From Longman Dictionary of Contemporary Englishscratch your headscratch your headinformalTHINK ABOUT to think carefully about a difficult question or problem This crisis has politicians scratching their heads and wondering what to do. → scratch
Examples from the Corpus
scratch your head• The last question really left us scratching our heads.• There's nothing more guiltily satisfying than watching the experts scratch their heads.• Where possible he would seize scraps of parchment and draw lines, muttering to himself and scratching his head.• He scratched his head, a sure sign of thought.• Budget directors are scratching their heads about how to deal with the shortfall.• One, and he scratched his head about this, appeared to be a sparrow.• When I scratched his head, he would not hold still as long as before.• You scratch your heads in selection, and can't find a proper balance with less than 12.• They studied the map for a while, scratched their heads, turned it upside down and studied it some more. 