Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Nine Colours

Age 11 to 16
Challenge Level Yellow starYellow starYellow star
Secondary curriculum
  • Problem
  • Getting Started
  • Student Solutions
  • Teachers' Resources

Why do this problem?

This problem challenges students to work in 3 dimensions and to use different representations of the cube. The task is easy to explain (though not so easy to solve). By using cubes or the interactivity, we hope students will become absorbed so that they are willing to think logically, work systematically and persevere.

Possible approach

"On a Rubik's cube, the challenge is to make each face a single colour. Today we're going to do the opposite of that, and build a cube where every face has nine different colours showing."


You may wish to offer students plastic cubes like in this picture.
Alternatively, they could work on squared paper, isometric paper, or on computers using this GeoGebra interactivity.

Give students plenty of time to work on the challenge in pairs. While they are working, listen to their conversations and share with the whole class any useful realisations and noticings (see the Key questions below).

The interactivity could be used as part of a plenary to discuss how to build a solution. It could also be used to explore the relationship between the layered representation and the 3D model of the cube, posing questions such as "If I colour this square on the left, what will change on the right?" and "If I want to colour this cube, where do I need to click on the left?"

 

Key questions

Some of the 27 cubes have faces that are invisible from the 'outside' of the large cube.

How many cubes have:

no 'visible' faces?
one face visible?
two faces visible?
three faces visible?


If one colour appears in a corner, where will the other two cubes of the same colour need to appear?

There will be a cube in the centre. Where else will cubes of that colour need to be positioned?

Possible support

Students could try Creating Cubes first.

 

Possible extensions

Painted Cube, Partly Painted Cube, and Marbles in a Box all require similar three-dimensional thinking.

 

 
 

Related Collections

  • Working Systematically - Lower Secondary

You may also like

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?

Three Cubes

Can you work out the dimensions of the three cubes?

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?

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo