In Sam and Jill's garden there are two sorts of ladybirds with 7 spots or 4 spots. What numbers of total spots can you make?

Take the number 6 469 693 230 and divide it by the first ten prime numbers and you'll find the most beautiful, most magic of all numbers. What is it?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

Exploring and predicting folding, cutting and punching holes and making spirals.

Use the interactivity to make this Islamic star and cross design. Can you produce a tessellation of regular octagons with two different types of triangle?

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

This number has 903 digits. What is the sum of all 903 digits?

Can you predict, without drawing, what the perimeter of the next shape in this pattern will be if we continue drawing them in the same way?

Nine squares with side lengths 1, 4, 7, 8, 9, 10, 14, 15, and 18 cm can be fitted together to form a rectangle. What are the dimensions of the rectangle?

Look at how the pattern is built up - in that way you will know how to break the final pattern down into more manageable pieces.

The number 2.525252525252.... can be written as a fraction. What is the sum of the denominator and numerator?

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

What is the total area of the first two triangles as a fraction of the original A4 rectangle? What is the total area of the first three triangles as a fraction of the original A4 rectangle? If. . . .

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle?

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

How many six digit numbers are there which DO NOT contain a 5?

When is 7^n + 3^n a multiple of 10? Use Excel to investigate, and try to explain what you find out.

Find the decimal equivalents of the fractions one ninth, one ninety ninth, one nine hundred and ninety ninth etc. Explain the pattern you get and generalise.

Follow instructions to fold sheets of A4 paper into pentagons and assemble them to form a dodecahedron. Calculate the error in the angle of the not perfectly regular pentagons you make.

A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.

Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.

Given an equilateral triangle inside an isosceles triangle, can you find a relationship between the angles?

The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.

The interval 0 - 1 is marked into halves, quarters, eighths ... etc. Vertical lines are drawn at these points, heights depending on positions. What happens as this process goes on indefinitely?

Two problems about infinite processes where smaller and smaller steps are taken and you have to discover what happens in the limit.