Obviously, since nobody can see the back person’s hat, its colour
cannot be determined. The nine members who answer correctly must
therefore be all except the one at the back.

The back person says “green” if he can see an odd number of
green hats in front of him, otherwise he says “purple”. The
second person can then use this information to determine the colour of her
hat. (For example, if the back person answered “green”, then
there is an odd number of green hats among the front nine. If the second
person sees an even number of green hats, she deduces that her own hat is
green.) Subsequent members therefore know how many green hats are behind
them (other than the back person) as well as in front, and can deduce their
colours by the same logic. (Put simply, it’s green if the number
of green hats in front plus the number of times “green” has been
said already is an odd number; otherwise it is purple.)

Another possible strategy is to use a code, such as the tone of voice or
the length of pause before answering, for each person to indicate the
hat colour of the one in front.