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.