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Revealing mistakes
While West is chasing a suspect, they crash into a wooden fence. The fence boards break into two pieces along perfectly straight edges, showing that they had been precut.
When Jim and Artie are trapped in the freezer they blow the door off by opening bullets and pouring the powder under the door then set it off with a bullet. Unless gunpowder is contained, as in a bomb, it will not explode but merely burn. Shooting into gunpowder will not cause it to burn/explode unless the muzzle blast ignites it. A bullet fired into it will only spread it around.
During a rehearsal for a tap-dance routine, a girl plays "Yankee Doodle" on a bugle which, having no valves like a trumpet or cornet, is incapable of playing it.
In the prologue, when the #1 ball 'explodes' the smoke obviously comes from below and in front of the pool table, not the #1 ball - which is visible through the smoke.
While West is chasing a suspect, they crash into a wooden fence. The fence boards break into two pieces along perfectly straight edges, showing that they had been precut.
At the end of the program when Vashti hits the cook with a hand full of cake, the actor playing the cook forcefully closes his eyes just before being hit, clearly demonstrating that the actor knew what was about to happen to him.
When James is knocked into the freezer there is no frost visible even in breath. He looks out a rectangular opening in the door watching Vashti and M'Guigan in the kitchen but from the outside the door has wooden paneling and no opening.
Judge M'Guigan quotes from Emma Lazarus' poem "The New Colossus" and shows an engraving of the Statue of Liberty atop its pedestal on Bledsoe's Island. At the time of the Grant administration (1869-1877), only the arm with the torch had been exhibited publicly, at the Philadelphia Centennial Exposition of 1876. The pedestal for the statue was not completed until 22 April 1886 and the statue itself was unveiled on 28 October 1886. The Lazarus poem was written in 1883 for a fund-raising event for building the pedestal. The poem was forgotten until friends of Lazarus finally had a plaque bearing the poem installed at the statue in 1903. On that plaque, the original line "Keep, ancient lands, your storied pomp!" is misquoted as "Keep ancient lands, your storied pomp!"
In the opening scenes, the judges (and others) speak of "this game of pool" as if it were a recently invented game. The games currently known collectively as pool (including eight-ball, nine-ball, straight pool, and one-pocket), were known from the early 1800s, but the term "pool" did not come into use until the 1890s. Thus, the games were already well-known (making the unfamiliarity feigned) but the terminology lay in the future.
Judge M'Guigan lists Lily Langtry among the "immortals of the theater," even though Langtry wouldn't debut on stage until 1881.
Jim is told that Artie has been burned. He finds "Artie" on the floor of the kitchen with bandages covering his entire head yet he is still wearing his chef's hat. Jim does not find this odd or suspicious.
In "The Night of the Flying Pie Plate" James says he knows [Arte's] groan anywhere. Yet here when Vashti brings him to the kitchen and sees a chef lying on the floor groaning he responds and reacts to Arte when it is in fact Judge M'Guigan in disguise.