who has received my Transmission, Danyuan. He is well versed in this matter. Go to him and ask him."

After the National Teacher's death, the emperor summoned Danyuan and asked him about it.

Danyuan said,
"South of Xiang and north of Dan,
(Xuedou says, "Soundless sound of one hand.")
in between, the gold of the nation.
(Xuedou says, "A mountain monk's staff.")
A ferryboat under the shadowless tree.
(Xuedou says, "Calm seas, clear rivers.")
In the jewelled palace, no one knows."
(Xuedou says, "That's it.")

translated by Yasuda Joshu Dainen roshi and Anzan Hoshin roshi; used with permission

I always looked at the answer like the is the cup half empty or half full question:

The pessimist says the sound of one hand clapping is a slap to another's face.

The optimist says it is the sound of one's hand clapping to another's hand in greeting and salutation.

The traditional answer to this koan is to:

face the questioner,
assume the position (upright, zazen posture),
and silently thrust out your hand.

I don't know though. Because I think, if I did that in front of a Zen master, I'd get bitch slapped.

Obviously, the only way to "answer" this, or any koan, is to fully embody the koan itself.

This is a classic Zen instruction.

This question was asked by Master Hakuin (1686-1769). Hakuin is credited with aiding in moving Zen into the urban life which was emerging at the time of his life. He writes;

What is the true meditation? It is to make everything: coughing, swallowing, waving the arms, motion, stillness, words, action, the evil and the good, prosperity and shame, gain and loss, right and wrong, into one single koan.

It is said that a young monk given this question to ponder responded by attempting to use his hand to strike a gong and play various musical instruments, all of which were declined as answers by the master. Finally, in a moment of satori, he exclaimed : " I have heard sound without sound!"

The master is conveying the point that sound and other physical phenomenon are just our mind's interpretation of outside stimuli or forces. The second hand is a metaphor for our senses and ego, the sound made is that which is produced when our mind and senses strike with these outside forces and translate them into something we as humans can perceive.

This also demonstrates that without such outside forces, our mind can create an appropriate reaction; the mental "sound" of one hand clapping.