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?

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

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?

What is the best way to shunt these carriages so that each train can continue its journey?

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 shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?

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?

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?

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?

A challenging activity focusing on finding all possible ways of stacking rods.

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?

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 rhythms can you make by putting two drums on the wheel?

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

Building up a simple Celtic knot. Try the interactivity or download the cards or have a go on squared paper.

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

Find your way through the grid starting at 2 and following these operations. What number do you end on?

In how many ways can you stack these rods, following the rules?

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

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

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

How many triangles can you make using sticks that are 3cm, 4cm and 5cm long?

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.

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?

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 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.

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 to balance this equaliser? You can put more than one weight on a hook.

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?

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?

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

My coat has three buttons. How many ways can you find to do up all the buttons?

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?

Use the information to describe these marbles. What colours must be on marbles that sparkle when rolling but are dark inside?

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

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

Can you find out in which order the children are standing in this line?

How many trains can you make which are the same length as Matt's, using rods that are identical?

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

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?

Investigate the different ways you could split up these rooms so that you have double the number.

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?

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