# Maths Item of the Month Archive 2009

### December 2009Become a Chartered Mathematics Teacher

 (From the ATM publication, Mathematics Teaching, Sept 2009) (From the MA publication, Mathematics in School, Nov '09)

Solution

### November 2009

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.)

Solution

### October 2009One of Doug French's favourite theorems

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".

Doug French 1941 - 2009

Solution

### September 2009

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?

Solution

### August 2009Safe combinations

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?

Solution

### July 2009The answer is 'Parabola'...now, what's the question?

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.

### June 2009What will A Level Mathematics and Further Mathematics be like from 2012?

Teachers were invited to give input into MEI's response to the QCA consultation on level 3 Mathematics.

### May 2009Chords in polygons

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?

Solution

### April 2009Doubling the Square

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.

Solution

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.

Solution

### February 2009Using only unit fractions

Can you answer the following four questions:

• Starting with and adding on each unit fraction in turn, is the sum ever an integer?
• Starting with and adding on each unit fraction in turn, does the sum grow without limit or is there a ceiling beyond which it cannot pass?
• What happens if you add all those unit fractions with denominators that are positive integer powers of a fixed number, such as ?
• Any rational number can be written as the sum of different unit fractions. How can you prove this?

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.

### January 2009Happy New Year

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?

Solution