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?
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?
Find your way through the grid starting at 2 and following these operations. What number do you end on?
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?
Use the clues to colour each square.
What happens when you try and fit the triomino pieces into these two grids?
How many triangles can you make on the 3 by 3 pegboard?
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?
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?
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?
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
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?
What is the best way to shunt these carriages so that each train can continue its journey?
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?
Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.
An activity making various patterns with 2 x 1 rectangular tiles.
Design an arrangement of display boards in the school hall which fits the requirements of different people.
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?
These practical challenges are all about making a 'tray' and covering it with paper.
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?
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 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.
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 many triangles can you make using sticks that are 3cm, 4cm and 5cm long?
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....?
How can you put five cereal packets together to make different shapes if you must put them face-to-face?
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?
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?
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 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. . . .
How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
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?
How many different triangles can you make on a circular pegboard that has nine pegs?
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?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
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 ways can you find of fitting five hexagons together? How will you know you have found all the ways?
These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.
A merchant brings four bars of gold to a jeweller. How can the jeweller use the scales just twice to identify the lighter, fake bar?
What is the smallest number of coins needed to make up 12 dollars and 83 cents?
Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?
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.
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
These are the faces of Will, Lil, Bill, Phil and Jill. Use the clues to work out which name goes with each face.
Can you cover the camel with these pieces?
If we had 16 light bars which digital numbers could we make? How will you know you've found them all?