These practical challenges are all about making a 'tray' and covering it with paper.

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

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?

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.

Imagine you have an unlimited number of four types of triangle. How many different tetrahedra can you make?

Use the interactivity to listen to the bells ringing a pattern. Now it's your turn! Play one of the bells yourself. How do you know when it is your turn to ring?

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?

Use the interactivity to play two of the bells in a pattern. How do you know when it is your turn to ring, and how do you know which bell to ring?

Design an arrangement of display boards in the school hall which fits the requirements of different people.

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?

What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?

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

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

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?

Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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?

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?

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?

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.

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

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

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?

The challenge here is to find as many routes as you can for a fence to go so that this town is divided up into two halves, each with 8 blocks.

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?

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?

If we had 16 light bars which digital numbers could we make? How will you know you've 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?

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

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

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

If you have three circular objects, you could arrange them so that they are separate, touching, overlapping or inside each other. Can you investigate all the different possibilities?

How many models can you find which obey these rules?

Ana and Ross looked in a trunk in the attic. They found old cloaks and gowns, hats and masks. How many possible costumes could they make?

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!

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

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

In this game for two players, you throw two dice and find the product. How many shapes can you draw on the grid which have that area or perimeter?

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Can you draw a square in which the perimeter is numerically equal to the area?

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?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

Place eight queens on an chessboard (an 8 by 8 grid) so that none can capture any of the others.

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

Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

Investigate all the different squares you can make on this 5 by 5 grid by making your starting side go from the bottom left hand point. Can you find out the areas of all these squares?

First Connect Three game for an adult and child. Use the dice numbers and either addition or subtraction to get three numbers in a straight line.

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?

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