Copyright © University of Cambridge. All rights reserved.

## 'Irrational Arithmagons' printed from http://nrich.maths.org/

In a multiplication arithmagon, the number on each edge of the arithmagon is the product of the numbers at the adjacent vertices.

*You may wish to look at Multiplication Arithmagons and develop a general strategy for working out the numbers at the vertices, given the edge numbers, before tackling the question below.*
Can you work out the numbers that belong in the circles to make this multiplication arithmagon correct?

If you're not sure where to start, click for a series of hints:

Each circle will be of the form $a+b\sqrt2$ for some values of $a$ and $b$.

In general, a multiplication arithmagon can be solved by multiplying, dividing, and square-rooting.

Dividing an expression by $a+b\sqrt2$ is the same as saying, "What must I multiply $a+b\sqrt2$ by to get that expression?"

To find the square root of an expression such as $12-8\sqrt2$, consider the equation $(x+y\sqrt2)^2=12-8\sqrt2$, expand the brackets, and deduce $x$ and $y$.