Question 137
In Cuccagna, the following solitaire is played with a finite string of zeros and ones. At each move one may (i) delete two consecutive ones; (ii) delete three consecutive zeros; or (iii) replace a substring 01 with 100. The moves are performed one at a time. One wins by reducing the string to length at most two. For example, 0101 becomes 01100 by move (iii) on the last two digits; deleting the two ones by (i), then the three zeros by (ii), gives the empty string. Among the 1024 strings of length ten, how many cannot be reduced to a winning string?