Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

An activity making various patterns with 2 x 1 rectangular tiles.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?

How many different triangles can you make on a circular pegboard that has nine pegs?

Start with three pairs of socks. Now mix them up so that no mismatched pair is the same as another mismatched pair. Is there more than one way to do it?

What happens when you try and fit the triomino pieces into these two grids?

How many different triangles can you draw on the dotty grid which each have one dot in the middle?

Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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

Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

Take 5 cubes of one colour and 2 of another colour. How many different ways can you join them if the 5 must touch the table and the 2 must not touch the table?

Using the cards 2, 4, 6, 8, +, - and =, what number statements can you make?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

My dice has inky marks on each face. Can you find the route it has taken? What does each face look like?

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Find all the numbers that can be made by adding the dots on two dice.

Moira is late for school. What is the shortest route she can take from the school gates to the entrance?

10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Two children made up a game as they walked along the garden paths. Can you find out their scores? Can you find some paths of your own?

This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .

What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

Can you find all the different triangles on these peg boards, and find their angles?

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

Arrange 9 red cubes, 9 blue cubes and 9 yellow cubes into a large 3 by 3 cube. No row or column of cubes must contain two cubes of the same colour.

This problem focuses on Dienes' Logiblocs. What is the same and what is different about these pairs of shapes? Can you describe the shapes in the picture?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Use the clues to find out who's who in the family, to fill in the family tree and to find out which of the family members are mathematicians and which are not.

Try out the lottery that is played in a far-away land. What is the chance of winning?

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

Can you find all the different ways of lining up these Cuisenaire rods?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

My briefcase has a three-number combination lock, but I have forgotten the combination. I remember that there's a 3, a 5 and an 8. How many possible combinations are there to try?

How can you put five cereal packets together to make different shapes if you must put them face-to-face?

Frances and Rishi were given a bag of lollies. They shared them out evenly and had one left over. How many lollies could there have been in the bag?

The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?