What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Draw a 'doodle' - a closed intersecting curve drawn without taking pencil from paper. What can you prove about the intersections?
I want some cubes painted with three blue faces and three red faces. How many different cubes can be painted like that?
Well done to Maulik aged 11 who sent in some nice work on this problem. Neil's solution is given below.
For a 2 by 2 square with column headings of x and x+1, and row headings of y and y+1, Neil says that:
So the diagonal from top right to bottom left is:
$(x+1)y+x(y+1) = xy+y+xy+x = 2xy+x+y$
Lets call that Z.
The diagonal from top left to bottom right is:
$xy+(x+1)(y+1) = xy+xy+x+y+1 = 2xy+x+y+1$
So the first diagonal is Z and the second Z+1 so the diagonal from top left to bottom right is always 1 more than the diagonal from top right to bottom left.
For a 3 by 3 square with column headings of x, x+1 and x+2, and row headings of y, y+1 and y+2, Neil says that:
The 3 by 3 square looks like this:
The diagonal from top right to bottom left is:
Let's say that $3xy+3x+3y = W$
The diagonal from top right to bottom left is W+1.