scratch one's head
Express puzzlement or perplexity, think hard, as in They scratched their heads over this vexing question, but no one knew the answer. Although literally scratching one's head may simply betoken the fact that it itches, it also may indicate mental mystification or bewilderment. The term's figurative use dates from the first half of the 1900s.
Words nearby scratch one's head
scratch coat,
scratch file,
scratch hardness,
scratch hit,
scratch line,
scratch one's head,
scratch pad,
scratch sheet,
scratch someone's back,
scratch test,
scratch the surface