Coloured Squares

Summary

• How many solutions are there?
• How do you know when you have found them all?

This cameo includes two Investigation Guides.

Materials

• Playing Board
• 16 coloured tiles - 4 in each of 4 colours
• Square line paper
• Electronic recording paper
  This is a DOC file. Colour the grids (tables) in the document by formatting the cells of the tables provided.
• Sample Solutions
  Five solutions are shown, but are they all solutions? Are the correct ones different solutions?

Content

• logical reasoning and problem solving strategies
• rotation
• visual patterns
• number patterns
• magic squares

Iceberg

Much of what we know about this problem is due to the work of Markus Bucher, Tasmania. According to Markus, the story behind the evolution of this activity goes something like this:

My Grade 3 son came home with some homework. The challenge was to place 16 cubes (4 each of 4 different colours) on a 4x4 grid so that no row, column or major diagonal (the two diagonals of four squares) contained two or more of the same colour. That was where the activity ended. There was no follow up to this in the classroom. There had to be more to this activity than that. 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. For the first session I asked that they attempt to solve the puzzle, transfer the solution onto another grid by colouring in the squares the appropriate colours and then look for patterns in the arrangement of the colours. When we came back as a group the children shared their strategies for solving the puzzle, either:

• row by row
• column by column
• top row and left-hand column first or
• diagonal first

And shared the patterns they had found by;

• comparing rows and, for example, found the colours reversed
• comparing columns
• dividing the grid into quarters and seeing each colour represented in each quarter
• looking at the 'movement' of any given colour

It was the last of these points that really made me stop and look. Owen a Grade 2 said, "They move like the horse in chess." We checked this and sure enough they did.

After having been to a workshop where we looked at visual patterns being expressed numerically, I asked the children to number the grid 1-16 and compare the sum of each colour. This led children to questions like:

• What do all the numbers from 1-16 add up to? (which led to work on adding consecutive numbers)
• What would the sum of each colour be in a 5x5 grid?
• Can you do this with smaller grids?

These were significant problems. The grids became addictive if not obsessive for some. During other lessons I would find grids balancing precariously on their laps under the desk. What appeared to be a one off task turned into a very rich series of lessons which the children found both challenging and enjoyable.

       

• How good are you at checking solutions?

As more solutions develop so do strategies for finding them:

• Some might notice the placement of each tile of the same colour is a knight move from the previous position.
• Some might use the four cells in the top left corner as a 'master' and attempt reflection and rotation strategies.
• Some might try breaking the problem into smaller parts using the following process:
  1. Fill the top line correctly first, eg: RYBG
  2. On the second line 'drive' Red in first from the left until it reaches the first correct position. This will be the third cell because using the first would have two reds vertically and using the second would have two reds in a major diagonal.
  3. Repeat for Y, B & G in turn on the second line.
  4. If the placement of a tile leads to conflict later in the line, undo what has just been done and try another position.
  5. Repeat the process for the third and fourth lines.

  In this case the RYBG first line leads to:

• Sets of four cells making a square in the corner must be four different colours.
• The four corner cells (top left, top right, bottom left, bottom right) must be four different colours.
• The central four cells must be four different colours.

As mentioned above, apart from the development of reasoning and spatial perception involved in this task, it can be extended further by asking:

• What happens if we number the squares 1 - 16?

And, if after all this adventure you want more, what happens if...:

• We explore 3 colours on a 3 x 3 grid?
• We explore 5 colours on a 5 x 5 grid?

Whole Class Investigation

All you need to explore this problem is plenty of coloured tiles or blocks such as those available in most schools. Once the students find one solution, the problem is very open. There is no particular end point other than illustrating again the process of working like a mathematician, which is really the only reason for learning mathematics. This Recording Sheet might assist in this whole class investigation because it is designed so students don't spend an excess amount of time colouring.

At this stage, Coloured Squares does not have a matching lesson on Maths300.