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

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How many models can you find which obey these rules?

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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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Explore this how this program produces the sequences it does. What are you controlling when you change the values of the variables?

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

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

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

This article for teachers suggests activities based on pegboards, from pattern generation to finding all possible triangles, for example.

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

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These practical challenges are all about making a 'tray' and covering it with paper.

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

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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?

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

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If you had 36 cubes, what different cuboids could you make?

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Just four procedures were used to produce a design. How was it done? Can you be systematic and elegant so that someone can follow your logic?

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Remember that you want someone following behind you to see where you went. Can yo work out how these patterns were created and recreate them?

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

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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?

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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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Pentagram Pylons - can you elegantly recreate them? Or, the European flag in LOGO - what poses the greater problem?

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When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.

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

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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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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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This activity investigates how you might make squares and pentominoes from Polydron.

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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 shapes can you build from three red and two green cubes? Can you use what you've found out to predict the number for four red and two green?

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

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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.

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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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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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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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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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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.

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

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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.

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

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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.?

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

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

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

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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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?