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

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

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

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?

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?

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?

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

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?

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

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

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?

How many different shaped boxes can you design for 36 sweets in one layer? Can you arrange the sweets so that no sweets of the same colour are next to each other in any direction?

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

How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?

How many models can you find which obey these rules?

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?

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.

Can you make dice stairs using the rules stated? How do you know you have all the possible stairs?

Can you order pictures of the development of a frog from frogspawn and of a bean seed growing into a plant?

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

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

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

Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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

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?

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

The planet of Vuvv has seven moons. Can you work out how long it is between each super-eclipse?

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.

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

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

Sitting around a table are three girls and three boys. Use the clues to work out were each person is sitting.

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.?

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

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?

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.

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

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

In how many ways can you stack these rods, following the rules?

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.

Sally and Ben were drawing shapes in chalk on the school playground. Can you work out what shapes each of them drew using the clues?

Can you rearrange the biscuits on the plates so that the three biscuits on each plate are all different and there is no plate with two biscuits the same as two biscuits on another plate?

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. . . .

Nina must cook some pasta for 15 minutes but she only has a 7-minute sand-timer and an 11-minute sand-timer. How can she use these timers to measure exactly 15 minutes?

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?

You cannot choose a selection of ice cream flavours that includes totally what someone has already chosen. Have a go and find all the different ways in which seven children can have ice cream.

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?

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

In this challenge, buckets come in five different sizes. If you choose some buckets, can you investigate the different ways in which they can be filled?

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?