OK as promised heres a complete breakdown of my Oxford interview. This is a very long post so don't read on if you don't like very long posts. Maybe just maybe someone will find this useful
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My first interview was at Teddy Hall (where I was staying), and he asked me about the function f(x) where if x is even f(x) = x/2 , and if x is odd f(x) = x + 1. I got to the point that it always ends up as 2,1,2,1,2... but then he asked me to prove it and I really screwed it up. Next I got asked about another function which takes 2 numbers and outputs the highest one (he drew it up like an electronics diagram with logic gates), This was fine when he asked me how many milliseconds you needed to get the highest from 4,8 and 16 numbers if each function takes 1ms, which I did, and then a general rule for n numbers, which I could also do (round up to the nearest power of 2, and then log this number to base 2). Then he asked me how to find the middle number for 3 inputs using only this function and another (which outputs the lowest input), this had me completly stumped and he seemed to enjoy telling me every idea I suggested was terribly wrong, thus ended the first interview.
Second one went a bit better (was at Oriel), first I got asked about the shortest path from opposite vertices of a cube length 1, which after much prompting I worked out to be root 5 (draw the net and use pythag). Then he showed me this "Christmas cracker" thing, as he called it, which was 5 groups of numbers and you said what groups a number was in and he told you what that number was. I worked out it was a sort of binary system, which sort of impressed him, the he asked me to write down a sum formula, which eventually after many a prompt, I got to. The tutors seemed nicer here as well and I had a long chat about evolutionary programming after the technical stuff, which seemed to impress again.
Third interview was at: wait for it, Balliol! Wow I thought this was one of THE most competitive colleges in Oxford but I was in only a handful of people asked for a third interview (which is good right?). This one went fairly well. First I got a question on probabilty (how many sets of partners are there in a 4 player doubles tennis game?, what about if there were 5 possible players? ie one doesn't play). I got both these right with no prompting, the guy seemed impressed. Then I had to work out a problem involving a machine which takes a list of numbers, and an integer. And if there is any combination of the numbers in the list that add to give the integer it outputs a yes, otherwise a no. He asked me to form an algorithm to use in the machine, which I did (try each number in turn, then each pair, then each trio etc...). He then helped me to work through my method for a few examples and find the maximum amount of comparisons which it could possibly do: (2^n) -1. He then said there was a solution in which the maximum number of comparisons was n, and if I could think what that might be. I thought it would be better to start with all the numbers and then work backwards. Amazingly this was the solution he had in mind and the professor seemed very impressed by this. We then worked through the complete method (which involved trying n numbers, and if there was a yes, trying n-1 and eliminating the number not used if n-1 was yes and so on. I still don't fully understand it, but it made sense at the time. So far so good, enter the second interviewer (a mathematician) to ask some "maths questions".
Differentiate y = 5^x 9 (in terms of x). Crap. I knew the answer, but I wasn't sure the complete method. Out of desperation I muttered "take logs of both sides". He did it on the board, wow I was right. Then he asked me if I could simplify ln(5^x). Crap again. I looked at it for about 10 seconds (seemed like 10 minutes) before finally remembering the laws of logs back from C2, thank you for the laws of logs, they may have saved my life. If I get an offer I will thank Napier and all he has done for the world (and his black cock, but thats another story). ln(5^x) = xln5. Yes I though now I could differentiate both sides!!! (1/y)dy/dx = ln5 therefore
dy/dx = y ln5. dy/dx = 5^x(ln5). He asked me if I learnt implicit differentiation at school and seemed disappointed when I said yes (as if I had just magically differentiated ln(y) with respect to x, without being taught how to do it lol!). Next I had to prove that the square of any odd number is a factor of 8 + 1. I could do this as I let 2k+1 equal an odd number, and then explained (2k+1)^2. I got 4k^2 + 4k +1, so take the one away, you get 4k(k+1), which after some help I figured out 4k always makes a multiple of 8 in the event of k being an even number, and k+1 always gives a multiple of 8 if k is an odd number (sort of like this anyway). The guy seemed happy and so ended my final interview.
Wow anyone still reading this. Yes I am bored