This problem offers students the opportunity to explore
numerical relationships algebraically, and use their insights to
make generalisations that can then be proved.

This problem could follow on from work on Arithmagons.

If a computer room is available, students could use the
interactivity to explore multiplication arithmagons and come up
with a strategy for deducing the vertex numbers from the edge
numbers.

Alternatively, students could create their own multiplication
arithmagons and then give their partner the edge numbers to see if
they can deduce the vertex numbers.

Start with vertex numbers in the range 1-12, then move on to
20-100, and finally simple fractions or decimals.

Once students have had time to explore a range of different
arithmagons, bring the class together to discuss the strategies
they have found to work out the vertex numbers.

"Can you see a relationship between the product of the three
edge numbers and the product of the three vertex numbers?"

If students haven't given this any thought, give
them time to try a few examples, and then encourage them to
use algebra to explain any generalisations they make.

When students have devised an efficient method for solving any
multiplication arithmagon, return to the more challenging
arithmagons that may have taken them some time to solve before, to
show the power of general thinking in solving problems.

Finally, the insights offered by algebraic thinking and
general methods can be used to tackle these questions:

- What must be true about the edge numbers for the vertex numbers to be whole numbers?
- How does the strategy for finding a vertex number given the edges on an addition arithmagon relate to the strategy for a multiplication arithmagon?
- 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?

Is it always possible to find numbers to go at the vertices
given any three numbers on the edges?

What is the relationship between the product of the edge
numbers and the product of the vertex numbers?

For solving the simpler multiplication arithmagons, finding
the factors of each number is a useful method. Why is there no
analagous method for addition Arithmagons?

Can students 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?

The stage 5 problem Irrational
Arithmagons takes some of these ideas
further.