pull something out of the hat
pull something out of the bag
pull something out of thin air

Their figure was not pulled out of a hat.

I just pulled that out of a hat, like a rabbit.

The hope in Washington is that he will pull a rabbit out of a hat.

I don't expect you to pull the killer out of a hat.

The first act involved pulling a rabbit out of a hat.

Was he able to pull something out of a hat for us?

Since when did receiving a good public education come down to having to take a gamble on getting one's name pulled out of a hat?

No one can really pull a rabbit out of a hat.

Once again, Riley had pulled a rabbit out of a hat.

Suddenly it emerged, like a rabbit pulled out of a hat.