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

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

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

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

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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 many trains can you make which are the same length as Matt's and Katie's, using rods that are identical?

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Use the interactivity to help get a feel for this problem and to find out all the possible ways the balls could land.

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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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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 ways can you find to join three equilateral triangles together? Can you convince us that you have found 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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Here are some rods that are different colours. How could I make a yellow rod using white and red rods?

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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?

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

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

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

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

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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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There are nine teddies in Teddy Town - three red, three blue and three yellow. There are also nine houses, three of each colour. Can you put them on the map of Teddy Town according to the rules?

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Can you work out how to make each side of this balance equally balanced? You can put more than one weight on a hook.

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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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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?

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How long does it take to brush your teeth? Can you find the matching length of time?

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Add or subtract the two numbers on the spinners and try to complete a row of three. Are there some numbers that are good to aim for?

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

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

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What do the numbers shaded in blue on this hundred square have in common? What do you notice about the pink numbers? How about the shaded numbers in the other squares?

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

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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?

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

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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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Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

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

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

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

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

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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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Jack has nine tiles. He put them together to make a square so that two tiles of the same colour were not beside each other. Can you find another way to do it?

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In this game for two players, take it in turns to shade one petal, or two petals next to each other. Is it better to go first or second?