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A Gallic shrug is like a Frenchman flipping you the bird with his shoulders. It means 'F*** you, it's not my mess.'
Your mom forgot your birthday. You ask her why. She gives you a Gallic shrug.
You're late for work. Your boss gives you a Gallic shrug and says, 'Not my problem.'
You ask your friend why he broke up with his girlfriend. He gives you a Gallic shrug and walks away.