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?

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

Exchange the positions of the two sets of counters in the least possible number of moves

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

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

How many winning lines can you make in a three-dimensional version of noughts and crosses?

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?

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 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?

How can the same pieces of the tangram make this bowl before and after it was chipped? Use the interactivity to try and work out what is going on!

An extension of noughts and crosses in which the grid is enlarged and the length of the winning line can to altered to 3, 4 or 5.

Find a cuboid (with edges of integer values) that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

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

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

A game for 2 players. Given a board of dots in a grid pattern, players take turns drawing a line by connecting 2 adjacent dots. Your goal is to complete more squares than your opponent.

How many moves does it take to swap over some red and blue frogs? Do you have a method?

Triangle numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

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?

Investigate how the four L-shapes fit together to make an enlarged L-shape. You could explore this idea with other shapes too.

How many different symmetrical shapes can you make by shading triangles or squares?

A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

Can you fit the tangram pieces into the outlines of the candle and sundial?

If you move the tiles around, can you make squares with different coloured edges?

Can you fit the tangram pieces into the outlines of Mai Ling and Chi Wing?

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?

Can you fit the tangram pieces into the outlines of the watering can and man in a boat?

Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Which of these dice are right-handed and which are left-handed?

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . .

Can you fit the tangram pieces into the outlines of the workmen?

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology.

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end?

What happens when you turn these cogs? Investigate the differences between turning two cogs of different sizes and two cogs which are the same.

Can you fit the tangram pieces into the outline of this goat and giraffe?

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

Can you fit the tangram pieces into the outline of Little Ming and Little Fung dancing?

Slide the pieces to move Khun Phaen past all the guards into the position on the right from which he can escape to freedom.

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs?

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?

Show that among the interior angles of a convex polygon there cannot be more than three acute angles.

Can you fit the tangram pieces into the outlines of the chairs?

Can you fit the tangram pieces into the outline of this shape. How would you describe it?

Can you fit the tangram pieces into the outlines of the lobster, yacht and cyclist?

Can you fit the tangram pieces into the outline of the child walking home from school?