IT-CESENATICO 2010 Problem 5

In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves: (i) cancel each two consecutive ones; (ii) delete three consecutive zeros; (iii) if the substring within the string is $01$ , one may replace this by substring $100$ . The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less. (For example, starting from $0101$ , one can win using move (iii) first in the last two digits, resulting in $01100$ , then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.) Among all the $1024$ possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?