cudgel (one's) brains

To try very hard to comprehend, solve, think of, or remember something. I was up all night cudgeling my brains for a way to pay off all my debts. She cudgeled her brains to remember the man's name. Well, you better cudgel your brains because we're gonna be stuck in this parking garage all night if you don't remember where you parked!

cudgel your brain (or brains)

think hard about a problem.

This expression was used by Shakespeare in Hamlet: ‘Cudgel thy brains no more about it’.

cudgel one's brains, to

To think hard; to make a vigorous attempt to solve or answer some question, or to remember something. The verb “to cudgel” means to beat with a cudgel (a short thick stick). Possibly the allusion here is to thrashing a schoolboy for failing to answer promptly or correctly. The word “cudgel” is hardly ever heard anymore except in this context, which dates from before 1600. Shakespeare had a clown say to another who was puzzling over a riddle, “Cudgel thy brains no more about it, for your dull ass will not mend his pace with beating” (Hamlet, 5.1). See also beat one's brains; rack one's brain.