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?

Sorry Stephen S.

Seven. Chop off two tails, chop off two heads. Now two heads and one tail remaining. Chop of remaining two heads. Chop off tail. Now two tails remaining. Chop off two tails. Now one head remaining. Chop off head. One more is added, Two more are added. You are now stuck with a one headed furious dragon, good luck. Chop off both heads.