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?

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

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

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

These activities focus on finding all possible solutions so if you work in a systematic way, you won't leave any out.

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?

These activities lend themselves to systematic working in the sense that it helps if you have an ordered approach.

These activities focus on finding all possible solutions so working in a systematic way will ensure none are left out.

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?

These activities lend themselves to systematic working in the sense that it helps to have an ordered approach.

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

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

The brown frog and green frog want to swap places without getting wet. They can hop onto a lily pad next to them, or hop over each other. How could they do it?

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

Can you find out in which order the children are standing in this line?

How will you go about finding all the jigsaw pieces that have one peg and one hole?

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?

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. . . .

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?

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

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

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

Find your way through the grid starting at 2 and following these operations. What number do you end on?

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?

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?

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?

Can you help the children find the two triangles which have the lengths of two sides numerically equal to their areas?

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?

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

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

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

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?

Chandra, Jane, Terry and Harry ordered their lunches from the sandwich shop. Use the information below to find out who ordered each sandwich.

Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?

Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?

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?

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

There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?

Can you fill in the empty boxes in the grid with the right shape and colour?

Find all the numbers that can be made by adding the dots on two dice.

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

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?

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

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.

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

This task depends on groups working collaboratively, discussing and reasoning to agree a final product.

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

The Vikings communicated in writing by making simple scratches on wood or stones called runes. Can you work out how their code works using the table of the alphabet?

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.

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