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?

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?

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

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?

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?

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

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?

Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

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?

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

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

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?

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?

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

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

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?

There are to be 6 homes built on a new development site. They could be semi-detached, detached or terraced houses. How many different combinations of these can you find?

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

In how many ways could Mrs Beeswax put ten coins into her three puddings so that each pudding ended up with at least two coins?

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?

Can you fill in this table square? The numbers 2 -12 were used to generate it with just one number used twice.

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!

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

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.

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

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

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

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

This challenge is about finding the difference between numbers which have the same tens digit.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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?

This dice train has been made using specific rules. How many different trains can you make?

Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

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?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

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.

Find the product of the numbers on the routes from A to B. Which route has the smallest product? Which the largest?

Can you put the numbers from 1 to 15 on the circles so that no consecutive numbers lie anywhere along a continuous straight line?