1963 AHSME Problem 36

A person starting with \$64 and making 6 bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is:

\textbf{(A)}\text{ a loss of } 27 \qquad \textbf{(B)}\text{ a gain of } 27 \qquad \textbf{(C)}\text{ a loss of } 37 \qquad \\ \textbf{(D)}\text{ neither a gain nor a loss}\qquad \\ \textbf{(E)}\text{ a gain or a loss depending upon the order in which the wins and losses occur}$

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.