a. Alice thinks up an array of N binary numbers
b. Bob guesses an array of N binary numbers
c. Alice responds only with ONE number,
which is the number of binary numbers in the correct position
An alternative to this problem is that Bob is allowed to include ? or non binary numbers in his guess as he will know they can't possibly be right. Which may (or may not) resolve this game faster.
Another alternative, is that you have prior probability estimates for each binary digit. (An interesting specific case, as for example you can get distribution estimates with each guess, right?)
There are some papers on random oracles which is similar to this, but I haven't found anything specific or useful. Trying to prove an optimal alg for solving this problem. Ulam's problem looks close, but couldn't find anything close enough.
Here's sort of something, http://www.geometer.org/mathcircles/bitmaster.pdf
But no references and terminology / formalization is a little weak.