The theme for this challenge is the triangle
and the shapes that it can be used to make. We'll also throw in a touch
of magic, looking at magic diamonds and magic hexagons.
We hope that you enjoy this diversion from the normal maths lesson!
Question 1
You may have seen puzzles called Su Dokus. They are
extremely popular in Japan where their alphabet makes it impossible to
create crosswords puzzles. Here is a Su Doku with a difference.
The numbers 1 - 8 have to be placed in the diamond in
such a way that there is one of each number in every line of triangles,
up, down and across as well as one in each of the four small diamonds that
make up the large diamond. When completed, the numbers make a magic
diamond because the total of the numbers in every row and small diamond
add to the same total, 36.
The diagrams below explain the requirements a bit more
clearly than words can.
There
are four lines going up, each made up of eight triangles.
There
are four lines going up, each made up of eight triangles.
The
four lines across have two parts, some of the triangles are in the
top half and the rest are in the bottom half of the diamond.
There
are four small diamonds, each made up of eight triangles.
Question 2
Another
shape that can be made with triangles is a hexagon and the one in our
diagram has two triangles along each side. We've added some counters to
the diagram to give you an idea as to how you might explore the following.
Question: In what different ways can you place
four counters in the triangles of a hexagon of side 2 so that there is
just one counter in each row of triangles.
Two arrangements are said to be different if you cannot
turn one into the other by rotating it or flipping it over. If that
restriction isn't included, there are rather too many arrangements for you
to find.
As with the diamond, the rows go up, down and across, as
shown below, but this time you will see that there are 7 triangles in some
of the rows and only 5 in others.
To help you explore this problem, you can click below
open the applet:
This enables you to click on a triangle to add a counter
to it or click on a counter to remove it. It's called 'transversals'
because that is the name given to arrangements of objects in a figure such
that there is just one object in each row.
Question 3
We
have found a way of positioning the numbers 1, 2, 3 and 4 in the across
rows of this hexagon so that the total in each across row is the same.
Check it out to see what we mean.
However we would much prefer it if the totals in the
rows that go up and those that go down were the same as well.
Question: What different ways can you find of
positioning the numbers 1, 2, 3 and 4 in the triangles of a hexagon such
that the totals along all the rows are the same?
As before, two arrangements are different if you cannot
turn one into the other by rotating it or flipping it over. We have
provided an applet that helps you experiment with this situation. Click
below to open the applet which will help you explore the problem.
Note: We would be very happy to hear how you used your
answers to Question 2 to help you with this question.
We're not sure how
you will go with this question, but if you do have some time left
over, you might like to send us one answer to the next challenge
in which we are looking for an arrangement of the numbers 1 - 6 in
a hexagon where the total along every row is the same. There is an
applet for that too - click below to set it going.
Question 1
Another
shape that can be made with triangles is a hexagon and the one in our
diagram has two triangles along each side. We've added some counters to
the diagram to give you an idea as to how you might explore the following.
We
have found a way of positioning the numbers 1, 2, 3 and 4 in the across
rows of this hexagon so that the total in each across row is the same.
Check it out to see what we mean.