I applied to Cambridge twice — I was rejected the first time, and accepted a year later.
My first interview was a train wreck. The journey to Cambridge (I travelled up there on the day) was a nightmare: driving rain, congested roads — I was stressed, my dad (who drove me) was stressed. The whole thing was a disaster. My dad said he could tell I’d be rejected based on the look on my face when I emerged from the college after the interview.
My second interview could not have gone better. I went down to Cambridge the night before and met up with friends who were undergraduates at the university. We went to see a movie, I had a couple of pints in a bar afterwards, and I went to bed fairly early. I slept like a baby and I woke up feeling genuinely refreshed and ready for anything. No stress about getting to the interview — I was already in the college: I just had to walk across the court.
The point of the interview is to assess whether you will respond well to one of the most important aspects of Oxbridge teaching: the tutorial (Oxford) or supervision (Cambridge). These are intense one-to-one or two-to-one, hour-long sessions where you discuss work you’ve been assigned in advance. If that mode of teaching doesn’t work for you, then maybe Oxbridge isn’t the right place for you — which is why they need to see how you respond to it.
Interviewers have no interest in asking you questions to which you already know the answer, or to which you have a prepared answer. That’s not helpful. They want to see how you deal with a problem or question you’ve never encountered before. They want to see how your brain works. In particular, whether you can articulate your thoughts in real time. (And that those thoughts are relevant, well-informed, and pretty clever.)
So here’s perhaps the bad news: I’m not sure you can prepare for an interview of this type. I think you have to trust that — if you are a suitable candidate for Oxbridge — your brain will deliver the goods when it counts.
Being well-rested is clearly helpful. Having a solid repository of facts to draw upon is important. (When I give mock interviews in mathematics I always start by asking what sin(π/6) is — I think that’s a basic fact that any self-respecting mathematician would know and would be able to tell me instantly without needing to think or work it out.) But the interview is not a test of facts, it’s a test of thinking. It’s how you use what you know to solve new problems.
One tip that’s often repeated which I do think is good advice — pause before answering. Don’t rush in. Give your brain a second or two to scan its database before you start talking. You’ll be much less garbled if you do.
And I know this is ridiculous, but insofar as this is possible — try to relax, try even to enjoy it and have fun. Talking about difficult problems with clever people is what Oxbridge is all about. The interview is a first opportunity to give that a try.