rack one's brain, to

Fig. to try very hard to think of something. I racked my brains all afternoon, but couldn't remember where I put the book. Don't waste any more time racking your brain. Go borrow the book from the library.

Also, cudgel one's brains. Strain to remember or find a solution, as in I've been racking my brain trying to recall where we put the key, or He's been cudgeling his brains all day over this problem. The first term, first recorded in 1583 as rack one's wit, alludes to the rack that is an instrument of torture, on which the victim's body was stretched until the joints were broken. The variant, from the same period, uses cudgel in the sense of "beat with a cudgel" (a short thick stick). Shakespeare used it in Hamlet (5:1): "Cudgel thy brains no more about it, for your dull ass will not bend his pace with beating." Also see beat one's brains out.

To think long and hard: I racked my brain for hours trying to recall her name.

To strain to remember something or discover a solution. The rack here is the medieval instrument of torture on which the victim’s body was stretched until it broke. The idea is old; “we break our brains for naught” comes from 1530. The word “rack” came into use about 150 years later.