### 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.

# Zeller's Birthday

##### Stage: 4 Challenge Level:

What day of the week were you born on? Do you know?

Here's a way to find out.

For example, if your date of birth was 6 July 1989 :

$D = 6$ , $M = 7$ , and $Y = 1989$

If $M$ had been a 1 or 2, subtract 1 from $Y$ and add 12 to $M$.

$Y_F$ is made from the first two digits of $Y$ and $Y_L$ is made from the last two digits of $Y$.

To begin, work out the sum of all the integer parts of $2.6M - 5.39$ , of $\frac{Y_F}{4}$ , and of $\frac{Y_L}{4}$

Add $D$ and $Y_L$ into that total, and then subtract 2 lots of $Y_F$

Divide that final answer by 7 and notice the remainder .

A remainder value of 0 means the date was a Sunday, 1 means it was a Monday, 2 for a Tuesday, and so on.

Can you follow the method (what you actually have to do) ?

You could check some dates you happen to know the answer for ?

When you are getting good at using the method start to ask yourself how it works and why does it give the right result?

Why 2.6 and why 5.39?