He has a magic sword that can, in one stroke, chop off either one head, two heads, one tail, or two tails.

This dragon is of a type related to the hydra; if one head is chopped off, a new head grows. In place of one tail, two new tails grow; in place of two tails, one new head grows; if two heads are chopped off, nothing grows.

What is the smallest number of strokes required to chop off all the dragon's heads and tails, thus killing it?

Start with 3Heads (H) and 3Tails (T)

you chop off 2H which leaves you with 1H 3T

next you chop of 2T which leaves you with 2H 1T

next you chop off 1T which leaves you with 2H 2T

then you chop off 1T which leaves you with 2H 3T

then you chop off 2H which leaves 0H 3T

then you chop off 2T which leaves 1H 1T

next you chop off 1T which leaves 1H 2T

then you chop off the 2T which leaves 2H that you chop off and you finish using 9 chops