What is the name of the musical instrument composed of a flat wooden sound box with 30 to 40 strings which are plucked with the fingertips.

This description perfectly fits the zither, a stringed instrument with ancient roots. The concert zither, which is the version most people recognize, typically has around five melody strings stretched over a fretted fingerboard, and another 25 to 35 open "accompaniment" strings. The player plucks the melody strings with a plectrum on their right thumb while using the other fingers to pluck the accompaniment strings. The instrument is laid flat on a table or across the player's lap. Its most famous and instantly recognizable performance is the haunting main theme for the 1949 film "The Third Man," composed and performed by Anton Karas.

While zither is the most precise answer, the dulcimer is a very close relative and is often considered an acceptable answer in a quiz setting. The word "zither" is actually a broad classification for any instrument with strings stretched across a sound box that has no neck or yoke. Under this definition, the dulcimer is technically a type of zither. The two most common forms are the hammered dulcimer, which is struck with small mallets, and the fretted Appalachian dulcimer, which is plucked or strummed. Because the Appalachian dulcimer is plucked and fits the general description, it's a common point of confusion, though it typically has far fewer strings than the concert zither.