For over half a century, in the middle of the night, or in the morning, mathematicians in Budapest or Berkeley, Prague or Sydney, would be summoned by a knock at their front door. There on their doorstep they would find one of the greatest mathematicians of the 20th century - Paul Erdos, a small suitcase in one hand and a bag full of papers in the other, announcing, My brain is open. Then with his host and other mathematicians assembled as needed, Erdos would begin another mathematical journey. Literally homeless, without even a bank account, Erdos would rely on his host to tend to his daily needs while he explored the realm of mathematics. Before long, fuelled by caffeine or Benzedrine, Erdos would exhaust his hosts. Brain still wide open, he would take off to mathematical conferences, or visit another colleague, logging hundreds of miles on his journeys. This text is an exploration of the world of mathematics in which Erdos moved, an exciting world vital to the technology of the 20th century but largely unknown to many.