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 is one of many ancient Indian sutras which involves a
cross subtraction method. Can you give a good explanation of WHY it
Scheduling games is a little more challenging than one might desire. Here are some tournament formats that sport schedulers use.
Well done Carol in Leeds and others.
This really is right at the top end of Stage 4 material and takes
some following. Most people would need to go over the argument
below several times, probably taking breaks and coming back again.
There are quite a lot of numbers to keep track of so stay patient
with yourself and here goes :
When a number is reduced or increased by a factor of ten the
square of that number is reduced or increased by a factor of one
$4000$ reduces by a factor of one hundred to become $40$ so a
factor of ten will connect their square roots.
So if I knew what the square root of $4000$ was, the square root
of $40$ would be the same but with the digits all one position
lower (to the right). Now to work.
This method is about deciding, one by one, what each digit
The square root of $40$ starts with a six and then becomes
decimal, so the square root of $4000$ starts with sixty-something,
before becoming decimal.
And it's that 'something' digit which I need to find.
This corresponds to the third stage of working in the method.
And the answer is two. $1262 \times2$ is $2524$