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

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

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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?

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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?

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

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How many models can you find which obey these rules?

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Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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Can you find a cuboid that has a surface area of exactly 100 square units. Is there more than one? Can you find them all?

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

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

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Can you recreate these designs? What are the basic units? What movement is required between each unit? Some elegant use of procedures will help - variables not essential.

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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?

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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.

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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?

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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?

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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?

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

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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?

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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?

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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?

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Let's say you can only use two different lengths - 2 units and 4 units. Using just these 2 lengths as the edges how many different cuboids can you make?

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

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

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

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

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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?

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

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In a bowl there are 4 Chocolates, 3 Jellies and 5 Mints. Find a way to share the sweets between the three children so they each get the kind they like. Is there more than one way to do it?

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

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?

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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?

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

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Two sudokus in one. Challenge yourself to make the necessary connections.

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

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

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

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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.

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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?

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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?

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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?

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

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Move your counters through this snake of cards and see how far you can go. Are you surprised by where you end up?

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

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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?

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How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

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

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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. . . .

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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?

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

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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.