Wessex Mathematics Masterclasses in association with the Royal Institution.

• 17-i-98: You have to be quite adult to stand a chance of doing this (adult enough to have done some Euclidean geometry at school). Problem: Given a pair of compasses, a straight-edge and the lengths of the three altitudes of a triangle, construct the triangle.
• 24-i-98: Write out Pascal's triangle in the usual way. Look at the entries which occur in the central column (there is only such an entry in alternate rows). These entries are 1, 2, 6, 20 etc. Take an initial fragment of this infinite sequence, reverse it, and form the sum of the term-by-term products with the original sequence. For example, taking the first three terms we have 1 times 6 + 2 times 2 + 6 times 1 = 16. What happens in general, and why?
• 31-i-98: You need to be able to do calculus to tackle this. Rotate a cube about one of its long diagonals. The shape you get looks like a munched apple with pointy ends. Calculate its volume, and compare that with the volume of the ball which circumscribes the original cube. If you get the answer correct, a chill should run down your spine.
• 7-ii-98: Suppose the universe was designed by Euclid and not Einstein. Take a rigid (i.e. non-foldable) chess board, and place it on Charon, the moon of Pluto. Show that wherever you are, and whenever you are, the sum of the squares of the distances from the tip of your nose to the centres of the black squares of this chess board is the same as the sum of the squares of the distances from the tip of your nose to the centres of the white squares.
• 28-ii-98: Our notation for the binomial coefficient (an entry in Pascal's triangle) is C(n,r) = n!/r!(n-r)!. Thus C(10,2)= 55. Let p be an odd prime number. Prove that C(2p-1,p-1) - 1 is divisible by p squared.
• 7-iii-98:Show that 999999999999 occur as the final digits of a number in the Fibonacci sequence
  0,1,1,2,3,5,8,13,21,34,55,89,144,...

  written in the ordinary (base 10) number system.

• Consider the Fibonacci sequence modulo $N$ where $N$ is any positive integer greater than $2.$ The resulting sequence is periodic. Show that the length of the fundamental period must be an even number.
• Consider whole numbers which have 4 digits when written in ordinary denary (base 10) notation. Say that such a number is of type $A$ if it can be written as the product of two numbers, each of which has two digits. Say that a 4 digit number is of type $B$ if it is not of type $A.$ Which are more numerous, type A numbers or type B numbers? No computers please!