Problem on finite state machines in the feynman textbook
The numbers are encoded LSB first. In your example, $(00)(01)(10)(11)(10)(00)$ corresponds to the addition $011100 + 001010$, and the answer should be $011001(0)$, corresponding to the binary answer $(0)100110$. The bit in parentheses is the carry bit, which I don't think Feynman intended you to output.

More generally, Feynman wasn't very accurate, so you should think of a reasonable concrete interpretation of the problem which will result in an automaton which has a small number of states; I can think of several, some of which requiring less than four states. (It depends whether $(00)$ is a single digit or a pair of consecutive digits.) I don't think it's very important to guess what Feynman intended exactly.