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This problem is a familar context involving perimeters and squares that requires careful analysis. There are opportunities to experiment but a need to think ahead.
Arrange nine large squares into a $3 \times 3$ array and colour the edges - counting from $1$ to $12$ this time.
Rearrange the nine squares in a haphazard way to enable you to colour the edges in a different colour (counting $1$ to $12$).
"Does this mean we cannot do it?"
When any pair is successful challenge them to repeat the task:
"Well done. Would you be able to do this again without making any "false moves"?"
"Can we use more colours when we have $16$ squares?"
"How about with $25$ squares...?"
Can you anticipate where certain squares will have to go in future rounds?
Lots of paper squares so students do not have to worry about making mistakes.
A $4 \times 4$ array is probably easier.
Can students do all three colourings without rearranging the squares?
Will it always be possible to add an extra colour as the squares get larger?
For a 3D version of this problem students could try Inside Out
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Points A, B and C are the centres of three circles, each one of which touches the other two. Prove that the perimeter of the triangle ABC is equal to the diameter of the largest circle.
Can you predict, without drawing, what the perimeter of the next shape in this pattern will be if we continue drawing them in the same way?