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?

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?

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?

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

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?

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?

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?

On a digital 24 hour clock, at certain times, all the digits are consecutive. How many times like this are there between midnight and 7 a.m.?

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

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

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

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

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?

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 greatest number of counters you can place on the grid below without four of them lying at the corners of a square?

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. . . .

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?

My cousin was 24 years old on Friday April 5th in 1974. On what day of the week was she born?

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?

Can you work out how to balance this equaliser? You can put more than one weight on a hook.

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?

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?

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

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

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

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?

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

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

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

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

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

Put 10 counters in a row. Find a way to arrange the counters into five pairs, evenly spaced in a row, in just 5 moves, using the rules.

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?

This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?

Place eight dots on this diagram, so that there are only two dots on each straight line and only two dots on each circle.

How many models can you find which obey these rules?

On a digital clock showing 24 hour time, over a whole day, how many times does a 5 appear? Is it the same number for a 12 hour clock over a whole day?

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

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.

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....?

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

Can you find all the different ways of lining up these Cuisenaire rods?

Stuart's watch loses two minutes every hour. Adam's watch gains one minute every hour. Use the information to work out what time (the real time) they arrived at the airport.