Explore the effect of reflecting in two parallel mirror lines.

Explore the effect of reflecting in two intersecting mirror lines.

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Explore the effect of combining enlargements.

Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?

Think of a number, square it and subtract your starting number. Is the number youâ€™re left with odd or even? How do the images help to explain this?

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

With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?

Take a counter and surround it by a ring of other counters that MUST touch two others. How many are needed?

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?

In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.

These squares have been made from Cuisenaire rods. Can you describe the pattern? What would the next square look like?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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

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

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

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

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

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

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?

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?

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

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?

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?

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

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.

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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

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

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

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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

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?

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?

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

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

What happens when you round these three-digit numbers to the nearest 100?

This activity involves rounding four-digit numbers to the nearest thousand.

What happens when you round these numbers to the nearest whole number?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.