### 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...)

### Tournament Scheduling

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

### Archimedes and Numerical Roots

The problem is how did Archimedes calculate the lengths of the sides of the polygons which needed him to be able to calculate square roots?

# Vedic Sutra - All from 9 and Last from 10

##### Stage: 4 Challenge Level:

With this method you only ever need multiplication tables up to 5 times 5. It is one of many ancient Indian sutras and this one involves a cross subtraction method which, according to old historical traditions, is responsible for the acceptance of the ´ mark as the sign of multiplication. Here is a very simple example of the method. Can you give a good explanation of WHY it works?

Suppose we want to multiply 9 by 7. We subtract each number from 10 and, using these differences (or deficiencies), write:

 9-1 7-3 6 3

The product has two parts, left and right.
To get the right part (or units digit) multiply the deficiencies (1×3)
The left hand digit (tens digit) of the answer can be found in four different ways. Why do they all give the same answer?

1. Subtract 10 from the sum of the two given numbers (9+7=16, 16-10=6)
2. Subtract the sum of the two deficiencies (1+3=4) from 10 and you get 6.
3. Cross subtract (9-3=6)
4. Cross subtract (7-1=6)

Here are some more examples. Try some of your own.

 9-1 8-2 9-1 8-2 6-4 7-3 9-1 5-5 5 4 5 6 8 1 4 0 Note: Here you have to express 5 times 2 as 1 ten and 0 units.