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? combinatorics proposedcombinatorics