All the characters in the movie are named after famous prisons around the world - Holloway, Wren, Leavenworth, San Quentin, etc. The characters match the prison types too - Alderson never meets the group, and isolation is the major form of punishment at Alderson Prison in West Virginia. Dr. Hollaway is a female doctor, and Holloway Prison is a women's prison in England, Kazan is autistic, and Kazan Prison is Russia is a "disorganized" prison, etc. See more...
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Deliberate "mistake": Leaven spends too much time thinking about if 645 and 372 are prime numbers to be believable. But she knows the factors of 649 and the result of 26 to the power of 3 almost immediately. Plus anyone with any maths knowledge whatsoever would know that any even number (or one divisible by 5) can't be prime, so 645 and 372 could be dismissed without even thinking about it.
Plot hole: Kazam is supposed to be a math genius, something that the plot hinges on, but he makes several mistakes when calculating the number of prime factors to find out whether a room is trapped or not. He says that 462 has three prime factors, when it has four, that 206 has four when it only has two and that 563 has two and 911 has three when both are actually prime numbers.
Other: Toward the beginning, when the characters wonder aloud what the rumbling noise is, someone suggests that it's the ventilation system, to which Rennes responds "No vents." However, in the passages between rooms there are panels with ventilation slits in them (which you can plainly see even as Rennes says the line), and it's pretty impossible that the characteristically level-headed and observant Rennes wouldn't have noticed that.
Continuity: Throughout the movie they are told to suck on the buttons from their shirts. In one scene, once Holloway climbs into the next room, Quentin is sucking on his button. He immediately speaks to Worth, muddling some words because of the button in his mouth. All of a sudden, the button disappears from his mouth and is in his hand again, and he then puts it back into his mouth.
Continuity: When Holloway is hanging outside the Cube from the rope of clothes, it would seem that it disappears at the moment she falls. We see her climbing up toward Quentin, and the rope is around her waist. Then they're holding hands face-to-face, and in the next shot when she slips, the rope stays still. Then, as Quentin goes back inside, it's disappeared entirely, and he doesn't take it back in with him.