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Broad Topics > Using, Applying and Reasoning about Mathematics > Visualising Dice, Routes and Pathways

Age 5 to 14

This article for teachers discusses examples of problems in which there is no obvious method but in which children can be encouraged to think deeply about the context and extend their ability to. . . . Troublesome Dice

Age 11 to 14 Challenge Level:

When dice land edge-up, we usually roll again. But what if we didn't...? Sliding Puzzle

Age 11 to 16 Challenge Level:

The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves. Cubes Within Cubes

Age 7 to 14 Challenge Level:

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used? Clocked

Age 11 to 14 Challenge Level:

Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours? How Many Dice?

Age 11 to 14 Challenge Level:

A standard die has the numbers 1, 2 and 3 are opposite 6, 5 and 4 respectively so that opposite faces add to 7? If you make standard dice by writing 1, 2, 3, 4, 5, 6 on blank cubes you will find. . . . Age 11 to 14 Challenge Level:

Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite? Buses

Age 11 to 14 Challenge Level:

A bus route has a total duration of 40 minutes. Every 10 minutes, two buses set out, one from each end. How many buses will one bus meet on its way from one end to the other end? The Triangle Game

Age 11 to 16 Challenge Level:

Can you discover whether this is a fair game? Coordinate Patterns

Age 11 to 14 Challenge Level:

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead? Bands and Bridges: Bringing Topology Back

Age 7 to 14

Lyndon Baker describes how the Mobius strip and Euler's law can introduce pupils to the idea of topology. Triangle Inequality

Age 11 to 14 Challenge Level:

ABC is an equilateral triangle and P is a point in the interior of the triangle. We know that AP = 3cm and BP = 4cm. Prove that CP must be less than 10 cm. Finding 3D Stacks

Age 7 to 11 Challenge Level:

Can you find a way of counting the spheres in these arrangements? Cubes Within Cubes Revisited

Age 11 to 14 Challenge Level:

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need? Concrete Wheel

Age 11 to 14 Challenge Level:

A huge wheel is rolling past your window. What do you see? Age 11 to 14 Challenge Level:

Can you mark 4 points on a flat surface so that there are only two different distances between them? There and Back Again

Age 11 to 14 Challenge Level:

Bilbo goes on an adventure, before arriving back home. Using the information given about his journey, can you work out where Bilbo lives? Cubist Cuts

Age 11 to 14 Challenge Level:

A 3x3x3 cube may be reduced to unit cubes in six saw cuts. If after every cut you can rearrange the pieces before cutting straight through, can you do it in fewer? Christmas Chocolates

Age 11 to 14 Challenge Level:

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes? Christmas Boxes

Age 11 to 14 Challenge Level:

Find all the ways to cut out a 'net' of six squares that can be folded into a cube. Polygon Rings

Age 11 to 14 Challenge Level:

Join pentagons together edge to edge. Will they form a ring? Tessellating Hexagons

Age 11 to 14 Challenge Level:

Which hexagons tessellate? Tetrahedra Tester

Age 11 to 14 Challenge Level:

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? Tourism

Age 11 to 14 Challenge Level:

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable. Zooming in on the Squares

Age 7 to 14

Start with a large square, join the midpoints of its sides, you'll see four right angled triangles. Remove these triangles, a second square is left. Repeat the operation. What happens? Hidden Rectangles

Age 11 to 14 Challenge Level:

Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard? Pattern Power

Age 5 to 14

Mathematics is the study of patterns. Studying pattern is an opportunity to observe, hypothesise, experiment, discover and create. Königsberg

Age 11 to 14 Challenge Level:

Can you cross each of the seven bridges that join the north and south of the river to the two islands, once and once only, without retracing your steps? Convex Polygons

Age 11 to 14 Challenge Level:

Show that among the interior angles of a convex polygon there cannot be more than three acute angles. Coloured Edges

Age 11 to 14 Challenge Level:

The whole set of tiles is used to make a square. This has a green and blue border. There are no green or blue tiles anywhere in the square except on this border. How many tiles are there in the set? Take Ten

Age 11 to 14 Challenge Level:

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube so that the surface area of the remaining solid is the same as the surface area of the original? Tetra Square

Age 11 to 14 Challenge Level:

ABCD is a regular tetrahedron and the points P, Q, R and S are the midpoints of the edges AB, BD, CD and CA. Prove that PQRS is a square. Sea Defences

Age 7 to 14 Challenge Level:

These are pictures of the sea defences at New Brighton. Can you work out what a basic shape might be in both images of the sea wall and work out a way they might fit together? Travelling Salesman

Age 11 to 14 Challenge Level:

A Hamiltonian circuit is a continuous path in a graph that passes through each of the vertices exactly once and returns to the start. How many Hamiltonian circuits can you find in these graphs? Framed

Age 11 to 14 Challenge Level:

Seven small rectangular pictures have one inch wide frames. The frames are removed and the pictures are fitted together like a jigsaw to make a rectangle of length 12 inches. Find the dimensions of. . . . Map Folding

Age 7 to 11 Challenge Level:

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? All in the Mind

Age 11 to 14 Challenge Level:

Imagine you are suspending a cube from one vertex and allowing it to hang freely. What shape does the surface of the water make around the cube? An Unusual Shape

Age 11 to 14 Challenge Level:

Can you maximise the area available to a grazing goat? Right Time

Age 11 to 14 Challenge Level:

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again? Picturing Triangular Numbers

Age 11 to 14 Challenge Level:

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers? Picturing Square Numbers

Age 11 to 14 Challenge Level:

Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153? Auditorium Steps

Age 7 to 14 Challenge Level:

What is the shape of wrapping paper that you would need to completely wrap this model? Triangular Tantaliser

Age 11 to 14 Challenge Level:

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles. Eight Hidden Squares

Age 7 to 14 Challenge Level:

On the graph there are 28 marked points. These points all mark the vertices (corners) of eight hidden squares. Can you find the eight hidden squares? 3D Stacks

Age 7 to 14 Challenge Level:

Can you find a way of representing these arrangements of balls? Seven Squares

Age 11 to 14 Challenge Level:

Watch these videos to see how Phoebe, Alice and Luke chose to draw 7 squares. How would they draw 100? Rati-o

Age 11 to 14 Challenge Level:

Points P, Q, R and S each divide the sides AB, BC, CD and DA respectively in the ratio of 2 : 1. Join the points. What is the area of the parallelogram PQRS in relation to the original rectangle? Problem Solving, Using and Applying and Functional Mathematics

Age 5 to 18 Challenge Level:

Problem solving is at the heart of the NRICH site. All the problems give learners opportunities to learn, develop or use mathematical concepts and skills. Read here for more information. Is There a Theorem?

Age 11 to 14 Challenge Level:

Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel? Chess

Age 11 to 14 Challenge Level:

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?