Without Head nor Tail

Problem:

On a distant island, a valiant knight must face dragons with several heads and tails. With a sword, the knight can cut off either: one head, two heads, one tail or two tails.

But these dragons have magical powers: cut off one head, and he will instantly regrow another one; cut off one tail, and he will regrow back two! On the other hand, if you cut off two heads with a single sword strike, nothing grows back... but two tails cut off at once are replaced by a new head. Of course, a dragon is only completely dead when it has no head or tail left.

How do you kill a magical dragon with 5 heads and 7 tails?
Are there any immortal dragons? Which ones?

Solution

To kill a dragon with 5 heads (H) and 7 tails (T):
1. Cut off 2 x 2H and 3 x 2T (two at a time). You are left with 4H and 1T
2. Cut off 3 x 1T (one T at a time). You are left with 4H and 4T
3. Cut off 2 x 2T (two T at a time). You are left with 6H
4. Cut off 3 x 2H (two H at a time). You have defeated the dragon!

Immortal dragons are those with an odd number of heads and no tail. If you have tails you can always cut them off and make heads to get an even number of heads.