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

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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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A thoughtful shepherd used bales of straw to protect the area around his lambs. Explore how you can arrange the bales.

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

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How many models can you find which obey these rules?

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

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These rectangles have been torn. How many squares did each one have inside it before it was ripped?

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

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

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

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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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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?

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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Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

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How many ways can you find of tiling the square patio, using square tiles of different sizes?

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Penta people, the Pentominoes, always build their houses from five square rooms. I wonder how many different Penta homes you can create?

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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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What is the smallest number of tiles needed to tile this patio? Can you investigate patios of different sizes?

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Cut differently-sized square corners from a square piece of paper to make boxes without lids. Do they all have the same volume?

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

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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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Can you draw a square in which the perimeter is numerically equal to the area?

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In this investigation, you must try to make houses using cubes. If the base must not spill over 4 squares and you have 7 cubes which stand for 7 rooms, what different designs can you come up with?

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What is the largest 'ribbon square' you can make? And the smallest? How many different squares can you make altogether?

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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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Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?

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

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

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Make a pair of cubes that can be moved to show all the days of the month from the 1st to the 31st.

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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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An investigation that gives you the opportunity to make and justify predictions.

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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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How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?

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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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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?