If not, what other idioms could be used to express that one is thinking?

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    I've not heard it before, but it sounds a very good metaphor to me. "I will have to put my thinking cap on" would be one that I might use. – WS2 Feb 5 '17 at 11:54
  • There's also see someone's wheels turning though IME it is usual to use this when refering to someone else rather than oneself. – Martin Smith Feb 5 '17 at 12:09
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    When gears turn, they are working properly. When gears grind, their relative speeds are incorrect and teeth are being broken off the gears, causing permanent damage that prevent them from working properly in the future. – jejorda2 Feb 5 '17 at 12:12

I'm not sure if this will be widely understood. It's likely to be confused with to grind someone's gears, which means

To greatly or specifically irritate or annoy someone

An alternative is to rack one's brain(s):

to think very hard, usually in order to remember something or to find a solution to a problem

(sources: The Free Dictionary)

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