Leah and Tom each have a number line. Can you work out where their counters will land? What are the secret jumps they make with their counters?

Annie and Ben are playing a game with a calculator. What was Annie's secret number?

If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?

These eleven shapes each stand for a different number. Can you use the multiplication sums to work out what they are?

Investigate this balance which is marked in halves. If you had a weight on the left-hand 7, where could you hang two weights on the right to make it balance?

Max and Mandy put their number lines together to make a graph. How far had each of them moved along and up from 0 to get the counter to the place marked?

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

Water freezes at 0°Celsius (32°Fahrenheit) and boils at 100°C (212°Fahrenheit). Is there a temperature at which Celsius and Fahrenheit readings are the same?

There is a particular value of x, and a value of y to go with it, which make all five expressions equal in value, can you find that x, y pair ?

Manufacturers need to minimise the amount of material used to make their product. What is the best cross-section for a gutter?

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

Use algebra to reason why 16 and 32 are impossible to create as the sum of consecutive numbers.

The equation a^x + b^x = 1 can be solved algebraically in special cases but in general it can only be solved by numerical methods.

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?

Find the equation from which to calculate the resistance of an infinite network of resistances.

George and Dominic worked really hard on this problem and reasoned it through very well.

Pupils from Stradbroke Primary School found that there are four points on the circle which are all the same distance from the horizontal axis.

Kkytha provided a couple of diagrams that helped explain why it is impossible to draw the triangle.

Simon from Elizabeth College found solutions to Two Regular Polygons by using a spreadsheet, efficiently combined with some very good reasoning to know when he had all the solutions. We liked the way Simon put the power of the spreadsheet to work under the control of some elegant mathematical thinking.

Find out about Magic Squares in this article written for students. Why are they magic?!