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...)
Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.
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?
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
Suppose we want to multiply 9 by 7. We subtract each number from
10 and, using these differences (or deficiencies), write:
The product has two parts, left and right.
To get the right part (or units digit) multiply the deficiencies
The left hand digit (tens digit) of the answer can be found in four
different ways. Why do they all give the same answer?
This gives the answer 63.
Here are some more examples. Try some of your own.