When Arlington finds the letter telling him to pay £2,000,000 for the incriminating Black Book, the words 'Leave the trolley at desk nine of the reading room...' are clearly visible. However, Barnaby and other characters all read 'desk three' when referring to the letter.
DCI Barnaby refers to "Sir Robert Peel - the founding father of the British police force" - but he only established the Metropolitan Police in 1829, nearly three decades after the City of Glasgow Police was established by Act of Parliament in 1800.
Felicity Lawd was tortured when her murderer placed her hand on an induction cook top to burn her hand. This type of cook top couldn't do that.
When Alan Best fires his shotgun at the fleeing thief, there is no recoil.
How did Arlington know to use a trolley of that exact kind to bring the money to the library? Or, alternatively, how did the murderer know exactly what kind of trolley Arlington would use? If they weren't identical, the switch would not be possible.
After Barnaby discovers that Neville Blackshaw does not have the two million pounds, DS Jones refers to Patricia Blackshaw as "the old woman at desk three." But the black book was left on desk three, and Patricia was clearly seated at desk four.