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Irrational Arithmagons

Age 16 to 18
Challenge Level

Why do this problem?

This problem offers students the opportunity for lots of practice at manipulating surds in order to complete an intriguing challenge. Students will need to make sense of the structure of the problem to find a route to the solution.

Possible approach

This problem follows on from Arithmagons and Multiplication Arithmagons.
The algebra required to analyse the simple addition arithmagons in the first problem is very straightforward and should not take long to deduce, but it is worth starting here in order to make the links between the additive structure and the multiplicative structure in the second problem.
In order to solve Irrational Arithmagons, students will need to spend time making sense of the structure of multiplication arithmagons - the interactivity in the second problem could be useful here. 
Once students have an analytical method for solving multiplication arithmagons in general, they can be given the irrational arithmagon to work on. Lots of important points for discussion might be raised, for example:
"What form does the product of $a + b\sqrt{2}$ and $c + d\sqrt{2}$ take?"
"How can we divide $a + b\sqrt{2}$ by $c + d\sqrt{2}$?" 
"How can we find the square root of an expression of the form $a + b\sqrt{2}$?"

Key questions

What is the relationship between the product of the edge numbers and the product of the vertex numbers?
Given a multiplication arithmagon with edge numbers $A$, $B$ and $C$, how can we calculate the vertex numbers?
If I multiply two numbers of the form $a + b\sqrt{c}$ together, what can you say about the form of the product?

Possible extension

Possible questions to extend students' thinking could be:
Is the solution unique?
Can any three numbers be placed on the edges of a multiplication arithmagon and yield a solution?
Is it possible to create arithmagons where some/all of the vertex numbers are irrational but the edges are rational?

Possible support

Spend lots of time developing a method for solving Multiplication Arithmagons first. Then create some irrational arithmagons using simple surd expressions in order to get a feel for the structure of arithmagons with surds.