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

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

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

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?

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

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?

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

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

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?

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

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

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

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

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

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

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?

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

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

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

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?

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?

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?

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

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

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

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.

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

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?

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

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

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

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

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

Seven balls are shaken. You win if the two blue balls end up touching. What is the probability of winning?

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

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

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?

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

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

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

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

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

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?

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

An equilateral triangle rotates around regular polygons and produces an outline like a flower. What are the perimeters of the different flowers?

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?

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

In this follow-up to the problem Odds and Evens, we invite you to analyse a probability situation in order to find the general solution for a fair game.

Imagine a room full of people who keep flipping coins until they get a tail. Will anyone get six heads in a row?