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, but not necessarily the same number of each colour. Prove that the blocks can be arranged so that at least one box contains blocks of the same colour and no box contains blocks with more than 2 colours.
(This problem is based on a a sample interview question published by the Department of Computer Science at the University of Oxford.)
ABC is an equilateral triangle inscribed in a circle. P is a point on the minor arc BC. Prove that |AP|=|BP|+|CP|
Charlie Stripp of MEI writes:
I first met Doug through the Mathematical Association nearly 20 years ago, very early in my teaching career. I worked with him extensively as a colleague and friend. He was one of my heroes and had a huge influence on my career. He was an accomplished mathematician and a superb teacher. He was also a passionate advocate for mathematics education. Through all this he always conducted himself as a perfect gentleman. His philosophy of teaching was summed up by one of his favourite quotes: "Know how, but also know why".
Keen to make a good impression at the start of term, you stand in front of the mirror to check you look OK. How long does the mirror need to be so that you can see yourself top to toe? How far from the mirror should you stand? If you wanted most people to be able to use the same mirror, how long should it be and how high up on the wall should you hang it?
A group of 11 scientists are working on a secret project, the materials of which are kept in a safe. They want to be able to open the safe only when a majority of the group is present. Therefore the safe is provided with a number of different locks, and each scientist is given the keys to certain of these locks. How many locks are required, and how many keys must each scientist have?
The new ATM online magazine, MTi, was launched at the MEI Conference on July 1. You can see one of the articles from the first issue.
Teachers were invited to give input into MEI's response to the QCA consultation on level 3 Mathematics.
A regular polygon is inscribed in a unit circle and all the different length chords connecting pairs of vertices are drawn.
What is the link between the number of sides of the polygon and the sum of the squares of the lengths of these chords?
Given a square, construct another square with double the area using just a straight edge. A straight edge would not allow you to measure a length and transfer this length, nor can you slide the straight edge to create parallel lines; all you can do is use the straight edge to draw a line between two given points.
Every road in a particular country is one-way. Every pair of cities is connected by exactly one direct road.
Prove by induction that there exists a city which can be reached from every other city either directly or via at most one other city.
Can you answer the following four questions:
This is an example of some of the questions from one of the GCSE Extension materials. The GCSE Extension materials are aimed at students who are working towards GCSE Mathematics and would benefit from exposure to mathematics beyond the GCSE specifications.
2009 = 452 – 42
Can 2009 be written as the difference of two squares in any other ways?
Are there any years which cannot be written as the difference of two squares?