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Multiplication Arithmagons

Age 14 to 16 Challenge Level:

Why do this problem?

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

Possible approach

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?

Key questions

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?

Possible extension

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.  

Possible support

Begin by spending some time looking closely at the structure of addition Arithmagons.