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The screen shot on the
Hints page shows some solution
When we looked at the solutions sent in,
some people had started by putting triangles together until they
could make a hexagon, others started with a hexagon and looked for
ways it might split up into the two types of triangle.
Well done to Daniel from Royston and to
Georgina from St. George's School for seeing the answer to the
first important question :
'Did that tell you something about yellow and green triangles,
about how they relate to each other' ?
Two yellow triangles make a rhombus of a size which can also be
made by two green triangles.
And that lets us see that the yellow and green hexagons are equal
because the yellow and green triangles have the same area (half
that rhombus) and both hexagons use six triangles. But the
equilateral triangles (below) do not use the same number of
triangles and so cannot be the same size.
Esther had a great approach using sequences
You can make equilateral triangle arrangements from yellow
The numbers needed follow the sequence : 1, 4, 9, 16 . . .
a² where a
the number in the sequence.
When I tried making equilateral triangles with the green triangles
I could see that one way to do it was by replacing each equilateral
space with three green triangles ( which would also of course scale
up the size of my arrangement ).
When working with green triangles the numbers follow the sequence:
3, 12, 27, 48 . . . 3b² where b
is the number in the sequence.
Which means I'm looking for a value of a
and of b
so that a² =
Keeping in mind that a
are whole numbers ( a
position in a sequence ) , a² would have to have a
factor of 3 to match 3b² , but any factor a²
has it will have twice because it's a number squared. So
a² can only contain an even numbers of 3
s in its prime factors and
3b² can only contain an odd numbers of 3
s. This means that a² and
3b² are never going to match, and an equilateral
arrangements made from yellow triangles is never going to match an
equilateral arrangements made from green triangles
This is great reasoning, but how do we
know that replacing each equilateral space with three green
triangles is the only way to build up to an equilateral triangle
?We hope you enjoy the brain-stretching this kind of reasoning
involves.Be encouraged, students a lot older than Stage 4 still
wrestle with this sort of thinking.