Why do 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 four large squares into a $2 \times 2$ array. Colour
the edges - counting from $1$ to $8$ as you go along.
Rearrange the four squares to enable you to colour the edges
in a different colour (without repeat and counting $1$ to $8$
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 $12$).
"I wonder whether we might be able
to colour the edge in a third colour?"
Rearrange the nine
squares again and "discover" that you are not going to be able to
do it - at least not on this occasion.
"Does this mean we cannot do
"What went wrong?"
"Does this mean it is
Look out for responses that develop the idea that you need to
be able to draw $12$ lines and there are $12$ "blank" edges.
However, not all of these are on the edge of the new array.
"So it feels like it should be
possible with a bit more thought".
Hand out sets of paper squares to pairs of students and
challenge them to complete the task successfully.
When any pair is successful challenge them to repeat the
"Well done. Would you be able to do
this again without making any "false moves"?"
After the first colouring, can students position the squares
for the second colouring knowing it will work for the third?
Move on to larger square arrays.
"Can we use more colours when we
have $16$ squares?"
"How about with $25$
Can you anticipate where certain squares will have to go in
Can students do all three colourings without rearranging the
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
Lots of paper squares so students do not have to worry about
A $4 \times 4$ array is probably easier.