Who composed a piano piece that needs both hands and the nose to strike the right notes: Beethoven, Bach, Mozart or Wagner?

The story goes that Mozart did.

"At an evening party, Mozart bet a case of champagne that Haydn could not play at sight a piece he had composed that afternoon. Haydn accepted the bet and proceeded to play it on harpsichord only to stop short after first few bars. It was impossible to continue because the composition required him to simultaneously strike notes at two ends of the keyboard and a note in the very center. Haydn exclaimed, 'Nobody can play this with only two hands.'

'I can,' Mozart said, and took his place at the keyboard. When he reached that problematic portion of his piece, Mozart bent forward and struck the central note with his nose.

Haydn conceded saying: 'With a nose like yours, it becomes easier.'"

--E. Van de Velde, Anecdotes Musicales; N. Slonimsky, Slonimsky's Book"

http://www.kalvos.org/creshess2.html

Seems to me the old John Schaum piano course for children had a piece based on this story.

Mozart. He had a competition with Haydn to see who could compose the most difficult piece. He did by playing a piece with so many notes he had to use his nose to play.

http://query.nytimes.com/gst/abstract.html?res=990CEFD8153FE533A25757C0A9669D94649FD7CF