This
problem offers students the opportunity to explore number
patterns with or without the use of symbols, and offers several
routes to generality.

Relating to this month's
theme, think of the process of putting numbers in the vertices
and then calculating the edge numbers as an action. Is it possible
to undo that action uniquely, that is, to 'solve' the
arithmagon?

Suggest that the students watch you in silence and then
describe what you have done.

Draw the skeleton of an arithmagon, fill in numbers at the
vertices and then the numbers on the edges.

Do this again, completing the numbers at the vertices and
waiting to be prompted by the class before completing the numbers
on the edges. You could introduce fractions, decimals and/or
negative numbers.

Once you are convinced that they know what's happening, draw
another skeleton and fill in the numbers on the edges. Can they
find the numbers at the vertices?

"Do you think you will be able to complete any arithmagon if I
give you the numbers on the edges?"

"Try it out yourselves - draw a skeleton and choose some
numbers to go on the edges. Can you now work out the numbers that
must go at the vertices?"

If a computer room is
available, students could work on the problem using the
interactivity.

Any arithmagon they can't complete could be displayed for
whole class consideration.

After some time draw the class together and share findings.
The following questions could be used as a focus for
discussion:

- Are there any arithmagons that you haven't been able to complete?
- When do you need negative numbers?
- When do you need fractions or decimals?
- Does anyone have a strategy for completing the arithmagons?
- Can algebra help us?

Further questions worth considering can be found at the end of
the problem.

It might be interesting to ask students to try
Sums of Pairs a few weeks later and see if they notice that it
is essentially the same problem.

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 total of the edge numbers
and the total of the vertex numbers?

Investigate the properties of the quadrilateral arithmagons
generated by putting two arithmagons side by side overlapping along
one edge.

Investigate square arithmagons.

Can all square arithmagons be solved? Are the solutions
unique?

What about pentagonal arithmagons?

What about Multiplication
Arithmagons?

Students might also like to explore the related concept of
magic graphs .

Here are some NRICH magic graph challenges:Magic W

Olympic Magic