rack one's brain(to think very hard in order to try to remember something, solve a problem, etc. (Merriam-Webster): I decided to be brave and opened the screen door just enough to stick my head outside. I held my son up to look at it also, still indoors. After it looked at me and yet showed no fear I had more time to study it and I was racking my brain to identify what it was that I was seeing. (mysteriousuniverse.org)ART Vancouver)
wild goose chase(The man led the police on a wild goose chase when he ran away from them downtown – Полиция тщетно пыталась разыскать этого человека, когда он убежал от них в центрTaras)