A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
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.
Most students tackle the problem by Guess & Check to find their first solution, which might be either of these:
However, closer inspection shows that a 90° clockwise rotation of the second one produces the first, so these are not unique solutions. As more solutions develop so do strategies for finding them:
We looked for;
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:
- visual patterns within the completed 4x4 grid.
- ways of solving larger grids and patterns between these
- numerical patterns (after numbering the squares 1-16)
And shared the patterns they had found by;
- row by row
- column by column
- top row and left-hand column first or
- diagonal first
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.
- 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
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:
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.
- 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?
The application of the strategy of breaking a problem into smaller parts might also lead to other observations:
- 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:
In this case the RYBG first line leads to:
- Fill the top line correctly first, eg: RYBG
- 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.
- Repeat for Y, B & G in turn on the second line.
- If the placement of a tile leads to conflict later in the line, undo what has just been done and try another position.
- Repeat the process for the third and fourth lines.
Perhaps this last observation is the most important in terms of deciding how many solutions there are. If the rest of the square could be built by deduction from the centre, then finding all the different arrangements of the centre would lead to finding all the different solutions of the 4 x 4 square. This reasoning is explored here in a document extracted from Maths With Attitude, Years 9 & 10.
- 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:
The sum of each colour is now the same! Now there is much more to investigate as Markus suggests in this document linking Coloured Squares and Magic Squares. Markus has also produced this Power Point presentation to summarise the iceberg Coloured Squares as he sees it so far.
- What happens if we number the squares 1 - 16?
And, if after all this adventure you want more, what happens if...:
Claire Campbell, St. Matthew's, Page, chose to extend her Year 2 students working on this task using this Investigation Guide.
- We explore 3 colours on a 3 x 3 grid?
- We explore 5 colours on a 5 x 5 grid?
Belinda Rayment, St. Francis Assisi, Calwell, chose to use this Investigation Guide with her students.