Solution for Adventure #1

The key to the puzzle is thinking where the dragon will go next. 

1. First we start by checking box #3 (the one in the middle). 
2. If it's not there that means its in box 1 or 5 (since the oracle said the dragon would be in  odd numbered box)
3. So let's suppose it was in box 1. Then it must go to box 2 since there's no box to it's left. So check box 2.
4. If it's not in box 2 it must have been in box 4. So it could move to box 3 or 5. Then we check box 3.
5. If it's not box 3, it must've been in box 5, meaning now it can only move to box 4. So finally we check box 4, and the dragon will definitely be in there.

The dwarf ruler is pleased by your clever problem solving skills, and informs you that there's a problem at their neighboring town too.