How many different ways can you find to join three equilateral triangles together? Can you convince us that you have found them all?

Find out what a "fault-free" rectangle is and try to make some of your own.

Arrange the shapes in a line so that you change either colour or shape in the next piece along. Can you find several ways to start with a blue triangle and end with a red circle?

Find all the different shapes that can be made by joining five equilateral triangles edge to edge.

Are all the possible combinations of two shapes included in this set of 27 cards? How do you know?

This cube has ink on each face which leaves marks on paper as it is rolled. Can you work out what is on each face and the route it has taken?

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?

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....?

A game for 2 people. Take turns placing a counter on the star. You win when you have completed a line of 3 in your colour.

This practical challenge invites you to investigate the different squares you can make on a square geoboard or pegboard.

How many trapeziums, of various sizes, are hidden in this picture?

Place the numbers 1 to 8 in the circles so that no consecutive numbers are joined by a line.

Try out the lottery that is played in a far-away land. What is the chance of winning?

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?

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

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 challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

This tricky challenge asks you to find ways of going across rectangles, going through exactly ten squares.

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

This challenge extends the Plants investigation so now four or more children are involved.

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

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

Solve this Sudoku puzzle whose clues are in the form of sums of the numbers which should appear in diagonal opposite cells.

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Here are some rods that are different colours. How could I make a dark green rod using yellow and white rods?

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?

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

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

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.

Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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

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.

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

Make your own double-sided magic square. But can you complete both sides once you've made the pieces?

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

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

Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

There is a long tradition of creating mazes throughout history and across the world. This article gives details of mazes you can visit and those that you can tackle on paper.

A Sudoku with clues given as sums of entries.

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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?

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

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.

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

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

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