Mother said: ‘I’d offer you coffee, Werner, but ours is ersatz, and I know you have the real thing at home.’

‘Shall I steal some from our kitchen for you, Frau von Ulrich?’ he said. ‘I think you deserve it.’

Mother blushed slightly, and Carla realized, with a twinge of disapproval, that even at forty-eight Mother was susceptible to Werner’s charm.

Werner glanced at a gold wristwatch. ‘I have to go,’ he said. ‘Life is completely frantic at the Air Ministry these days.’

Frieda said: ‘Thank you for the lift.’

Carla said to Frieda: ‘Wait a minute – if you came in Werner’s car, where’s your bike?’

‘Outside. We strapped it to the back of the car.’

The two girls belonged to the Mercury Cycling Club and went everywhere by bike.

Werner said: ‘Best wishes for the interview, Carla. Bye, everyone.’

Carla swallowed the last of her bread. As she was about to leave, her father came down. He had not shaved or put on a tie. He had been quite plump, when Carla was a girl, but now he was thin. He kissed Carla affectionately.

Mother said: ‘We haven’t listened to the news!’ She turned on the radio that stood on the shelf.

While the set was warming up, Carla and Frieda left the house, so they did not hear the news.

The University Hospital was in Mitte, the central area of Berlin where the von Ulrichs lived, so Carla and Frieda had a short bicycle ride. Carla began to feel nervous. The fumes from car exhausts nauseated her, and she wished she had not eaten breakfast. They reached the hospital, a new building put up in the twenties, and found their way to the room of Professor Bayer, who had the job of recommending a student for the scholarship. A haughty secretary said they were early and told them to wait.

Carla wished she had worn a hat and gloves. That would have made her look older and more authoritative, like someone sick people would trust. The secretary might have been polite to a girl in a hat.

The wait was long, but Carla was sorry when it came to an end and the secretary said the professor was ready to see her.

Frieda whispered: ‘Good luck!’

Carla went in.

Bayer was a thin man in his forties with a small grey moustache. He sat behind a desk, wearing a tan linen jacket over the waistcoat of a grey business suit. On the wall was a photograph of him shaking hands with Hitler.

He did not greet Carla, but barked: ‘What is an imaginary number?’

She was taken aback by his abruptness, but at least it was an easy question. ‘The square root of a negative real number; for example, the square root of minus one,’ she said in a shaky voice. ‘It cannot be assigned a real numerical value but can, nevertheless, be used in calculations.’

He seemed a bit surprised. Perhaps he had expected to floor her completely. ‘Correct,’ he said after a momentary hesitation.