Can you find any two-digit numbers that satisfy all of these statements?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Can you find a way to identify times tables after they have been shifted up or down?

Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

Can you find a relationship between the number of dots on the circle and the number of steps that will ensure that all points are hit?

How many pairs of numbers can you find that add up to a multiple of 11? Do you notice anything interesting about your results?

Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

Imagine you were given the chance to win some money... and imagine you had nothing to lose...

My two digit number is special because adding the sum of its digits to the product of its digits gives me my original number. What could my number be?

Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?

Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

15 = 7 + 8 and 10 = 1 + 2 + 3 + 4. Can you say which numbers can be expressed as the sum of two or more consecutive integers?

Here is a machine with four coloured lights. Can you make two lights switch on at once? Three lights? All four lights?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

Can you find the values at the vertices when you know the values on the edges?

Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

In 15 years' time my age will be the square of my age 15 years ago. Can you work out my age, and when I had other special birthdays?

Can you find rectangles where the value of the area is the same as the value of the perimeter?

A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?

There are nasty versions of this dice game but we'll start with the nice ones...

Think of a number and follow my instructions. Tell me your answer, and I'll tell you what you started with! Can you explain how I know?

How many winning lines can you make in a three-dimensional version of noughts and crosses?

If you move the tiles around, can you make squares with different coloured edges?

I'm thinking of a rectangle with an area of 24. What could its perimeter be?

Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

What's the largest volume of box you can make from a square of paper?

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?

Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

There are lots of different methods to find out what the shapes are worth - how many can you find?

Which set of numbers that add to 10 have the largest product?

Start with two numbers and generate a sequence where the next number is the mean of the last two numbers...

Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

If you have a large supply of 3kg and 8kg weights, how many of each would you need for the average (mean) of the weights to be 6kg?

Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

Who said that adding, subtracting, multiplying and dividing couldn't be fun?

Six balls are shaken. You win if at least one red ball ends in a corner. What is the probability of winning?

Play around with sets of five numbers and see what you can discover about different types of average...

An aluminium can contains 330 ml of cola. If the can's diameter is 6 cm what is the can's height?

Have a go at creating these images based on circles. What do you notice about the areas of the different sections?

Chris is enjoying a swim but needs to get back for lunch. If she can swim at 3 m/s and run at 7m/sec, how far along the bank should she land in order to get back as quickly as possible?

Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?

Explore the relationships between different paper sizes.

Charlie likes to go for walks around a square park, while Alison likes to cut across diagonally. Can you find relationships between the vectors they walk along?

Explore when it is possible to construct a circle which just touches all four sides of a quadrilateral.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?