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There are **71** NRICH Mathematical resources connected to **Cubes and cuboids**, you may find related items under 3D geometry, shape and space.

Problem
Primary curriculum
Secondary curriculum
### Always, Sometimes or Never? Shape

Are these statements always true, sometimes true or never true?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Changing Areas, Changing Volumes

How can you change the surface area of a cuboid but keep its volume the same? How can you change the volume but keep the surface area the same?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Triple Cubes

This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Next Size Up

The challenge for you is to make a string of six (or more!) graded cubes.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Partly Painted Cube

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Holes

I've made some cubes and some cubes with holes in. This challenge invites you to explore the difference in the number of small cubes I've used. Can you see any patterns?

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubestick

Stick some cubes together to make a cuboid. Find two of the angles by as many different methods as you can devise.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Sending a Parcel

What is the greatest volume you can get for a rectangular (cuboid) parcel if the maximum combined length and girth are 2 metres?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cuboids

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Spider and the Fly

A spider is sitting in the middle of one of the smallest walls in a room and a fly is resting beside the window. What is the shortest distance the spider would have to crawl to catch the fly?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painted Cube

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### The Third Dimension

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

Age 5 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### A Puzzling Cube

Here are the six faces of a cube - in no particular order. Here are three views of the cube. Can you deduce where the faces are in relation to each other and record them on the net of this cube?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Marbles in a Box

How many winning lines can you make in a three-dimensional version of noughts and crosses?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Nine Colours

Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes Cut Into Four Pieces

Eight children each had a cube made from modelling clay. They cut them into four pieces which were all exactly the same shape and size. Whose pieces are the same? Can you decide who made each set?

Age 5 to 7

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Making Cuboids

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?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubes

How many faces can you see when you arrange these three cubes in different ways?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Classic Cube

The net of a cube is to be cut from a sheet of card 100 cm square. What is the maximum volume cube that can be made from a single piece of card?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Inside Out

There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you can colour every face of all of the smaller cubes?

Age 14 to 16

Challenge Level

Article
Primary curriculum
Secondary curriculum
### Dice, Routes and Pathways

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 think mathematically, especially geometrically.

Age 5 to 14

Article
Primary curriculum
Secondary curriculum
### Thinking 3D

How can we as teachers begin to introduce 3D ideas to young children? Where do they start? How can we lay the foundations for a later enthusiasm for working in three dimensions?

Age 7 to 14

Problem
Primary curriculum
Secondary curriculum
### All Tied Up

A ribbon runs around a box so that it makes a complete loop with two parallel pieces of ribbon on the top. How long will the ribbon be?

Age 14 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Classifying Solids Using Angle Deficiency

Toni Beardon has chosen this article introducing a rich area for practical exploration and discovery in 3D geometry

Age 11 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubist Cuts

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Three Cubed

Can you make a 3x3 cube with these shapes made from small cubes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Dicey

A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Green Cube, Yellow Cube

How can you paint the faces of these eight cubes so they can be put together to make a 2 x 2 x 2 cube that is green all over AND a 2 x 2 x 2 cube that is yellow all over?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### More Christmas Boxes

What size square should you cut out of each corner of a 10 x 10 grid to make the box that would hold the greatest number of cubes?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Christmas Boxes

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

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cubic Conundrum

Which of the following cubes can be made from these nets?

Age 7 to 16

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Icosian Game

This problem is about investigating whether it is possible to start at one vertex of a platonic solid and visit every other vertex once only returning to the vertex you started at.

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Counting Triangles

Triangles are formed by joining the vertices of a skeletal cube. How many different types of triangle are there? How many triangles altogether?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Take Ten

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plutarch's Boxes

According to Plutarch, the Greeks found all the rectangles with integer sides, whose areas are equal to their perimeters. Can you find them? What rectangular boxes, with integer sides, have their surface areas equal to their volumes?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### How Many Dice?

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 there are 2 and only 2 different standard dice. Can you prove this ?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Boxed In

A box has faces with areas 3, 12 and 25 square centimetres. What is the volume of the box?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Tic Tac Toe

In the game of Noughts and Crosses there are 8 distinct winning lines. How many distinct winning lines are there in a game played on a 3 by 3 by 3 board, with 27 cells?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Four Points on a Cube

What is the surface area of the tetrahedron with one vertex at O the vertex of a unit cube and the other vertices at the centres of the faces of the cube not containing O?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Wrapping Gifts

A box of size a cm by b cm by c cm is to be wrapped with a square piece of wrapping paper. Without cutting the paper what is the smallest square this can be?

Age 16 to 18

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### All in the Mind

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?

Age 11 to 14

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Double Your Popcorn, Double Your Pleasure

We went to the cinema and decided to buy some bags of popcorn so we asked about the prices. Investigate how much popcorn each bag holds so find out which we might have bought.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Presents

We need to wrap up this cube-shaped present, remembering that we can have no overlaps. What shapes can you find to use?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Little Boxes

How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Cereal Packets

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

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Construct-o-straws

Make a cube out of straws and have a go at this practical challenge.

Age 7 to 11

Challenge Level

Problem
Primary curriculum
Secondary curriculum
### Multilink Cubes

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

Age 7 to 11

Challenge Level