The way Jupiter bluffs his way in past both the gate guard and the secretary to see Hitchcock at World Studios.
Aside from the fact this is widely considered the quintessential Three Investigators book, the blueprint for every well-written treasure-hunting plot to follow, and the favorite of most fans, it is also one of the more complex, tightly-written, suspenseful, and atmospheric books, particularly the entire journey to the Merita Valley graveyard at the climax. It's no wonder that it's this case which puts the boys on the map as investigators, even more so than their first. (Helped along, of course, by the great value of the lost masterpiece and all the press they get in the papers; since Hitchcock ends up abandoning his search for a real haunted house, their work for him in Terror Castle really doesn't get revealed to the public, and it seems their names were kept out of Stephen Terrill's "return from the dead".)
Bob faking going under hypnosis to fool Mr. Won, then leaving notes out of the trunk of the car so Jupiter and the police can track him, Pete, and Chang down.
Hans and Bob breaking down the gate with one of the salvage yard trucks to get to the pier in time to save Jupe and Pete from Rawley's gang.
Also, Jupe's plan for catching the midgets at the end: after finding the belt and making them think he has changed his mind about Rawley's offer and wants to become a criminal, he gets them to come to Headquarters to buy the belt from him and has Bob and Pete demonstrate all the secret entrances except one so that they can escape when the midgets attack—where the cops are waiting to catch them.
How they get Gus his inheritance, even when Three-Dots threatens them: Jupe found it so he is safe, he gives it to Gus so he is safe, but if Three-Dots takes it from him, the curse will be invoked. So to prevent this, Three-Dots has to buy it.
Ringing the bell of Prince Paul to get the people of Varania to rise up against Duke Stefan.
The discovery of what the clock was meant to do, and where the paintings were hidden.
The Potter's Dynamic Entry, complete with shotgun, to save the boys, Mrs. Dobson, and Tom from Demetrieff and Kaluk.
Uncle Titus vs. Olsen. The latter fellow comes to the salvage yard to try and procure the cages from the junkyard next to Jungle Land; Jupiter, suspicious over how badly the man wants them (and also remembering his uncle had wanted the cages to sell to a circus) quotes an exorbitantly high price. Furious, Olsen departs, only to return when Uncle Titus does to lambaste him about the "con job" Jupe tried on him. After stating he wanted to sell the cages to the circus as Jupiter had pointed out, Titus listens to the price Olsen says Jupiter quoted him...and then notes his nephew got it wrong before quoting an even higher price.
[Jupiter] felt like hugging his uncle.
George being used to capture Bo Jenkins.
Jim Hall whistled softly as the animal handler grabbed the black bag.
Heavy, padding steps sounded across the room. An ominous rumbling made Jenkins turn and stare. His jaw sagged and he paled.
A large, yellow-eyed lion stood there, head down and long tail twitching restlessly. The growling continued.
"Forget it, Jenkins," Jim Hall said calmly. "You're not going anyplace. One more step and George will have you for dinner." Hall turned to the lion. "Won't you, George?"
The lion slowly opened its cavernous mouth and moved forward.
There was a clattering sound as the gun fell from Bo Jenkins's fingers.
Pete using Queenie to take out Shaitan.
As annoying as the boys found her at first, it can't be denied that Allie is an amazing character, especially for a girl in a boys' adventure series of the time. She's brave, bold, and brash; she's smart and insightful, and her snark is often very funny; she's willing to do whatever it takes to help her aunt; she manages to figure out where the boys' secret entrances and hideout are; and she's determined to help them save the day even if it puts her in danger to do it. Jupiter by the end is forced to admit she is a valuable ally and worth befriending; even Pete has some grudging praise for her (probably helped along by her having praised him for his tackle of the bomb-carrying bad guy), and Worthington has nothing but respect and admiration for her. It's a shame she became much more of The Scrappy in Death Trap Mine as she was a genuinely good role model in this book.
Even in Death Trap Mine, Allie makes some good deductions, and helps solve the case with her local knowledge. She gets another Awesome moment near the end of that book when she proves more knowledgeable about desert survival than either Pete or the henchmen, calls the henchmenís bluff about shooting them, and keeps Pete and herself alive long enough to be rescued.
Jupiter's use of his Photographic Memory to track down the kidnapped Jeff: having heard both the chimes of an ice cream truck going by and the crossing-arms signal at the railroad tracks where a train was going by, he is able to find out from the ice cream company dispatcher where his different routes went, learn which one would have been at a railroad crossing at the time the kidnapper had Jeff call, and so find the house where he was being held.
Diego Alvaro, immediately after hearing Skinny Norris use an ethnic slur against him and suggesting "why don't you go back where you came from", demands an apology. When Skinny refuses to give it, he hauls off and slaps him. They get into a fight, where no matter how times he gets knocked down, Diego gets right back up again. They end up falling into traffic, where Diego saves Skinny's life. And after this, when Skinny thanks him and finally, grudgingly apologizes, Diego immediately walks away as if nothing had happened, because his pride and honor were satisfied. That is how you do an Establishing Character Moment.