### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

### Napoleon's Theorem

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

### Twizzle Wind Up

A loopy exploration of z^2+1=0 (z squared plus one) with an eye on winding numbers. Try not to get dizzy!

# Twizzles Venture Forth

##### Stage: 4 Challenge Level:

In this animation, the you can change the value of the blue twizzle z . The red twizzle takes the value (z-i) . The green twizzle takes the value (z+i) . The grey twizzle takes the value (z-i)(z+i) . You can check that by multiplying the red and green twizzles using Twizzle Arithmetic . You need to know that i is the name we give the twizzle which has a number arrow equal to the unit arrow but rotated through 90 degrees.

Full Screen Version
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There are lots of things to investigate with this animation, and lots of things to think about...

Which values of the blue twizzle make the grey twizzle equal to zero?

When you set the blue twizzle to one of these values, what happens to the red and green twizzles?

When the grey twizzle is zero, (z-i)(z+i)=0 . This should suggest to you the two values of z for which this is true. Multiply out the expression (z-i)(z+i) and so write the equation in a simpler form.

Explain why you can't solve this equation using ordinary numbers.

If you get stuck, look at the hints. Before you progress to Twizzle Wind Up, look at the notes.