Coloured Squares
Years 2 - 10

That weekend our family spent our idle moments playing with the grid and some of its possibilities.

We looked for:

  • visual patterns within the completed 4x4 grid.
  • ways of solving larger grids and patterns between these
  • numerical patterns (after numbering the squares 1-16)
Satisfied that the original task was just the tip of the iceberg I took the activity to my Grade 2 class.

Markus Bucher


You need a colour printer.
  • Print the Playing Board & Squares in colour and cut out the tiles.
    Printing on thin card is great. Your local print shop might do that for you.
  • Print the Square Line Paper for recording.
  • Collect a red, green, yellow and blue coloured marker or pencil.
  • Write the title of this challenge and today's date on a fresh page in your maths journal.
When the activity is over your recording sheets should be put into your journal beside any other notes you make.

Investigating Coloured Squares

  • Open this Coloured Squares Starter.
    You can read it on screen or print it.
  • Investigate Questions 1 - 5 on the Starter.
  • You can return to this problem and continue at another time.
    Or if you get 'hooked' keep on going for as long as you want to.
Have fun exploring Coloured Squares.

Digging Deeper

The big challenge for a mathematician in this problem is written on the Starter.
  • How many solutions are there?
  • How do you know when you have found them all?
We don't know if anyone has found the answer to that yet.
Perhaps it will be you who finds a way to find them all.
But first you will need to dig deeper into the problem to get more data.
  • First find at least four solutions and record on the square line paper. It saves time if you just put a cross of colour in each square.
  • Check these Sample Solutions to see how they compare with yours. (You can read it on screen or print it.)
    This sheet also mentions another way to record. If you know how to use tables in a word processor or a spreadsheet you can colour the squares by formatting the colour of the cells.

... Solutions or not? ...

  • When you have at least four solutions, examine them carefully to see if you can find any patterns, or rules about where the colours go.
  • Record anything you discover.
  • One student said that in correct solutions, the way the colours moved was like a move in chess.
    What do you think this student meant?
  • In your journal, write and draw to show someone else how to find a solution.
    Doing this is explaining your strategy for finding a solution.
  • Discuss your strategy with someone else. Perhaps together you can find a different strategy.
Keep on playing around with the problem whenever you feel like - just like the family in story above. All the problems mathematicians solve take lots of time. Sometimes days, or weeks, or even years. The hardest maths problem ever solved took about 300 years!

Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
  • Read again what it means to work like a mathematician.
  • How did you work like a mathematician during this investigation?
  • How does it feel to work like a mathematician and not completely solve the problem?


Answers & Discussion

These notes were originally written for teachers. We have included them to support parents to help their child learn from Coloured Squares.

This investigation offers success for everyone. You are successful if you find only one solution. You are successful if you discover a way to know all the solutions and know why there are no more. And you are successful if you decide to stop somewhere in between.

  • Notes for Coloured Squares, where you will find Markus Bucher's full letter, more ideas that others have explored, without anyone yet claiming that they have found all the solutions and an investigation guide designed to lead Year 9 & 10 students further towards finding all solutions.

Send any comments or photos about this activity and we can start a gallery here.


Maths At Home is a division of Mathematics Centre