Two Christmases ago, William said, 'we were sitting down watching Ali G on TV. We were laughing when she came in. She couldn't understand what was going on, so we explained to her.

'She saw Ali G click his fingers and say "Respec", and Harry and I showed her what to do. After three goes she had it. Later that day, we were all in the dining room, having Christmas lunch, when she tried it out.'

His brother Harry, who is currently on holiday from Eton, said: 'It was the end of the meal, and she stood up and said, "Darling, lunch was marvellous - respec", and clicked her fingers.'

Apparently the whole royal family, including the Queen, burst out laughing.

The princes paid lavish tribute to the Queen Mother's sense of humour - about which Prince Charles had already spoken fondly earlier last week.

'Anything that was meant to be formal and went wrong, she enjoyed,' recalled William. 'She laughed herself stupid about it. It kept us all sane.'

William, now a student at St Andrews, revealed: 'She loved to hear about all my friends and all they got up to. And she loved to hear about how much trouble I got into at school.'

Until now, courtiers have always insisted that William enjoyed an unblemished school career.

William described an early memory of the Queen Mother on a visit to Birkhall, one of her Scottish homes. 'As I was getting out of the car, I tripped and fell flat on my face, right in front of her. I must have been about six or seven. She helped me up again. It was so funny because she was so old, and there was me being hauled off the ground by her.'

He also recalled her inspiring in him a sense of public duty. 'My favourite photograph of us together is a picture of me aged about nine or 10 helping the Queen Mother up the steps of Windsor Castle. I remember the moment because she said to me: "Keep doing that for people and you will go a long way in life."

'She loved seeing what we got for Christmas and asking what was the latest trendy thing. I was given a mobile phone and showed it to her. I invited her to press a few buttons and it kept blinking and flashing at her.'

It was at Birkhall that the Queen Mother gave William a farewell lunch before he set off to St Andrews last September. 'As she said goodbye, she said: "Any good parties, invite me down." But there was no way. I knew full well that if I invited her down, she would dance me under the table.'