Instructions
Pick up the music box from Guybrush's vault on Lucre Island.
Go to the prosthesis shop nearby. Use the music box on shopkeeper Deadeye Dave to distract him. While "MUSIC PLAYING" flashes on-screen, quickly walk to the barrel by the door and look at it. Take the wooden prosthetic hand inside.
Travel to the Bait Shop. Once inside, use the wooden prosthetic hand on the termite circus in the bottom-left corner of the store. This will leave the prosthetic hand full of talented termites.
Pick up the bottle of cologne from the "Scents and Sensibilities" perfume stand on Lucre Island's main street.
Go to Ozzie's mansion and stateroom, and use the bottle of cologne on one of his many fine taxidermy specimens. Any one will do. Ozzie will get so furious at what you've done that he'll break his walking stick, which he now must get repaired.
Enter the House of Sticks, and walk to the rack next to the "Choose the Right Walking Stick" sign. Here, you'll find Ozzie's new cane. Use the now termite-infested wooden prosthetic hand with his new cane. Ozzie will come to collect his walking stick, and will leave a trail of sawdust behind him.
Go back to Ozzie's mansion and talk to him. Accuse him of setting you up with Pegnose Pete, and respond to his questions any way you want. Ozzie will then get nervous and go to check on the loot. Follow his trail of sawdust into the Lucre Island forest, to find the hiding place for the stolen booty. You've now outsmarted Ozzie without using your sword.