### Triangle Incircle Iteration

Start with any triangle T1 and its inscribed circle. Draw the triangle T2 which has its vertices at the points of contact between the triangle T1 and its incircle. Now keep repeating this process starting with T2 to form a sequence of nested triangles and circles. What happens to the triangles? You may like to investigate this interactively on the computer or by drawing with ruler and compasses. If the angles in the first triangle are a, b and c prove that the angles in the second triangle are given (in degrees) by f(x) = (90 - x/2) where x takes the values a, b and c. Choose some triangles, investigate this iteration numerically and try to give reasons for what happens. Investigate what happens if you reverse this process (triangle to circumcircle to triangle...)

### Vedic Sutra - All from 9 and Last from 10

Vedic Sutra is one of many ancient Indian sutras which involves a cross subtraction method. Can you give a good explanation of WHY it works?

### Tournament Scheduling

Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.

# Unusual Long Division - Square Roots Before Calculators

##### Stage: 4 Challenge Level:

The next digit of the answer is 5.

There are a number of ways of exploring around this area, don't rush.

You might try 'trial and improvement' as one way to work towards a square root of say 30, 50 or any other number.

Try this new method to find square roots for 30, 50, etc., then check with a calculator to see that it's working out correctly. Listen again to the audio as many times as you need. There's a lot of calculation to follow.

When you are ready, think about the harder questions:
• Why multiply by 2 and by 10?

• And why look for an end digit that is also the digit you use to multiply?