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?
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?
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?
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?
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?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
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.
An activity making various patterns with 2 x 1 rectangular tiles.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
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?
How many different triangles can you draw on the dotty grid which each have one dot in the middle?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
Can you work out how to balance this equaliser? You can put more than one weight on a hook.
Use the clues to colour each square.
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 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. . . .
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?
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
How many triangles can you make on the 3 by 3 pegboard?
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?
What happens when you try and fit the triomino pieces into these two grids?
Can you cover the camel with these pieces?
These practical challenges are all about making a 'tray' and covering it with paper.
Find your way through the grid starting at 2 and following these operations. What number do you end on?
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
How many models can you find which obey these rules?
Imagine that the puzzle pieces of a jigsaw are roughly a rectangular shape and all the same size. How many different puzzle pieces could there be?
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.
Ben and his mum are planting garlic. Use the interactivity to help you find out how many cloves of garlic they might have had.
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.
Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
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.
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?
When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?