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?
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?
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?
Use the clues to colour each square.
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
How will you go about finding all the jigsaw pieces that have one peg and one hole?
What happens when you try and fit the triomino pieces into these two grids?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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 cover the camel with these pieces?
What is the best way to shunt these carriages so that each train can continue its journey?
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.
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?
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?
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?
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?
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?
Design an arrangement of display boards in the school hall which fits the requirements of different people.
These practical challenges are all about making a 'tray' and covering it with paper.
How many triangles can you make on the 3 by 3 pegboard?
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?
How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.
Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.
Find your way through the grid starting at 2 and following these operations. What number do you end on?
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
How many different triangles can you make on a circular pegboard that has nine pegs?
Alice's mum needs to go to each child's house just once and then back home again. How many different routes are there? Use the information to find out how long each road is on the route she took.
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?
How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?
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?
Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?
Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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?
Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
An activity making various patterns with 2 x 1 rectangular tiles.
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?
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?
Can you find all the different ways of lining up these Cuisenaire rods?
How many models can you find which obey these rules?
How many different rhythms can you make by putting two drums on the wheel?
Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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.
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. . . .
If you hang two weights on one side of this balance, in how many different ways can you hang three weights on the other side for it to be balanced?
In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?