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## 'Multiplication Arithmagons' printed from http://nrich.maths.org/

An arithmagon is a polygon with numbers at its vertices which
determine the numbers written on its edges. An introduction to
arithmagons can be found

here.

Usually, we add the numbers at the vertices to find the numbers on
the edges, but these arithmagons follow a different rule.

Can you work out how the values at the vertices determine the
values on the edges in the arithmagons generated by the
interactivity below?

If you are given the values on the edges, can you find a way to
work out what values belong at the vertices? Use the interactivity
below to test out your strategies. There are three different
challenge levels to try.

Once you are confident that you
can work out the values at the vertices efficiently, here are some
questions you might like to consider:
- Can you describe a strategy to work out the values at the
vertices irrespective of the values given for the
edges?

- Is there a relationship between the product of the values at
the vertices and the product of the values on the
edges?

- What happens to the numbers at the vertices if you double (or
treble, or quadruple...) one or more of the numbers on the
edges?

- Can you create a multiplication arithmagon with fractions at
some or all of the vertices and whole numbers on the edges?

A challenging extension to think
about:
Can you create a multiplication arithmagon where the numbers at the
vertices are all irrational but the numbers on the edges are all
rational?

What about where just one or two numbers at the vertices are
irrational but the numbers on the edges are rational?