#Question
2Algebraic references with respect to summation formulae and proofs by induction.
3It is a fact that, apart from the peripherals, the whole of a computer can be made from NAND gates. The Egyptians created NAND gates using marbles rolling down shutes and used the them for booby trapping pyramids. Did, then, the Egyptians invent the computer? If not, explain fundamentally why not.
4The game of chess will be played perfectly by the computers of 2010. What is the meaning of this statement and is it likely to be true?
5Why is the pole vaulting world record about 6.5m, and why can’t it be broken?
6How would you ensure security between two people, A and B?
7What happens when light has to pass through a medium denser than air?
8What is the one fundamental difference between a spreadsheet and a database? Surely both hold information, so perhaps there is no fundamental difference?
9Why is the number 2.7182818… used in mathematics?
10Explain the principle of the global positioning system (GPS). What factors contribute to its accuracy?
11How do pirates divide their treasure? A group of 7 pirates has 100 gold coins. They have to decide amongst themselves how to divide the treasure, but must abide by pirate rules: 1) The most senior pirate proposes the division. 2) All of the pirates (including the most senior) vote on the decision. if half or more vote for the division, it stands. If less than half vote for it, they throw the most senior pirate overboard and start again. 3) The pirates are perfectly logical, and entirely ruthless (only caring about maximizing their own share of the gold). so what division should the most senior pirate suggest to the other six?
12Tidy boxes.  You are given 10 boxes, each large enough to contain exactly 10 wooden building blocks, and a total of 100 blocks in 10 different colours. There may not be the same number in each colour, so you may not be able to pack the blocks into the boxes in such a way that each box contains only one colour of block. Show that it is possible to do it so that each box contains at most two different colours.
13Searching for the maximum.  The real-valued function f(x), defined for 0 ≤ x ≤ 1, has a single maximum atx = m. If 0 ≤ u < v ≤ m then f(u) < f(v), and if m ≤ u < v ≤ 1 then f(u) > f(v). You are told nothing else about f, but you may ask for the value of f(x) for any values of x you choose. How would you find the approximate value of m?  How accurately could you find m if you could choose only 10 values of x for which to evaluate f(x)?
14Death by chocolate.  You are locked in a room with your worst enemy. On a table in the centre of the room is a bar of chocolate, divided into squares in the usual way. One square of the chocolate is painted with a bright green paint that contains a deadly nerve poison. You and your enemy take it in turns to break off one or more squares from the remaining chocolate (along a straight line) and eat them. Whoever is left with the green square must eat it and die in agony. You may look at the bar of chocolate and then decide whether to go first or second. Describe your strategy.
15Monkey beans.  An urn contains 23 white beans and 34 black beans. A monkey takes out two beans; if they are the same, he puts a black bean into the urn, and if they are different, he puts in a white bean from a large heap he has next to him. The monkey repeats this procedure until there is only one bean left. What colour is it?
16Lily-pad lunacy. Eleven lily pads are numbered from 0 to 10. A frog starts on pad 0 and wants to get to pad 10. At each jump, the frog can move forward by one or two pads, so there are many ways it can get to pad 10. For example, it can make 10 jumps of one pad, 1111111111, or five jumps of two pads, 22222, or go 221212 or 221122, and so on. We'll call each of these ways different, even if the frog takes the same jumps in a different order. How many different ways are there of getting from 0 to 10?
17Missing numbers.  Imagine you are given a list of slightly less than 1,000,000 numbers, all different, and each between 0 and 999,999 inclusive. How could you find (in a reasonable time) a number between 0 and 999,999 that is not on the list?
18Scribble.  The game of Scribble is played with an inexhaustible supply of tiles, and consists of forming 'words' according to certain rules. Since each tile bears one of the letters P, Q, or R, these are not words that will be found in an ordinary dictionary. The game begins with the word PQ on the board; each move consists of applying one of the following rules:
f the word on the board is Px, for some shorter word x, you may change it to Pxx. For example, if the word is PQRRQ then you may change it to PQRRQQRRQ.
If the word on the board is xQQQy, for some shorter words x and y, then you may change it to xRy, replacing the sequence QQQ with a single R.
If the word on the board is xRRy, for some shorter words x and y, then you may change it toxy, deleting the sequence RR entirely.
(i) For each of the following words, say whether you can make it or not by following the rules of the game:QPR, PQQ, PQR, PR. (ii) Given any word, how can you decide whether it can be made or not?
19Any negative impacts computers might have on society
20Firstly I was asked about the towers of Hanoi. If I had 64 blocks How many moves required to move the pyramid.
21 I was given a sequence and asked to figure out a general formula.
22Resistance
23Calculus
24Probability
25How does a router work
26Show me a sort algorithm. How efficient is it?
27Show me a search algorithm, how efficient is it? How can it be improved? Whats the efficiency of the new one?
28What is the maximum number of pieces you can cut a cake into, using four, straight cuts?
29Why does light pass through glass and not metal?
30What distinguishes metals from other materials
31How modern microprocessors work
32Why don’t manufacturers like intel just make their chips bigger, instead of smaller, and what is so good about small? She then asked about the limitations in making them small.
33Where would you like to see computing going in the next 20-30 years?
34He asked me to look at a piece of paper which had on it a scalene triangle. In it was a square; flush along the bottom and touching one of the sides. He asked me if, without measuring, I could draw another perfect square which was also flush against the bottom but touched both sides.
35If I were to study there, what natural sciences option I would select.
36Why are there tides on both sides of the Earth at the same time?
37What potential energy was
38Where potential energy is 0
39Where potential energy can be negative
40Why your voice sounds higher when you suck helium
41Why a violin sounds different to a piano or a saxophone
42How many comparisons must a bubblesort algorithm make when sorting a list of N items?
43How can a computer calculate the square root of 2?
44Simplify [sqrt(3)/2 + 0.5i]^6 without using a binomial expansion.
45Sketch y = sin(x)^2 and explain its relation to the curve y = -0.5*cos(2x).
46Sketch the graph y=A[1-e^(-Bx)]^2, explain the graph in terms of energy and distance between two atoms.
47Integrating and differentiating x^3+x^2+x+(1/x)
49You have two jugs, an infinite supply of water, and you only know how much water the jugs hold when full. Ther first jug could hold 7 litres of water, the second could hold two. By filling the jugs, emptying the jugs, or pouring water from one jug to the other, try and end up with 4 litres of water on the first jug and 2 litres of water in the second. Which combinations are possible?
50You have two positively charged particles with charge +k and mass m. They start infinitely far apart. One starts by moving towards the other at speed v, the other is stationary. Find the smallest distance between them.
511,1,(2),,5,(8),13,…. I was asked wh every number in brackets() was even and why every number in [] brackets was a multiple of 3, I was then asked if the pattern continued for the next numbers.
52He asked me a question about laying dominoes on a chess board, and whether it was possible to cover all the squares (domino covers 2 squares) if you cut off two diagonally opposite corners.
53Differentiate e^x^2, x^x and to draw the graph of (x^3)-x
54Towers of Hanoi
55What sin 5x is equal to?
56Write down 3 consecutive numbers and spot a pattern.
57Minimum number of breaks needed to break a choc bar into single pieces.
58Number of rectangles that can be fitted into a n by m rectangle
59Why d2y/dx2 is negative when a maximum turning point occurs
60What Group Theory was
61What I understood by differentiation
62Differentiate x^2.
63Differentiate sin(x)
64How many vertices and how many sides a square has. Then he asked about a cube. Then he asked about a four-dimensional cube.