Irrational Arithmagons

Age 16 to 18

Challenge Level Yellow star

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?

arithmagon with -6 + 2 root 2, 7 + 7 root 2, 2 - 4 root 2

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

$b$ .

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

$y$ .

Number and algebra

Geometry and measure

Probability and statistics

Working mathematically

Advanced mathematics

For younger learners

Irrational Arithmagons

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