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?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
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?
If you split the square into these two pieces, it is possible to fit the pieces together again to make a new shape. How many new shapes can you make?
Using different numbers of sticks, how many different triangles are you able to make? Can you make any rules about the numbers of sticks that make the most triangles?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
What is the best way to shunt these carriages so that each train can continue its journey?
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
Hover your mouse over the counters to see which ones will be removed. Click to remover them. The winner is the last one to remove a counter. How you can make sure you win?
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
An activity making various patterns with 2 x 1 rectangular tiles.
How many triangles can you make on the 3 by 3 pegboard?
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
A challenging activity focusing on finding all possible ways of stacking rods.
Use the clues to colour each square.
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. . . .
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
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?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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 many different triangles can you draw on the dotty grid which each have one dot in the middle?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
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?
In how many ways can you stack these rods, following the rules?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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?
What happens when you try and fit the triomino pieces into these two grids?
Can you cover the camel with these pieces?
How many different rhythms can you make by putting two drums on the wheel?
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 practical challenges are all about making a 'tray' and covering it with paper.
How can you arrange the 5 cubes so that you need the smallest number of Brush Loads of paint to cover them? Try with other numbers of cubes as well.
Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
Investigate the different ways you could split up these rooms so that you have double the number.
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
How many shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?
This activity investigates how you might make squares and pentominoes from Polydron.
When newspaper pages get separated at home we have to try to sort them out and get things in the correct order. How many ways can we arrange these pages so that the numbering may be different?
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?
There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Can you find all the different triangles on these peg boards, and find their angles?
Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.