CWMO 2013 Problem 4

There are $n$ coins in a row, $n\geq 2$ . If one of the coins is head, select an odd number of consecutive coins (or even 1 coin) with the one in head on the leftmost, and then flip all the selected coins upside down simultaneously. This is a $move$ . No move is allowed if all $n$ coins are tails. Suppose $m-1$ coins are heads at the initial stage, determine if there is a way to carry out $\lfloor\frac {2^m}{3}\rfloor$ moves