Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?

Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The winner is the player to take the last counter.

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?

It starts quite simple but great opportunities for number discoveries and patterns!

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

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?

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

Charlie has moved between countries and the average income of both has increased. How can this be so?

Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

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

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

How many centimetres of rope will I need to make another mat just like the one I have here?

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

This task follows on from Build it Up and takes the ideas into three dimensions!

Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Here are some arrangements of circles. How many circles would I need to make the next size up for each? Can you create your own arrangement and investigate the number of circles it needs?

An investigation that gives you the opportunity to make and justify predictions.

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

Can you see how to build a harmonic triangle? Can you work out the next two rows?

Find out what a "fault-free" rectangle is and try to make some of your own.

Use the interactivity to investigate what kinds of triangles can be drawn on peg boards with different numbers of pegs.

Here are two kinds of spirals for you to explore. What do you notice?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

Can you find all the ways to get 15 at the top of this triangle of numbers?

It's easy to work out the areas of most squares that we meet, but what if they were tilted?

Are these statements always true, sometimes true or never true?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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?

Are these statements always true, sometimes true or never true?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

It would be nice to have a strategy for disentangling any tangled ropes...

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

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

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?

Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

Polygonal numbers are those that are arranged in shapes as they enlarge. Explore the polygonal numbers drawn here.