I first heard this from my supervisor but always felt confused. Sinologist Dr. Josheph Needman called it a sixty-four thousand question that why modern science and technology revolution did not first happen in then much advanced China but in Europe. For more, please read 李约瑟之谜. Here is the answer to this “sixty-four dollar question” from everything2.com.
A sixty-four dollar question (also written $64 question) is the critical question about a problem or a crucial issue, something difficult if not impossible to answer.
This comes from a popular 1940s U.S. radio quiz show, Take It Or Leave It, which offered $64 as the largest prize. The first question had a prize of $1 and the prize total doubled with each successive question: $2, 4, 8, 16, 32, culminating in the $64 question. (Like Who Wants To Be A Millionaire if they actually stuck to the same ratio between questions.) There was later a CBS television quiz show in the U.S., The $64,000 Question, based on the radio show. There the first question was the $64 dollar one, and the contestants then worked their way up to the top prize, $64,000. It and its first spin-off, The 64,000 Challenge, were hosted by Hal March. A later, syndicated spinoff was “The $128,000 Question.”
Because of this change between the radio and the TV versions, as well as inflation, the expression is also heard as “sixty-four thousand dollar question” and even “sixty-four million dollar question,” (as well as the numeric representations of those amounts) and has led to such take-offs as “sixty-four cent question,” “sixty-four franc question” and “sixty-four bit question” (the latter referring specifically to using 64-bit technology).
A vocal version of the TV show’s instrumental theme music, sung by Hal March, was released as a 45 rpm single called “Love is the Sixty-Four Thousand Dollar Question” in 1956. Its lyrics were by Fred Ebb, and the music by Norman F. Leyden. A songwriter called Dave Keats has written a different song called “Sixty-Four Dollar Question.”