This activity offers students the opportunity to explore number
patterns with or without the use of symbols and offers several
routes to generality. Arithmagons can be very hands-on but can
also be analysed using algebra, either formally using symbols
or visual pattern spotting.
Possible approach
This problem is often best started with a period of
experimentation to get a feel for the problem. Students should
then be encouraged to try to solve the three arithmagons and
then look to see patterns to try to work out a general rule for
solving arithmagons.
Key questions
- How do the squares relate to the circles?
- Is there a biggest number which you can put in a circle? A
smallest number?
- Can you see any patterns?
- What is the total sum of all of the numbers in a completed
arithmagon?
Possible
extensions
Extension work could involve the investigation of square
arithmagons.
- Can you solve a square arithmagon?
- Is there always a solution? Are there many solutions?
You might also like to explore the related concept of a
magic graph s. These are
sets of circles (vertices) joined by straight lines edges. On a
magic graph the vertices and edges can be labelled by numbers.
There are different ways of choosing the numbers
Edge Magic Graphs :
any sum of connected vertex-edge-vertex in the graph is the
same
Vertex Magic Graphs :
the sum of the number at a vertex and on all the edges attached
to that vertex is the same for all vertices.
There is still current research on magic graphs, mathematicians
are writing research papers and there is a website which
lists all the latest results discovered .
NRICH has some magic graph challenges which you may like to try:
Olympic Magic
Magic W
W Mates
Magic W Wrap Up
Plum Tree
Magic Caterpillars
Possible support
To help students get started you might
provide arithmagons with one of the answers filled in for a few
examples.