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

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

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?

The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Delight your friends with this cunning trick! Can you explain how it works?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Can you explain the strategy for winning this game with any target?

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

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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?

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

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

What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =

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

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

Can all unit fractions be written as the sum of two unit fractions?

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

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

Nim-7 game for an adult and child. Who will be the one to take the last counter?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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?

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

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

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

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

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

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

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?

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?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

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

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

In how many ways can you arrange three dice side by side on a surface so that the sum of the numbers on each of the four faces (top, bottom, front and back) is equal?

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

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

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