Continuity mistake: Danno boards the helicopter without his briefcase, but arrives to conduct the interview carrying the case and tape recorder. He takes the copter back home again with nothing in hand, getting into his car empty handed as well. But he enters McGarrett's office carrying the briefcase again.
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Character mistake: After distracting the guards, Danno tosses a grappling hook onto a ledge and climbs the side of the building. He then goes about his business, carelessly leaving the hook and rope dangling from the ledge for all to see. For a seasoned 5-0 team member, that's pretty sloppy covert operations.
Plot hole: As a security test, McGarrett secretly has several dummy bombs planted in the court building, then later shows the governor where they are. Each time he pulls one of the devices from its hiding place, citizens in the courthouse stroll casually by without so much as glancing at the loudly-ticking "bomb." No one knew about the test, so McGarrett's bombs should be attracting some attention, if not causing outright panic.
Character mistake: When Ben talks to the old guy about Spider Brown, Ben says, "Somebody that got rich the past week." The time between the shooting and trial was probably months, not just one week. Spider would have been flashing money right after the hit, and then disappeared.
Revealing mistake: A loud, fierce gun battle rages between dozens of 5-0 sharpshooters, Army MPs and the desperadoes inside the payroll office. Amidst all the high-powered rifle fire, a woman in a blue dress strolls casually by in the background, well within bullet range, and never even turns her head to see what all the ruckus is about. Gee, you'd think colossal shooting wars broke out on her block every day.
Character mistake: McGarrett tells Pereno that the sabotaged race car went over a cliff. But the car crashed into a wall, with no cliff in sight. It's very unlike Hawaii's top cop to get the facts wrong - especially when he was an eye witness to the event.