Find the missing integer from 0 to n

Note - there are a total of bits, so we are not allowed to read the entire input!  

Also note, the problem is asking us to minimize the number of bits we read. We can spend as much time as we want doing other things provided we don't look at extra bits.

How can we find the last bit of the missing integer?

Ask all the n integers what their last bit is and see whether 0 or 1 is the bit which occurs less often than it is supposed to. That is the last bit of the missing integer!

How can we determine the second-to-last bit?

Ask the numbers which ended with the correct last bit! By analyzing the bit patterns of the numbers from 0 to n which end with this bit.  

By recurring on the remaining candidate numbers, we get the answer in T(n) = T(n/2) + n =O(n), by the Master Theorem.

  http://www.cs.sunysb.edu/~algorith/lectures-good/node5.html