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?

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?

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.

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?

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?

How many models can you find which obey these rules?

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 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 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 shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

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

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?

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?

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?

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

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?

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

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

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

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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

Here are four cubes joined together. How many other arrangements of four cubes can you find? Can you draw them on dotty paper?

This challenge is to design different step arrangements, which must go along a distance of 6 on the steps and must end up at 6 high.

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

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

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?

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

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

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

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

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.

Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?

Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

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

When intergalactic Wag Worms are born they look just like a cube. Each year they grow another cube in any direction. Find all the shapes that five-year-old Wag Worms can be.

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

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?

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

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.

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

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

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?